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+Anonymous authors Paper under double-blind review + +# ABSTRACT + +Large amounts of data are a common requirement for many deep learning approaches. However, data is not always equally available at large scale for all classes. For example, on highly optimized production lines, defective samples are hardly acquired while non-defective samples come almost for free. The defects however often seem to resemble each other, e.g., scratches on different products may only differ in few characteristics. In this work, we propose to make use of the shared characteristics by transferring a stylized defect-specific content from one type of background product to another. Moreover, the stochastic variations of the shared characteristics are captured, which also allows generating novel defects from random noise. These synthetic defective samples enlarge the dataset and increase the diversity of defects on the target product. Experiments demonstrate that our model is able to disentangle the defect-specific content from the background of an image without pixel-level labels. We present convincing results on images from real industrial production lines. Also, we show consistent gains of using our method to enlarge training sets in classification tasks. + +# 1 INTRODUCTION + +Automated Visual Inspection (AVI) is vital for quality control in modern production lines. Despite the fact that AVI has been studied for decades, it remains a challenging task with many open research questions await to be answered. One of the main challenges in data-driven AVI is the acquisition of suitable training data. This is for two reasons: First, collecting a vast amount of labelled data is usually labor-intensive and time-consuming. In many cases, even experts are required to identify where and what to look for. However, the acquired label information is task-specific and cannot be reused or transferred to a new task in most cases. Thus, the tedious labelling process must be repeated for each new product, even if its defect is similar to other products in people’s eyes. Second, in real-world scenarios such as highly optimized production lines, a more severe problem emerges: data imbalance. Only very few defective parts are produced by design. Moreover, the acquired anomaly images from a single product are lacking diversity and may not capture the full defect distribution. Training a robust deep neural network model in such conditions is very challenging. + +Since collecting sufficient real-world defective samples is impractical, algorithms to synthesize required images became a focus in research. Image synthesis through Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) has shown promising performance in recent years. But it also requires large amounts of balanced data which are not available in most industrial use cases, in particular for irregular defect patterns and large variation. Therefore, GANs tend to overfit to the training examples when trained with little data (Karras et al., 2020a). + +In this work, we tackle these issues by exploiting cross-domain information: we first define two sets of domains—foreground domains and background domains. The foreground domain describes a set of images that contains a specific foreground content to be grouped into a distinctive category, and each content has a different style. The background domain instead is considered as a group of images that shares similar structural appearance over the whole image. For example, we can set foreground domains as defect types and background domains as product types while the styles of defects indicate their artistic looks such as light or heavy strokes. Building upon StarGAN v2 (Choi et al., 2020), the concept underlying this work is to transfer and generate foreground contents with a variety of styles across different background domains, as illustrated in Figure 1. + +![](images/7375134a93e4d6c487812a69c420b54b7f33ccdc499b7470357d6f8e1d3d2531.jpg) +Figure 1: The underlying concept of DT-GAN is to transfer and generate foreground contents with a variety of artistic styles (e.g., light / heavy strokes) across different background domains. + +The contributions of this work are three-fold: First, we introduce Defect Transfer GAN (DT-GAN), a model that learns transferring existing foreground content and generating novel contents onto different backgrounds at the same time. In the real-world scenario, it allows defect inspection networks to learn from a variety of synthetic defective images by composing the foreground defects together with various non-defective images from different products. Second, DT-GAN is able to disentangle the foreground defect-specific content and the defect-irrelevant background in a weakly-supervised manner. Third, extensive experiments show that our method can generate diverse and real-looking defective samples even for products with only 20 real defective images. These defective images generated by DT-GAN boost the performance in defect inspection networks significantly. + +# 2 RELATED WORK + +GANs have shown their power in many computer vision tasks such as image synthesis (Luciˇ c et al., ´ 2019), style translation (Johnson et al., 2016), super-resolution (Ledig et al., 2017), image impainting (Pathak et al., 2016) and many other applications. To quantify the performance of GANs, visual quality and the diversity of generated images are considered as two of the most important criteria. Recent models address these requirements either by dedicated loss functions (Mao et al., 2019b; Yang et al., 2019) or architectural design (Brock et al., 2019). StyleGAN v2 (Karras et al., 2020b), the latest state-of-the-art model in image synthesis, introduces stochastic variation in image generating process by adding per-pixel noise after each convolution. However, it is non-trivial to adapt the model to transform given input images due to the design of the generator. + +In contrast, image-to-image translation methods (Isola et al., 2017) provide a way to recover the connection between inputs and the generated images while encouraging diversity. For example, Zhu et al. (2017b) and Huang et al. (2018) impose consistent mappings in latent space to achieve the goal. Some approaches (Ma et al., 2019; Park et al., 2019) use reference images as guidance to generate diverse outputs. Mokady et al. (2020) further extends the translation task from styles to contents. It learns to identify a specific content in a given input (e.g., a specific pair of glasses) and transfer it to the target image. However, aforementioned methods only consider the translation between two domains and their extension to multiple domains is non-trivial. + +Surface defect detection is one of the important tasks in real-world industrial manufacturing. It aims at identifying and classifying defects with the help of machine vision. Traditional methods (Ngan et al., 2011) build models upon hand-crafted feature extractors, which are unstable and outperformed by deep learning based models. However, the performance and generalization ability of deep learning approaches are restricted due to limited number of defective samples in real-world scenarios. Data augmentation aims to enrich the training dataset by introducing different kinds of invariance for the model to capture. Several recent works (Niu et al., 2020; Zhang et al., 2021) have proposed to adopt GANs as a data augmentation method to generate realistic defective samples. Among them, Defect-GAN (Zhang et al., 2021) tries to capture the stochastic variation within defects by mimicking the defacement and restoration processes. However, it still learns a deterministic mapping between inputs and outputs while DT-GAN achieves multi-modality by varying styles. Moreover, our method can generate realistic defects with sophisticated patterns copied from real-world defective samples. + +![](images/83af523dcfeb8e486bbe482fe2fcb6e531192ec8f0ab542aeb5d211eff025025.jpg) +Figure 2: Overview of all modules in DT-GAN. + +# 3 METHODOLOGY + +Our primary aim is to perform unpaired image-to-image translation across multiple foreground domains within a single model. In our use case, the foreground domains refer to the defect types, which means we want to achieve translations between different types of defects while the background remains unaffected. We assume that there is always an adequate amount of normal samples (e.g., non-defective) available, while anomaly samples are rare and hard to acquire. + +# 3.1 PROPOSED FRAMEWORK + +Our framework builds on StarGAN v2, a multimodal image-to-image translation model. Given an input image $\textbf { x } \in { \mathcal { X } }$ and an arbitrary domain $y \in \mathcal { V }$ , StarGAN v2 generates a domain specific style code in a learned style space and outputs an image that is stylized to fit the domain of $y$ . Its network architecture consists of four modules: a generator, a mapping network, a style encoder and a discriminator. We modify and extend all four modules (see Figure 2) and describe the key differences in details as below. + +Style-Content Separation. Given a latent code $\mathbf { z }$ and a domain $y$ , the mapping network $M$ (Figure 2(b)) generates a style code $\mathbf { s } = M _ { y } ( \mathbf { z } )$ and a domain specific content $\mathbf { c } = M _ { y } ( \mathbf { z } )$ in different branches. It is worth mentioning that $M _ { y }$ here denotes an output of $M$ corresponding to the domain $y$ . This feature allows our method to separately model the structural appearance (i.e. content) and its artistic looks (i.e. style), which is essential because applying different styles to the same content enriches the diversity of outputs. By randomly sampling $\mathbf { z }$ from a standard normal distribution and $y$ from all available foreground domains, $M$ is able to produce diverse style codes and domain specific contents. + +The encoder $E$ (Figure 2(c)) extracts the style code $\mathbf { s } = E _ { y } ( \mathbf { x } )$ and the domain specific content $\mathbf { c } = E _ { y } ( \mathbf { x } )$ from an given image $\mathbf { x }$ , which reflect the characteristics of reference images instead of randomly sampled noise. + +Foreground/Background (FG/BG) Disentanglement. The generator $G$ (Figure 2(a)) translates an input image x into an output image $G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } )$ according to given domain specific style code es and content ec, which are provided either by the mapping network $M$ when generating from random noise or by the style-content encoder $E$ when transferring an existing content from a reference image. To achieve a FG/BG disentanglement, we split the channels of the three-dimensional feature map (i.e., $H \times W \times C$ ) at the bottle neck of $G$ into two parts. The model is then forced to encode the background into the first channels and the domain specific content cˆ into the latter channels by classification losses as discussed in Section 3.2. cˆ can then be replaced with content ec from the target domain. The adaptive instance normalization (AdaIN) (Huang & Belongie, 2017) is then used to inject es into ec during the decoding process while the background $B G _ { G } ( \mathbf { x } )$ is decoded separately. StarGAN v2 learns $\mathbf { F G }$ and BG together which leads to a conditional relationship between both. Our disentanglement and separate encoding break this conditioning and therefore enable our method to freely combine FG and BG as well as learn the full variation of FG content. Finally, $B G _ { G } ( \mathbf { x } )$ and ec are concatenated together and then fused before output. + +Multi-task discriminator with auxiliary classifiers. The discriminator $D$ (Figure 2(d)) is a multitask discriminator with two auxiliary classifiers: a foreground domain classifier and a background domain classifier. This feature strengthens the disentanglement of FG and BG by first ensuring the input image $\mathbf { x }$ contains a domain specific content that can be recognized by the foreground domain classifier independent of the background. Later, each branch $D _ { y }$ in the multi-task discriminator $D$ is trained to determine if an image $\mathbf { x }$ is a real image of its foreground domain or a fake image $G ( \mathbf { x } , \mathbf { s } , \mathbf { c } )$ generated by $G$ . Apart from that, one extra branch $B G _ { \mathrm { c l s } }$ is attached to decide whether the background information of the input images is well preserved. + +Content Transfer. Mokady et al. (2020) introduced a concept that a model should be able to identity the difference between two domains when one of the domains contains a feature that the other does not have. We refer to this concept as ‘anchor’ and extend to multiple domains $( > 2 )$ by the FG/BG disentanglement, the multi-task discriminator and the foreground content classifier in $D$ . We treat domain Normal as the anchor domain i.e. set the domain specific content to zero, because a normal image has no domain specific content in our definition. As a result, we can now transfer contents between all combination of FG and BG domains (see Figure 10). + +Compared to StarGAN v2, our method not only models style codes and contents separately but also disentangles the foreground and background of an image in a weakly-supervised manner. These features allow explicit control over output images by combining desired style codes and contents from one of the subnetworks with the input images. Therefore, it leads to higher variance regarding the location, structural pattern and artistic style of defects in the synthetic images of DT-GAN. + +# 3.2 TRAINING OBJECTIVES + +Given an image $\mathbf { x } \in \mathcal { X }$ , its original foreground domain $y \in \mathcal { V }$ and its background domain $p \in \mathcal { P }$ , the following objectives are used to train our framework. + +Adversarial loss. In the training phase, a noise vector $\mathbf { z } \in { \mathcal { Z } }$ and a target foreground domain $\widetilde y \in \mathcal { V }$ are sampled randomly. Both of them are fed to $M$ , producing a target style code $\widetilde { \mathbf { s } }$ and a target content ec as follows: $\widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } = M _ { \widetilde { y } } ( \mathbf { z } )$ . Goal of the training is to ensure that $\widetilde { \mathbf { s } }$ and ec are sampled from the distribution over styles and contents of the target domain $\widetilde { y }$ . The generator $G$ then combines an image $\mathbf { x }$ with $\widetilde { \mathbf { s } }$ and $\widetilde { \mathbf c }$ and learns to generate an output image $G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } )$ that is indistinguishable from real images in the target domain $\widetilde { y }$ . We encourage this behavior by using an adversarial loss same as in Choi et al. (2020) + +$$ +\mathcal { L } _ { \mathrm { a d v } } = \mathbb { E } _ { { \mathbf { x } } , y } \big [ \log D _ { y } ( { \mathbf { x } } ) \big ] + \mathbb { E } _ { { \mathbf { x } } , \widetilde { y } , { \mathbf { z } } } [ \log \left( 1 - D _ { \widetilde { y } } ( G ( { \mathbf { x } } , \widetilde { { \mathbf { s } } } , \widetilde { { \mathbf { c } } } ) ) \right) ] , +$$ + +where $D _ { y }$ and $D _ { \widetilde { y } }$ are the output branches of $D$ that correspond to the source domain $y$ and the target domain $\widetilde { y }$ , respectively. + +Style-content reconstruction loss. Similar to StarGAN v2, to enforce the generator $G$ takes the style code $\widetilde { \mathbf { s } }$ and the domain specific content ec into consideration during the generation process, we employ a style-content reconstruction loss + +$$ +\begin{array} { r } { \mathcal { L } _ { \mathrm { s t y . c o n } } = \mathbb { E } _ { \mathbf { x } , \widetilde { y } , \mathbf { z } } \big [ \| \widetilde { \mathbf { s } } - S _ { E } ( G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } ) ) \| _ { 1 } \big ] + \mathbb { E } _ { \mathbf { x } , \widetilde { y } , \mathbf { z } } \big [ \| \widetilde { \mathbf { c } } - C _ { E } ( G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } ) ) \| _ { 1 } \big ] . } \end{array} +$$ + +This objective urges the style-content encoder $E$ to recover $\widetilde { \mathbf { s } }$ and c from $G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } )$ . Here, the stylecontent encoder $E$ learns a mapping from an image to its style and content domains, which allows $G$ to synthesize an image with given s and c from reference images at test time. + +Diversity loss. In order to further boost the diversity of output images from $G$ , we introduce a loss that encourages diversity as follows: for a pair of random latent codes $\mathbf { z } _ { 1 }$ and $\mathbf { z } _ { 2 }$ we compute $\widetilde { \mathbf { s } } _ { i } , \widetilde { \mathbf { c } } _ { i } = M _ { \widetilde { y } } ( \mathbf { z } _ { i } )$ for $i \in \{ 1 , 2 \}$ and enforce a different outcome of the generator $G$ for differently mixed style and content input pairs: + +$$ +\begin{array} { r l } & { \mathcal { L } _ { \mathrm { d s } } = \mathbb { E } _ { \mathbf { x } , \widetilde { y } , \mathbf { z } _ { 1 } , \mathbf { z } _ { 2 } } \left[ \| G ( \mathbf { x } , \widetilde { \mathbf { s } } _ { 1 } , \widetilde { \mathbf { c } } _ { 2 } ) - G ( \mathbf { x } , \widetilde { \mathbf { s } } _ { 2 } , \widetilde { \mathbf { c } } _ { 1 } ) \| _ { 1 } \right] } \\ & { \quad \quad + \mathbb { E } _ { \mathbf { x } , \widetilde { y } , \mathbf { z } _ { 1 } , \mathbf { z } _ { 2 } } \left[ \| G ( \mathbf { x } , \widetilde { \mathbf { s } } _ { 1 } , \widetilde { \mathbf { c } } _ { 1 } ) - G ( \mathbf { x } , \widetilde { \mathbf { s } } _ { 2 } , \widetilde { \mathbf { c } } _ { 2 } ) \| _ { 1 } \right] } \\ & { \quad \quad + \sum _ { m , n , o } \left[ \mathbb { E } _ { \mathbf { x } , \widetilde { y } , \mathbf { z } _ { 1 } , \mathbf { z } _ { 2 } } \left[ \| G ( \mathbf { x } , \widetilde { \mathbf { s } } _ { m } , \widetilde { \mathbf { c } } _ { n } ) - G ( \mathbf { x } , \widetilde { \mathbf { s } } _ { o } , \widetilde { \mathbf { c } } _ { o } ) \| _ { 1 } \right] \right] , } \end{array} +$$ + +where $m , n \in \{ 1 , 2 | m \neq n \}$ and $o \in \{ 1 , 2 \}$ . Driven by this term, the generator $G$ is forced to discover meaningful style features and contents that eventually lead to diversity in generated images. + +We ignore the denominator ${ \left\| { \bf z } _ { 1 } - { \bf z } _ { 2 } \right\| } _ { 1 }$ of the original diversity loss (Mao et al., 2019a) for stable training as in StarGAN v2. + +Cycle consistency loss. To ensure that the generated image $G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } )$ preserves the domaininvariant properties of its input image $\mathbf { x }$ , we impose the cycle consistency loss (Zhu et al., 2017a) + +$$ +\mathcal { L } _ { \mathrm { c y c } } = \mathbb { E } _ { { \mathbf { x } } , y , \widetilde { y } , { \mathbf { z } } } \big [ | | { \mathbf { x } } - G ( G ( { \mathbf { x } } , \widetilde { { \mathbf { s } } } , \widetilde { { \mathbf { c } } } ) , \widehat { { \mathbf { s } } } , \widehat { { \mathbf { c } } } ) | | _ { 1 } \big ] , +$$ + +where $\hat { \bf s } , \hat { \bf c } = E _ { y } ( { \bf x } )$ is the extracted style code and domain specific content of the input image $\mathbf { x }$ , and $y$ is the original domain of $\mathbf { x }$ . By learning to reconstruct the input image $\mathbf { x }$ with given style code ˆs and content cˆ, the generator $G$ is then further encouraged to disentangle the background, the domain specific content and the style code. + +Content consistency loss. Besides the cycle consistency loss, we apply another constraint to enforce that the detached domain specific content from $G$ is consistent with the one retrieved from $E$ according to + +$$ +\begin{array} { r } { \mathcal { L } _ { \mathrm { c o n . c y c } } = \mathbb { E } _ { \mathbf { x } , y , \widetilde { y } , \mathbf { z } } \left[ \left\| F G _ { G } ( \mathbf { x } ) - \widehat { \mathbf { c } } \right\| _ { 1 } \right] + \mathbb { E } _ { \mathbf { x } , y , \widetilde { y } , \mathbf { z } } \left[ \left\| F G _ { G } ( G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } ) ) - \widetilde { \mathbf { c } } \right\| _ { 1 } \right] , } \end{array} +$$ + +where $\hat { \mathbf { c } } = E _ { y } ( \mathbf { x } ) , \widetilde { \mathbf { c } } = E _ { \widetilde { y } } ( \mathbf { x } ) , F G _ { G } ( \mathbf { x } )$ and $F G _ { G } ( G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } ) )$ are the pop-out domain specific content from input image $\mathbf { x }$ and generated image $G ( \mathbf { x } , \widetilde { \mathbf { s } } , \widetilde { \mathbf { c } } )$ , respectively. + +Classification losses. We employ two classification losses: the first one is the foreground content classification loss + +$$ +\mathcal { L } _ { \mathrm { F G . c l s } } = \mathbb { E } _ { \mathbf { x } _ { \mathrm { r e a l } } , y } \Big [ - \log D _ { \mathrm { F G . c l s } } \big ( y | \mathbf { x } _ { \mathrm { r e a l } } \big ) \Big ] + \mathbb { E } _ { \mathbf { x } _ { \mathrm { f a k e } } , \widetilde { y } } \Big [ - \log D _ { \mathrm { F G . c l s } } \big ( \widetilde { y } | \mathbf { x } _ { \mathrm { f a k e } } \big ) \Big ] \ , +$$ + +which aims to ensure that the domain specific content is properly encoded and carries enough information from the target domain. The second one is the background classification loss + +$$ +\mathcal { L } _ { \mathrm { B G } . \mathrm { c l s } } = \mathbb { E } _ { \mathbf { x } _ { \mathrm { r e a l } } , p } \big [ - \log D _ { \mathrm { B G } . \mathrm { c l s } } ( p | \mathbf { x } _ { \mathrm { r e a l } } ) \big ] + \mathbb { E } _ { \mathbf { x } _ { \mathrm { f a k e } } , p } \big [ - \log D _ { \mathrm { B G } . \mathrm { c l s } } ( p | \mathbf { x } _ { \mathrm { f a k e } } ) \big ] \ , +$$ + +where $p$ is the corresponding background type of $\mathbf { x } _ { \mathrm { r e a l } }$ and $\mathbf { x } _ { \mathrm { f a k e } }$ . With the help of this objective, the generator $G$ learns to preserve the domain-invariant characteristics of its input image $\mathbf { x }$ while dissociating the foreground domain specific part. + +Full objective. Our full objective functions can be summarized as + +$$ +\begin{array} { r l } { \underset { G , F , E } { \operatorname* { m i n } } \underset { D } { \operatorname* { m a x } } } & { \mathcal { L } _ { \mathrm { a d v } } + \lambda _ { \mathrm { s t y } . \mathrm { c o n } } \mathcal { L } _ { \mathrm { s t y } . \mathrm { c o n } } - \lambda _ { \mathrm { d s } } \mathcal { L } _ { \mathrm { d s } } + \lambda _ { \mathrm { c y c } } \mathcal { L } _ { \mathrm { c y c } } + } \\ & { \lambda _ { \mathrm { c o n . c y c } } \mathcal { L } _ { \mathrm { c o n . c y c } } + \lambda _ { \mathrm { F G . c l s } } \mathcal { L } _ { \mathrm { F G . c l s } } + \lambda _ { \mathrm { B G . c l s } } \mathcal { L } _ { \mathrm { B G . c l s } } \ , } \end{array} +$$ + +where $\lambda _ { \mathrm { s t y } }$ , $\lambda _ { \mathrm { d s } }$ , $\lambda _ { \mathrm { c y c } }$ , $\lambda _ { \mathrm { c o n \mathrm { { - } c y c } } }$ , $\lambda _ { \mathrm { F G \mathrm { - } c l s } }$ and $\lambda _ { \mathrm { B G \mathrm { { - } c l s } } }$ are the hyperparameters for each term. + +# 4 EXPERIMENTS + +We evaluated the images generated by DT-GAN through a series of experiments both quantitatively and qualitatively. Finally, we demonstrate the benefits of our generated images when being used as data augmentation for a defect classification task on limited data. + +Dataset. All experiments were performed on a real industrial dataset: a Surface Defect Inspection (SDI) dataset that contains three different kinds of products from production lines and samples from each product are classified into three mutually exclusive classes: Normal, Scratch and Spot. All of the images are grayscale. Detailed statistics of the dataset are summarized in Appendix A. Note that only the training set was used in GAN training, the test set was left untouched for final evaluation in classifier training. For a fair comparison, all images were resized to $1 2 8 \times 1 2 8$ resolution for both GAN training and classifier training, which was also the highest resolution used in the baselines for image generation. For comparison, we also conducted experiments on the widely used MVTec Anomaly Detection dataset (Bergmann et al., 2019) in Appendix E.4. + +# 4.1 DEFECT GENERATION + +Baselines. As discussed in Section 3, DT-GAN can either use the mapping network to randomly generate styles and defects, or it can use the style-content encoder to extract both from reference images. We refer to these cases as ‘latent-guided’ and ‘reference-guided’, respectively. + +Since the two ways of guidance are fundamentally different, we evaluated them against two sets of baselines: Our reference-guided image generation was compared to Mokady et al. (2020) and StarGAN v2, because both of them can perform a reference-guided translation. Note that Mokady et al. (2020) can only translate between two domains while StarGAN v2 and DT-GAN can achieve multi-domain translation within a single model. Images generated through the latent-guided part of DT-GAN were compared to state-of-the-art GANs in image synthesis: BigGAN (Brock et al., 2019) and StyleGAN v2 (Karras et al., 2020b). We set BigGAN to condition on defect types during training while StyleGAN v2 was trained unconditionally. All baselines were trained from scratch with the public implementations provided by the authors1. + +# 4.1.1 QUANTITATIVE EVALUATION + +Metrics. We employed the commonly used frechet inception distance (FID) (Heusel et al., 2017) to evaluate both the visual quality and the diversity of the generated images. We also report the kernel inception distance (KID) (Binkowski et al., 2018) which is a more stable metric for small sets of images like our SDI dataset. Lower FID and KID scores indicate better performance. + +Both scores are shown in Table 1. We observe that methods like BigGAN and StyleGAN v2, which perform defect synthesis purely based on latent codes, generally provide unsatisfactory results on the SDI dataset, presumably due to the small number of defective samples that were available. These methods then struggle to capture the complex and irregular patterns of defects. We also experimented with augmentation methods for GAN training (Karras et al., 2020a; Zhao et al., 2020) but did not find a consistent improvement (see Appendix E.2). We thus only report the best scores. + +Reference-guided synthesis methods like Mokady et al. (2020) and StarGAN v2 seem to generate more realistic images. The scores of StarGAN v2 on a single product are omitted here because generating images with specified background is not possible due to its network design—the product type changes in output images, which we refer to as ‘identity-shift’. As seen in Table 1, our method achieves better scores in all cases. We believe this is due to the fact that our method allows free combination of foreground defects and backgrounds, making the generated images more diverse even with a small number of training samples. + +Table 1: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID and KID. Note that the reported values are not comparable between columns, because they were calculated on different training sets. + +
MethodFID↓KID↓
ABCAllABCAll
Mokady (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v21137.70-110.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
+ +# 4.1.2 QUALITATIVE EVALUATION + +We present a qualitative comparison with the baseline methods in latent-guided image synthesis in Figure 3. To make a fair comparison, we trained StyleGAN v2 and BigGAN on each product separately to have control on background products. Note however, that images from DT-GAN were always obtained from a single model. We can see that some generated samples from StyleGAN v2 do not contain clear defects, and samples from BigGAN present abnormal grid patterns. Both methods do not take images as inputs but generate synthetic images according to a given latent code which contains information for both FG and BG. This conditioning leads to limited diversity in the output images. On the other hand, StarGAN v2 performs translation based on input images but suffers from the same entanglement issue. Thus, it fails to preserve the background, which results in artifacts or identity-shift in its outputs. Our network architecture that disentangles foreground and background seems to mitigate these issues. See Appendix E.4 for more images. + +![](images/e8844ad801507fcb58625110b5ddd1e08b278ccc393d4b6551f6e75e4f93f6ca.jpg) +Figure 3: Qualitative comparison of latent-guided image synthesis results. In each subfigure: on the left, defective images are fully generated from random noise. On the right, random defects are synthesized onto given normal samples. Note that BigGAN\* denotes it was trained with DiffAug. + +![](images/c8d4f138dd7a663cb4fbcdd6d95ca7b5d88bc4505014c289d8d78e678a6ef857.jpg) +Figure 4: Qualitative comparison of reference-guided image synthesis results on the SDI dataset. Each method transforms the given source images into target foreground domains (e.g., Scratches) with the styles and contents extracted from the reference images. + +Also for reference-guided image synthesis, where we used different background and foreground reference images as illustrated in Figure 4, only our method produces high quality images with preserved background from the source and transferred foreground defect from the reference. + +Ablation study. We visually demonstrate the effect of each component we added to DT-GAN compared to StarGAN v2 in Figure 5, using the examples of both latent- and reference-guided image synthesis from Normal to Scratches. The quantitative evaluation can be found in Appendix E.3. + +Column (a) corresponds to StarGAN v2 and highlights the drawback of entangled FG/BG again (i.e. the identity-shift in the background). We first tackle this problem by modeling the style code and foreground content explicitly and feeding them separately to the generator. This leads to a better preservation of the background structure in column (b) for the reference-guided subnetwork, but not for the latent-guided synthesis on the bottom of Figure 5. Thus, we add a foreground classifier in the discriminator in (c) to ensure the output image contains the desired foreground content (scratch). Similarly, we introduce a background classifier to the discriminator in column (d). Note that the additional product type labels can be acquired automatically from production lines. + +For column (e), we add the separate decoders for foreground and background in the generator which are fused only in the end. This enhances the preservation of background characteristics like lighting even more. Imposing an additional penalty for foreground content extracted from a normal sample as described in Section 3.1 leads to another visual improvement of the foreground edges for reference-guided synthesis in column (f). Finally, inspired by StyleGAN, we incorporate adaptive noise injection to the mapping network, which significantly boosts the performance of our latentguided image synthesis as shown in column (g). + +Styling. We visually demonstrate the effect of style codes in our method by randomly sampling those and combining them with fixed reference background and foreground images in Figure 6, where a variety of artistic styles can be seen on the output columns. + +![](images/91696fc85e55662030167a35ef4aae436fa0857c2b3a62e62f0e14e5967749b8.jpg) +Figure 5: Ablation study. (a) The baseline StarGAN v2. (b) $^ +$ Style-Content branches. (c) $^ +$ Foreground classifier. (d) $^ +$ Background classifier. (e) $^ +$ Separately decoding foreground and background in $G$ . (f) $^ +$ Anchor foreground domain (e.g. Normal). (g) $^ +$ Noise injection in Mapping Network. + +![](images/10b4554eb02cdbab48f0a3b29e5fed2318aee10f044aacf5703500d8bb21eef0.jpg) +Figure 6: Visual effect of randomly sampled style codes on fixed pairs of reference background (Source) and foreground (Content) images. + +# 4.2 DT-GAN FOR DATA AUGMENTATION + +We also evaluated our method as a data augmentation method for defect classification on the SDI dataset. We defined one task ‘general’, where the classifier was trained on images from all products at once, while task ‘single product’ only used the subset of images for one product. + +Besides, we incrementally varied the amount of real Normal data available for classifier training: 4500, 6600, 12000 and 18600. In the case of defective images, all of them were always used due to the small amount unless otherwise specified. As backbone we used a ResNet-50 (He et al., 2016a) with ImageNet pretrained weights. For experiments with synthetic data, we attached an auxiliary domain classifier to the network through a Gradient Reversal Layer (Ganin & Lempitsky, 2015). + +Table 2: Quantitative comparison of the baseline methods on defect classification task at the scale of 12000 images/class. The reported values are the achieved error rates $( \% )$ over five runs. + +
MethodResNet-50EfficientNet-b4
No-Aug21.64±1.2412.06±0.64
Trad-Aug12.58±0.819.33±0.73
Mokady (2020)11.11±1.1913.26±1.13
StarGAN v213.07±1.3012.25±0.79
StyleGAN v211.55±1.7911.68±0.76
BigGAN+DiffAug11.45±0.6112.06±0.50
Ours9.9±0.699.14±1.02
+ +Since the SDI dataset is highly imbalanced, we oversampled the minority classes (Ling et al., 1998) unless the data was balanced through synthetic images. Additionally, we always applied traditional data augmentation techniques like random horizontal flips, jittering and lighting (Shorten & Khoshgoftaar, 2019) except where noted. All following results were evaluated by the achieved error rates over five runs with different random seeds. + +Effectiveness of synthetic data. We first compare classifier performance for no augmentation (NoAug), traditional data augmentation (Trad-Aug), and a combination of traditional augmentation with synthetic images for GAN methods including DT-GAN. We also introduce a stronger backbone, EfficientNet-b4 (Tan & Le, 2019), to demonstrate that our results are not confined to a specific network. Table 2 shows that our method is the only one that improves performance for both backbones, presumably due to the combination of high visual image quality and diversity in our samples. + +Table 3: Experimental results on using different amount of synthetic images generated by DT-GAN to train classifiers. The left-most column stands for number of samples per class to be classified. The training set of the baselines is balanced by oversampling while ours is by synthetic images. + +
Dataset Size20AAll
Trad-AugOursTrad-AugOurs
450015.55±0.6314.28±1.2512.75±0.6111.04±0.76
660016.69±0.7614.41±3.1213.07±1.5710.60±0.48
1200016.95±1.0214.22±1.5312.05±0.819.90±0.69
1860016.12±2.1915.36±0.8612.37±0.3210.21±0.96
+ +Impact of dataset size. Motivated by the limited availability of data in real-world production scenarios, we therefore evaluated DT-GAN for data augmentation on a subset of the full SDI dataset (All), which only contains 20 defective samples in product A for each defect type (20A). In this case, DT-GAN was also trained on the reduced subset. As shown in Table 3, there is a clear improvement when synthetic images from DT-GAN are used as data augmentation, even for the extremely limited data subset. Further results on single product classifiers can be found in Appendix E.1. + +Table 4: Cross-domain effect on single product classifiers trained with reference-guided synthetic images at the scale of 12000 images/class. + +
Trad-AugvAVBvCvABC
A13.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
B6.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
C16.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
+ +Cross-domain effect. We hypothesized that limited data can be counteracted by transferring defects across multiple background products, if there are at least some defects that occur on multiple products (See Appendix E.1 for further discussion). We tested this approach by comparing the performance of classifiers trained on synthetic images with defects from a specific source (vA, vB, vC) to classifiers trained on images with defects from all products (vABC). As we can see in Table 4, the best performances are reached by the models that take over defects from other products. We interpret this as support for our hypothesis and its practical usefulness. + +# 5 CONCLUSION + +We propose a novel method, DT-GAN, which allows diverse defect synthesis both by generating from randomly sampled noise and by following the guidance of given reference images. Due to explicit style-content separation and FG/BG disentanglement, DT-GAN achieves higher image fidelity, better variance in defects and full control over background and foreground while being sample-efficient. We demonstrated the feasibility and benefits of DT-GAN on a real industrial defect classification task and the results show our method provides consistent gains even with limited data and boosts the performance of classifiers compared to state-of-the-art image synthesis methods. For future investigation, we aim to represent defects more explicitly (e.g., localization) to improve the explainability of the model and also enhance the model transferability to unseen products. + +# REPRODUCIBILITY STATEMENT + +We aim for full reproducibility by publishing the source code and dataset with the final version of the paper. 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In NIPS, 2017b. + +# A THE SURFACE DEFECT INSPECTION DATASET + +The Surface Defect Inspection (SDI) dataset consists of 20,414 images at $1 2 8 \times 1 2 8$ resolution. It contains three background domains—product A, product $\mathbf { B }$ and product C, each can be further classified into three foreground domains—Normal, Scratches and Spots. Figure 7 shows example images of the SDI dataset. To be noticed that the dataset is highly imbalanced not only between normal and defective samples but also between different products as shown in Table 5. This sets a more challenging task when training deep neural networks like GANs and downstream classifiers. + +For each foreground and background domains, we randomly select 50 images for a joint validation/test set, which is then further split into separate sets in the ratio of 3:7, and use all remaining images as training sets for GAN and classifier training. We present the distribution of the training set when training DT-GAN in Table 6. Note that the normal samples used in GAN training are only a subset of all available samples in Normal and we keep the rest of them for generating defective samples at test time. For classifier training, we show the statistics in Table 7, where the number of normal samples involved in classifier training increase incrementally. The validation set is used to select the best model during classifier training while the test set is left untouched until the final evaluation. Both of the validation and test set are inaccessible by DT-GAN. + +Table 5: Distribution of the full SDI dataset. + +
Overview
ABC
Normal625062506250
Scratches340167121
Spots108670258
+ +Table 6: The training set for DT-GAN and the baseline image synthesis methods. + +
Overview
ABC
Normal700700700
Scratches29011771
Spots58620208
+ +Table 7: The training, validation and test set for classifier training, where $N$ increases incrementally—1500, 2200, 4000 and 6200. + +
TrainValidationTest
ABCABCABC
NormalNNN121815383235
Scratches29011771141615363435
Spots58620208141615363435
+ +# B TRAINING DETAILS + +DT-GAN. We follow the training scheme as described in StarGAN v2 with minor modifications. To fit the model on a single Nvidia GTX TITAN X, the batch size is reduced to four while the model is still trained for 100,000 iterations. The training time is about three and a half days on the dedicated GPU with the modified network architecture2 and loss functions mentioned in Section 3 in PyTorch (Paszke et al., 2017). We set $\lambda _ { \mathrm { s t y } } = 1$ , $\lambda _ { \mathrm { d s } } = 1$ , $\lambda _ { \mathrm { c y c } } = 1$ , $\lambda _ { \mathrm { c o n . c y c } } = 1$ , $\lambda _ { \mathrm { c l s } } = 1$ and $\lambda _ { \mathrm { B G . c l s } } = 1$ for the SDI dataset. All other design choices remain the same as in StarGAN v2. + +Classifiers. We train all the classifiers that use ResNet-50 as backbone for 100 epochs with the SGD optimizer (Ruder, 2016) and batch size 256. The initial learning rate is 0.001, momentum is 0.9 and weight decay is 1e-4. A learning rate scheduler is set to reduce the learning rate by factor of 0.1 when the validation loss stops decreasing for 5 epochs. The same setting also applies to EfficientNet-b4, except the batch size is reduced to 128. Although DT-GAN can synthesize realistic defective samples, we notice that there still exists a domain gap between the generated samples and the real samples. To explore the full potential of the generated samples, we attach an auxiliary source classifier to distinguish between synthetic and real samples. Then, this classifier is connected to the backbone (e.g. ResNet-50) through a Gradient Reversal Layer. With the help of the Gradient Reversal Layer, the backbone is forced to extract the shared features between synthetic and real samples, which ensures all training samples are effectively learned. + +![](images/2b5df2b8d264369f1ee04ca1e3d7919ee1d3d69e7370aa9deec145bb6941e15e.jpg) +Figure 7: Overview of the SDI dataset. + +We design a two-layer perceptron that connects to the average pooling layer in ResNet-50 as shown in Figure 8. Note that the usual fully connected layer after the average pooling in ResNet-50 remains the same and is not affected by the extra branch we added. Inspired by Chen et al. (2018), a threelayer perceptron is used for EfficientNet-b4 instead as shown in Figure 9. Its layers are initialized with a random normal distribution, where the standard deviation is set to 0.01 for the first two layers and 0.05 for the output layer. The biases for all layers are set to 0. + +![](images/5876fcb93af3f0aad7fc6536e733bff52799290b671bdeef4fd81bf21ab172fb.jpg) +Figure 8: ResNet-50 with GRL. + +![](images/79d423c36937100d387ec0e803940f1adbc96236a1a0b915320ec3741d5f7095.jpg) +Figure 9: EfficientNet-b4 with GRL. + +# C EVALUATION SETUP + +Generated samples from DT-GAN. DT-GAN requires images as input for generating synthetic data. At test time, we translated each Normal image in the SDI dataset into four defective images: two with Scratches and two with Spots. The translations were performed by two subnetworks: by the mapping network $M$ using random noise (‘latent-guided’) and by the style-content encoder $E$ using a reference image (‘reference-guided’). We first randomly sampled one latent code for each defective foreground domain. Similarly, we also randomly sampled one reference image from the training set for each defective foreground domain. The corresponding style codes and defect contents were then produced by the two subnetworks respectively and fed to the generator for target image generation. + +We conducted classification experiments separately on images generated from the two subnetworks and a mixture set of both (i.e. $50 \%$ from each subnetwork). Experiments show consistent gains of using synthetic images generated from DT-GAN (Table 8). We observe that the latent-guided synthetic images in general perform better than the reference-guided one, while the mixture set provides more stable results with regard to the standard deviation. Presumably the mixture set benefits from the combination of samples from reference-guided synthesis, which are well aligned with the original defect distribution, and the samples from latent-guided synthesis, i.e. from random noise, which adds novel but plausible defects to the dataset. In the main text, we report the results of the mixture set for all experiments, including the quantitative evaluation of DT-GAN. + +Table 8: Classification results with regard to the synthetic images generated from the two subnetworks and the mixture set. + +
Dataset SizeAll
Trad-AugLatentReferenceMix
450012.75±0.6110.72±0.9611.48±0.8811.04±0.76
660013.07±1.5710.34±1.8611.55±1.6410.60±0.48
1200012.05±0.819.90±1.2610.40±0.999.90±0.69
1860012.37±0.3211.04±1.2612.12±0.7510.21±0.96
+ +Frechet inception distance (FID) and Kernel inception distance (KID). ´ We used the feature vectors from the last average pooling layer of the ImageNet pretrained Inception-V3 to calculate both scores. For each test image from the Normal domain, we translated it into a synthetic defective image of each defect domain. The style codes and contents for the translation were acquired in two ways: by randomly sampling from the standard normal distribution and by randomly sampling a reference image from the train set of a defect domain. To calculate the FID and KID score, we generated 4000 defective samples per product per defect domain for each way of guidance, and formed the mixture set by randomly sampling 2000 images per product per defect domain from each way. The reported FID and KID scores were then computed between the defective images in the training set and the mixture set of synthetic defective images. The same procedure was applied when computing scores on single product subsets of the SDI dataset. For example, for product A, we calculated the scores between the defective image of product A in the training set and the mixture set of synthetic defective images of product A. + +# D NETWORK ARCHITECTURE + +In this section, we provide the architectural details of all four modules in DT-GAN. + +Table 9: Generator architecture. + +
(a)Encoder
LayerResampleNormOutput Shape
Image x-128 × 128×3
Conv 1×1-128 ×128 ×128
ResBlkAvgPoolIN64× 64×256
ResBlkAvgPoolIN32 × 32 × 512
ResBlkAvgPoolIN16 × 16 × 512
ResBlkIN16 ×16× 512
ResBlkIN16 ×16 × 512
(b) Background Decoder(c) Foreground Decoder
LayerResampleNorm Output ShapeLayerResample NormOutput Shape
Input-16 × 16 × 448Input ResBlk16 × 16 × 64
ResBlkIN 16 ×16× 448 16 ×16× 512ResBlkAdaIN16 ×16× 64
ResBlkIN=AdaIN16 × 16 × 256
ResBlk=IN16 ×16 × 512ResBlkAdaIN16 ×16× 256
ResBlkUpsampleIN32 × 32×512ResBlk Upsample ResBlkAdaIN32 × 32 × 256
ResBlk ResBlkUpsample UpsampleIN IN64 × 64× 256 128 × 128× 448UpsampleAdaIN64×64×128
ResBlkUpsampleAdaIN128 ×128 × 64
(d) Fusion
LayerResampleNormOutput Shape
Input128 × 128 × (448 + 64)
Conv 1×1128 × 128× 3
+ +Table 10: Mapping network architecture. + +
LayerActivationOutput Shape
Latent z=16
LinearReLU512
LinearReLU512
LinearReLU512
LinearReLU512
(b) Style Code(c) Content
LayerActivation Output ShapeLayerResample ActivationNoise Output Shape
Input=512Input=512
LinearReLU512Reshape--1×1×512
LinearReLU512ResBlkUpsampleINTrue2×2×512
LinearReLU512ResBlkUpsampleINTrue4×4×512
Linear164ResBlkUpsampleINTrue8×8×256
ResBlkUpsampleINTrue16 ×16×128
Conv 1×1=INTrue16 × 16× 64
+ +Generator (Table 9). For the SDI dataset, the encoder part of the generator consists of three downsampling blocks and two intermediate blocks (Table 9 (a)), all of them are pre-activation residual units (He et al., 2016b). Then the encoded feature map is split channel-wise into background (Table 9 (b)) and foreground (Table 9 (c)). Both of them are then carried through separate decoders. We use the instance normalization (IN) and the adaptive instance normalization (AdaIN) as indicated. The style code is injected into all AdaIN layers to modulate the affine transformations. Note that + +Table 11: Style-content encoder and discriminator architectures. + +
(a)SharedLayers
LayerResampleNormOutput Shape
Input x128 × 128 ×3
Conv 1×11128 × 128 × 64
ResBlkAvgPool64 × 64× 256
ResBlkAvgPool32 × 32×512
ResBlkAvgPool16 ×16 × 512
(b) Style Code /Discriminator and BG Classifier(c) Content /FG Classifier
LayerResample NormOutput ShapeLayerResample NormOutput Shape
Input-16 ×16× 512Input16 ×16× 512
ResBlkAvgPool8×8×512LReLU16 ×16 × 512
ResBlkAvgPool4×4×512Conv 1×1*K16×16×64*K
LReLU4×4×512
Conv 4×41×1×512
LReLU1×1×512
Reshape512
Linear *KD*K
+ +AdaIN is only used in the foreground decoder. The outputs of both decoders are only fused in the end (Table 9 (d)). + +Mapping Network (Table 10). The mapping network consists of four shared linear layers (Table 10 (a)) and two separate branches: one for generating style codes (Table 10(b)) and one for contents (Table 10(c)). Each of them is further divided into $K$ output branches, where $K$ denotes the number of domains. The dimension of the input, the output style code and the output content is set to 16, 64, and $1 6 \times 1 6 \times 6 4$ , respectively. The latent code is sampled from the standard normal distribution. Note that we apply per-pixel noise after each convolution in the content branch, which we have observed to increase the diversity of generated defects significantly (cf. Figure 5 (g)). + +Style-Content Encoder (Table 11). The style-content encoder consists of a CNN (Table 11 (a)) with two branches (Table 11 (b) and (c)) as in the mapping network. Each branch has $K$ outputs, where $K$ is the number of domains. Three pre-activation residual blocks are shared among two branches, followed by a specific structure for each branch. The output dimension $D$ in Table 11 is set to 64, which denotes the dimension of the style code. + +Discriminator (Table 11). The discriminator is a multi-task discriminator with two auxiliary classifiers for the foreground content and the background. The structure is almost identical to the stylecontent encoder, except $D$ is set to 1 for real/fake classification. The background classifier acts in parallel to final linear layer in Table 11 (b) and provides the logits for background classification. The foreground classifier instead acts on top of the output in Table 11 (c) and four more pre-activation residual layers are applied to encode the content into logits for foreground content classification. + +# E ADDITIONAL RESULTS + +# E.1 ADDITIONAL RESULTS ON THE SDI DATASET + +We provide additional reference-guided image synthesis results on the SDI dataset in Figure 10. We demonstrate all the possible transfers among all foreground domains. Both style codes and contents are extracted from the reference images. To be noted that DT-GAN can append and remove foreground defects not only onto Normal samples but also to defective samples. For example, in the fifth column of Figure 10, the original scratch in the source image is removed and only the defects from the reference images are presented in the output images. + +Besides, we present additional evaluations showing the effectiveness of our synthetic data according to Table 3. As seen in Table 12, the synthetic images from DT-GAN also boost the performance in single product classifiers, where the classifiers were trained on the subset of images for one product (A, B, C) instead of the full dataset (ABC). + +![](images/ba0e2234e87b25b63cc95ac4e5644f8da2a00a1e319667a7cf9fa0a32eed56c7.jpg) +Figure 10: Reference-guided image synthesis results on the SDI dataset. The first row and the first column are the real images sampled from the dataset, while the rest are synthetic images generated by the proposed DT-GAN. Our model provides translations between different foreground domains (Normal, Scratches and Spots) with styles and contents extracted from reference images while the backgrounds from source images are well preserved. + +As discussed in Section 4.2, we assumed that the data-insufficiency problem can be mitigated by transferring defects across multiple background products. To examine if this assumption holds, we compared the performance of classifiers trained on synthetic images with defects from a specific source (vA, vB, vC) to classifiers trained on images with defects from all products (vABC). The results on the cross-domain effect with regard to different sizes of the training set are shown in Table 13. We again notice that using our synthetic data is beneficial. Moreover, in most cases the performance is further improved by exploiting cross-domain information (i.e. by transferring defects from other products). We interpret this as support for our assumption and the practical usefulness of our method in the real-world scenario. The case of cross-domain image synthesis when the desired combination is not presented in the training set is covered in the study on the MVTec Anomaly Detection dataset (Bergmann et al., 2019) (see Appendix E.4). + +Table 12: Quantitative results for DT-GAN as a data augmentation method to train general and single product classifiers. The left-most column indicates the number of samples per class, including all images from the training set plus increasing amounts of synthetic images. In the first row, 20A refers to the case of 20 real defective samples for product A, while All refers to the full training set. + +
Dataset Size20A
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450035.09±2.62 27.64±3.127.8±1.485.6±1.6715.24±1.90 13.14±1.7015.55±0.63 14.28±1.25
660039.64±2.28 27.64±1.658.8±1.646.2±1.6415.81±1.73112.38±1.6516.69±0.76 14.41±3.12
1200034.18±4.39 28.55±7.325.8±0.455.6±1.1416.19±1.17 10.86±1.28 16.95±1.02 14.22±1.53
1860039.45±7.06 32.55±5.047.2±0.845.2±1.1014.86±0.85 13.14±2.06 16.12±2.19 15.36±0.86
Dataset SizeAll
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450016.00±1.04110.18±1.75 8.79±0.455.60±1.5117.13±6.62 14.09±2.2712.75±0.61 11.04±0.76
660014.90±1.3810.54±1.22 7.60±1.516.80±3.1115.23±2.33 11.42±013.07±1.57 10.60±0.48
1200013.81±2.366.72±1.65 6.80±1.644.60±016.57±3.20 13.90±2.5712.05±0.819.90±0.69
1860013.63±2.22 10.54±2.45 6.80±1.794.99±1.87 15.62±0.85 11.61±1.24 12.37±0.32 10.21±0.96
+ +Table 13: Cross-domain effect on single product classifiers trained with reference-guided synthetic images at all scales. Note that here A, B and C stand for 3 products in the SDI dataset while vA, vB, vC and vABC indicate the defects are copied from which reference set. + +
Dataset SizeA
Trad-AugvAvBvCvABC
450016.00±1.0412.90±2.6113.08±1.6514.90±2.4615.27±3.49
660014.90±1.3813.99±1.8911.26±1.0414.36±4.0416.00±2.85
1200013.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
1860013.63±2.2212.72±5.2214.36±3.8314.18±5.0513.81±8.56
DatasetB
SizeTrad-AugvAvBvCVABC
45008.79±0.457.80±2.155.60±1.1410.19±0.846.79±1.30
66007.60±1.516.80±1.657.80±1.108.00±2.346.00±1.41
120006.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
186006.80±1.796.19±1.784.40±1.146.60±1.955.99±1.58
DatasetC
SizeTrad-AugvAVBvCvABC
450017.14±4.6214.85±0.5216.76±2.5813.90±1.9812.00±1.59
660015.23±2.3313.14±1.2413.90±2.2914.28±1.3412.57±1.57
1200016.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
1860015.62±0.8513.71±1.7315.99±6.7512.57±3.2612.95±2.98
+ +# E.2 ADDITIONAL FID AND KID RESULTS ON THE SDI DATASET + +We provide additional results in the case of training GANs with augmentation methods in Table 14. Augmentation methods like ADA (Karras et al., 2020a) or DiffAug (Zhao et al., 2020) are proposed to adapt GAN training to limited data. We applied these augmentation methods to StyleGAN v2 and BigGAN, because these state-of-art image synthesis methods are not optimized for small dataset. However, incorporating the augmentation methods in training GANs on the SDI dataset is not always beneficial. The performance of StyleGAN v2 is largely degraded when using ADA, potentially due to the conflict between augmentation methods and the decentralized location of defects—in the SDI dataset, defects can occur anywhere on the surface. This is in contrast to datasets that were used to evaluate the aforementioned augmentation methods in GANs, where the objects are centralized (e.g., ImageNet (Deng et al., 2009), Cifar (Krizhevsky & Hinton, 2009)) and their attributes (e.g., beard, eye glasses in CelebA (Liu et al., 2015)) only occur in specific images parts. + +Table 14: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID and KID. Note that the reported values are not comparable between columns, because they are calculated on different training sets. The scores of StarGAN v2 on single products are omitted because generating images with specified background is not possible due to its network design. + +
MethodFID↓KID↓
ABCAllABCAll
Mokady et al. (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v2--137.701-10.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
StyleGAN v2 + ADA149.6642.75135.6976.160.1380.0190.1910.055
BigGAN235.66192.89193.61151.430.2480.1990.2760.115
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
+ +# E.3 ABLATION STUDY WITH REGARD TO FID AND KID SCORES + +We report the FID and KID scores of the ablation study in Table 15. We notice that both subnetworks show positive correlation to each modification except for structural change as in (a) and (e) . Among the two subnetworks, the reference-guided subnetwork outperforms the latent-guided one in the beginning, which is due to the fact that transferring existing contents is easier than generating them from random noise. This effect is also observed in Figure 5. However, the performance of the latentguided subnetwork improves significantly after applying per-pixel noise injection. The subnetwork can now output non-deterministic foreground contents even for a fixed input vector which results in better visual quality and higher diversity of generated defects. In the main text, the scores of the mixture set are reported. + +Table 15: Ablation study with regard to FID and KID scores. + +
FID↓KID↓
LatentReferenceMixLatentReferenceMix
(a) Baseline StarGAN v237.7337.9937.700.0130.0130.013
(b)+ Style-Content branches43.9032.6133.360.0170.0110.011
(c)+Foreground classifier37.1432.3427.690.0140.0110.008
(d) + Background classifier34.1232.5030.230.0110.0110.010
(e)+ Separately decoding foreground and background in G48.5238.1134.790.0170.0150.011
(f) + Anchor foreground domain (e.g. No rmal)43.6637.4532.150.0190.0150.011
33.050.0090.011
(g)+ Noise injection in Mapping Network34.4229.730.009
+ +# E.4 ADDITIONAL RESULTS ON THE MVTEC ANOMALY DETECTION DATASET + +The MVTec Anomaly Detection dataset (Bergmann et al., 2019) contains 15 different object and texture categories for anomaly detection. The dataset is formed of non-defective image for training and both non-defective and defective images with various kinds of defects for testing. The pixel-level annotations of all defective images are also provided. It is worth noting that the MVTec Anomaly Detection dataset is relatively small scale in number of images, where the number of training images is ranging from 60 to 391. Moreover, the number of defective images for each defect category in the test set is varying only from 8 to 30, which is relatively limited considering the sophisticated pattern of defects. + +We conducted image synthesis experiments on a subset of MVTec Anomaly Detection dataset, where we selected four texture categories: Carpet, Leather, Wood and Tile for our targeted scenario i.e. surface defects. Furthermore, we aggregated some of the original defect types defined in the MVTec Anomaly Detection dataset into scratches and spots according to their visual appearance. We then simply added the subset of the MVTec Anomaly Detection dataset to the training set together with the SDI dataset for training DT-GAN. Details of the resulting dataset are shown in Table 17. Note that the small scale of available data posts a major challenge for training generative models. + +Quantitative Evaluation. We present additional quantitative results on the subset of the MVTec Anomaly Detection dataset in Table 16, following the same evaluation setup as described in Appendix C. As shown in Table 16, our method achieves the best scores in Carpet and Wood, which supports our claim that DT-GAN generates synthetic images with higher fidelity and more diverse defect. However, we also observe that StyleGAN v2 seems to outperform our method in Leather and Tile. + +Please note that FID and KID are not optimized to evaluate such a small dataset, there the results should only be interpreted together with the qualitative results. + +Note the we again omit the FID and KID of StarGAN v2 because it is not cable of generating images for a specified product due to the ‘identity-shift’, which is also explained in detail in the qualitative evaluation. + +Table 16: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID and KID. Note that the reported values are not comparable between columns, because they were calculated on different training sets. + +
MethodFID↓KID↓
CarpetLeatherTileWoodCarpetLeatherTileWood
Mokady (2020)41.8760.26275.1281.710.040.030.290.04
StarGAN v211=1
StyleGAN v251.3751.60225.96140.010.050.030.230.12
BigGAN + DiffAug34.47101.70391.54113.320.030.070.420.07
Ours22.7986.13321.3575.830.010.070.360.03
+ +Qualitative Evaluation. For qualitative results, we again discuss the ‘latent-guided’ and ‘referenceguided’ synthesis separately. + +We present the ‘latent-guided’ image synthesis results of StyleGAN v2 in Figure 11 and Figure 12 and BigGAN in Figure 13 and Figure 14. The results are acquired by training one model for each product and then generating 16 images from randomly sampled latent codes from each of them. As pointed out in Section 4.1.2, both methods can not adapt well on small dataset. They suffer from model collapsing and show signs of overfitting by generating images similar to the training data. For example, StyleGAN v2 generates images either with no clear defect or identical to the training set (e.g., Leather in Figure 11 and Product B in Figure 12). The overfitting we observe here also explains the better FID and KID scores in Table 16. For Tile, we can see clear signs of mode collapse in the generated Tile images of StyleGAN v2. Similarly, BigGAN produces images with single mode and abnormal patterns (e.g., grid structure and gray edges). Unlike StyleGAN v2 and BigGAN, StarGAN v2 and our method both require images as input (i.e. Source). Therefore, we randomly sampled two Normal images and applied eight defects which are generated from randomly sampled latent codes to each of them. As seen in Figure 15 and Figure 16, StarGAN v2 fails to preserve the background from the given input images due to the highly entangled FG and BG. Also it fails to generates legit and diverse defects without separately modeling the style and the content. In contrast to aforementioned methods, our DT-GAN produces images with higher fidelity and more diversity in defect patterns as shown in Figure 17 and Figure 18. We believe this again prove the importance of style-content separation and FG/BG disentanglement, which we introduce in Section 3.1. + +For ‘reference-guided’ image synthesis, the results of Mokady et al. (2020) are shown in Figure 19 and Figure 20 while the results of StarGAN v2 are in Figure 21 and Figure 22. We can observe a clear shift in color in all the outputs from Mokady et al. (2020). Moreover, Mokady et al. (2020) can only transfer content between two domains. In order to perform translation from a non-defective sample to a defective one, we trained a model for each type of defect and for each product. This sums up to be 13 models (Scratches and Spots for 6 categories and Scratches only for Tile). The results from the intended use within one background domain can be found on the diagonal and are marked in red in both Figure 19 and Figure 20. We still show the images that we feed in images from other background domains. As expected, the model then fails to preserve the background of given source images and introduce artifacts to the outputs. Similarly, StarGAN v2 does not preserve the background from the input images. Without style-content separation and FG/BG disentanglement, we observe that StarGAN v2 encodes the background characteristics together with the foreground content of the reference images, which results in identity-shits in its output images. Moreover, the output images either show no clear defect or contain abnormal patterns which sabotage the fidelity. On the contrary, our method can faithfully transfer the foreground content of reference images across given background of different products as shown in Figure 23 and Figure 24, which demonstrate the effectiveness of the style-content separation and FG/BG disentanglement we introduced in Section 3.1. + +It is also worth noting that our method can perform cross-domain image synthesis even the desired combination is not presented in the training set. We demonstrate this on product Tile, which only has images with Scratches but no Spots. As shown in Figure 18 and Figure 24, DT-GAN can generated spots one given Tile images. However, this kind of transformation is most useful when the desired combination is reasonable for the downstream applications. + +Limitation and Future Work. We have demonstrated the feasibility of the proposed DT-GAN by incorporating more products from the MVTec Anomaly Detection dataset in our training procedure. Intensive experiments have shown that the generated images from DT-GAN yielded better results compared to the baseline image synthesis methods. However, we noticed that despite the diverse patterns of the generated defects, DT-GAN tends to apply the styles learned from the SDI dataset also to the samples from the MVTec Anomaly Detection dataset. For example, we can observe some ”halo” effects in Leather and Wood in Figure 18 and some of the generated scratches in Figure 17 and Figure 23 are rather weakly pronounced. We hypothesize this can be counteracted by explicitly localizing the defect and enforcing the model to learn conditional relationships between ‘styles’ and different backgrounds. We aim to address these issues in future work. + +Table 17: Overview of our formation of the MVTec Anomaly Detection sub-dataset. The first column represents the original defect types in the MVTec Anomaly Detection dataset while the first row stands for the defect types in our targeted scenario. We list the ID of samples we took from the MVTec Anomaly Detection dataset and show the number of samples in row Sum. + +
(a) Carpet
ScratchesSpots
Color011,012,014,016, 017000,003,004,007,015,018
Thread000-018
Hole-000 - 016
Sum2423
(b) Leather
ScratchesSpots
Color001,003,005,007,009,011,013,015,018000,002,006,008,010,012,014
Cut000 -018 000 - 006,009 - 016-
Fold Glue000 - 002,005-009,011-015,018
Poke003,009,010,016,017000-017
Sum39
48
(c) Tile
ScratchesSpots
Crack000 - 016二
Sum170
(d) Wood
ScratchesSpots
Color003,005
Scratch001-006,008- 010,013 -016,018- 020000 - 016
Hole=000 - 004,006-009
Combined008001,002,009
Sum1912
+ +# Randomly Sampled Defects (Scratches) + +![](images/567f75bf12cbd5a5119634c75240da3d88366422eb7484cba51a63409149e476.jpg) +Figure 11: Latent-guided image synthesis results of StyleGAN v2 on the SDI dataset and the MVTec AD dataset. We train a model for each product and generate 16 Scratches images from randomly sampled latent codes. + +![](images/3f55519405e1df2af7deb5998aea2cd1baa9a5f4eca9816aac99ce93d6637aef.jpg) +Figure 12: Latent-guided image synthesis results of StyleGAN v2 on the SDI dataset and the MVTec AD dataset. We train a model for each product and generate 16 Spots images from randomly sampled latent codes. + +![](images/9af2881df4cd89bcd7f83e40ecac133c4486b383aca6a5f80167e218d4861e0f.jpg) +Figure 13: Latent-guided image synthesis results of BigGAN with DiffAug on the SDI dataset and the MVTec AD dataset. We train a model for each product and generate 16 Scratches images from randomly sampled latent codes. + +![](images/6502d31fa4830920177c7a1201405a5d38fd61293a22fa5cc40cc805b4888a23.jpg) +Figure 14: Latent-guided image synthesis results of BigGAN with DiffAug on the SDI dataset and the MVTec AD dataset. We train a model for each product and generate 16 Spots images from randomly sampled latent codes. + +![](images/753dedb2a88181dd1a02122d6d4012b9e01f9fd382ccd9580c8645207ed06d8d.jpg) +Figure 15: Latent-guided image synthesis results of StarGAN v2 on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Scratches. Note that without the style-content separation and the FG/BG disentanglement, StarGAN v2 not only fails to preserve the background from the given Source image but also fail to generates legit defects. + +![](images/49e2bc6b1e4d4de05a60e2ef2fb4992d1dc00bdd8bcc62cfa38819c364a1a72e.jpg) +Figure 16: Latent-guided image synthesis results of StarGAN v2 on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Spots. Note that without the style-content separation and the FG/BG disentanglement, StarGAN v2 not only fails to preserve the background from the given Source image but also fail to generates legit defects. + +![](images/d7665bc9189f97191cb363ff43243d45d9c0eed07ddb8d132a1899df30a87129.jpg) +Figure 17: Latent-guided image synthesis results of DT-GAN on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Scratches. Note the our model takes input Source images as background and only synthesizes the foreground defects from randomly sampled latent code compared to StyleGAN v2 and BigGAN. + +![](images/6dc33723b61a8a29ea99523303f759e2ec80cca613f5db714539c271f27d80a8.jpg) +Figure 18: Latent-guided image synthesis results of DT-GAN on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Spots. Note the our model takes input Source images as background and only synthesizes the foreground defects from randomly sampled latent code compared to StyleGAN v2 and BigGAN. + +![](images/6fbc28db81f825e8f35749ff8c9692af1a0d34358c15a1ea8e335765a76b1ab6.jpg) +Figure 19: Reference-guided image synthesis results of Mokady et al. (2020) on the SDI dataset and the MVTec AD dataset. We train a model for each product and each defect type. Then we translate Normal images to Scratches by taking the Source as background and applying the foreground defect from Reference to it. + +![](images/52103c7943c67c62f890f5864b5e34df9520df52349661b703bc363d3ec05466.jpg) +Figure 20: Reference-guided image synthesis results of Mokady et al. (2020) on the SDI dataset and the MVTec AD dataset. We train a model for each product and each defect type. Then we translate Normal images to Spots by taking the Source as background and applying the foreground defect from Reference to it. + +![](images/cb73ee815eca7466279848e34aba2a7625dd3b27655d8dcff88d57a806fdbab6.jpg) +Figure 21: Reference-guided image synthesis results StarGAN v2 on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Scratches by taking the Source as background and applying the foreground defect from Reference to it. Note that without the style-content separation and the FG/BG disentanglement, StarGAN v2 not only fails to preserve the background from the given Source image but also fail to generates legit defects. + +![](images/c0757dd5081906d67150a0b1e7071d90d4070ef9a33fedd3004d5d77e8301ddd.jpg) +Figure 22: Reference-guided image synthesis results of StarGAN v2 on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Spots by taking the Source as background and applying the foreground defect from Reference to it. Note that without the style-content separation and the FG/BG disentanglement, StarGAN v2 not only fails to preserve the background from the given Source image but also fail to generates legit defects. + +![](images/8f5416bea6a57fe53694b7b81ca32fab188b10b5307dd5743801f95cfe3dc61b.jpg) +Figure 23: Reference-guided image synthesis results DT-GAN on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Scratches by taking the Source as background and applying the foreground defect from Reference to it. + +![](images/038273ee81067405ed8edb30707628ce21e36f0f73845fa8004a02f19865a69d.jpg) +Figure 24: Reference-guided image synthesis results of DT-GAN on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Spots by taking the Source as background and applying the foreground defect from Reference to it. \ No newline at end of file diff --git a/parse/dev/2hMEdc35xZ6/2hMEdc35xZ6_content_list.json b/parse/dev/2hMEdc35xZ6/2hMEdc35xZ6_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..05c17659d4568f8d59933aeea5e0079057ce7b6d --- /dev/null +++ b/parse/dev/2hMEdc35xZ6/2hMEdc35xZ6_content_list.json @@ -0,0 +1,2269 @@ +[ + { + "type": "text", + "text": "DEFECT TRANSFER GAN: DIVERSE DEFECT SYNTHESIS FOR DATA AUGMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 236, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Large amounts of data are a common requirement for many deep learning approaches. However, data is not always equally available at large scale for all classes. For example, on highly optimized production lines, defective samples are hardly acquired while non-defective samples come almost for free. The defects however often seem to resemble each other, e.g., scratches on different products may only differ in few characteristics. In this work, we propose to make use of the shared characteristics by transferring a stylized defect-specific content from one type of background product to another. Moreover, the stochastic variations of the shared characteristics are captured, which also allows generating novel defects from random noise. These synthetic defective samples enlarge the dataset and increase the diversity of defects on the target product. Experiments demonstrate that our model is able to disentangle the defect-specific content from the background of an image without pixel-level labels. We present convincing results on images from real industrial production lines. Also, we show consistent gains of using our method to enlarge training sets in classification tasks. ", + "bbox": [ + 233, + 267, + 764, + 474 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 502, + 336, + 518 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Automated Visual Inspection (AVI) is vital for quality control in modern production lines. Despite the fact that AVI has been studied for decades, it remains a challenging task with many open research questions await to be answered. One of the main challenges in data-driven AVI is the acquisition of suitable training data. This is for two reasons: First, collecting a vast amount of labelled data is usually labor-intensive and time-consuming. In many cases, even experts are required to identify where and what to look for. However, the acquired label information is task-specific and cannot be reused or transferred to a new task in most cases. Thus, the tedious labelling process must be repeated for each new product, even if its defect is similar to other products in people’s eyes. Second, in real-world scenarios such as highly optimized production lines, a more severe problem emerges: data imbalance. Only very few defective parts are produced by design. Moreover, the acquired anomaly images from a single product are lacking diversity and may not capture the full defect distribution. Training a robust deep neural network model in such conditions is very challenging. ", + "bbox": [ + 174, + 535, + 825, + 700 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since collecting sufficient real-world defective samples is impractical, algorithms to synthesize required images became a focus in research. Image synthesis through Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) has shown promising performance in recent years. But it also requires large amounts of balanced data which are not available in most industrial use cases, in particular for irregular defect patterns and large variation. Therefore, GANs tend to overfit to the training examples when trained with little data (Karras et al., 2020a). ", + "bbox": [ + 174, + 708, + 823, + 791 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we tackle these issues by exploiting cross-domain information: we first define two sets of domains—foreground domains and background domains. The foreground domain describes a set of images that contains a specific foreground content to be grouped into a distinctive category, and each content has a different style. The background domain instead is considered as a group of images that shares similar structural appearance over the whole image. For example, we can set foreground domains as defect types and background domains as product types while the styles of defects indicate their artistic looks such as light or heavy strokes. Building upon StarGAN v2 (Choi et al., 2020), the concept underlying this work is to transfer and generate foreground contents with a variety of styles across different background domains, as illustrated in Figure 1. ", + "bbox": [ + 174, + 799, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/7375134a93e4d6c487812a69c420b54b7f33ccdc499b7470357d6f8e1d3d2531.jpg", + "image_caption": [ + "Figure 1: The underlying concept of DT-GAN is to transfer and generate foreground contents with a variety of artistic styles (e.g., light / heavy strokes) across different background domains. " + ], + "image_footnote": [], + "bbox": [ + 258, + 103, + 735, + 237 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The contributions of this work are three-fold: First, we introduce Defect Transfer GAN (DT-GAN), a model that learns transferring existing foreground content and generating novel contents onto different backgrounds at the same time. In the real-world scenario, it allows defect inspection networks to learn from a variety of synthetic defective images by composing the foreground defects together with various non-defective images from different products. Second, DT-GAN is able to disentangle the foreground defect-specific content and the defect-irrelevant background in a weakly-supervised manner. Third, extensive experiments show that our method can generate diverse and real-looking defective samples even for products with only 20 real defective images. These defective images generated by DT-GAN boost the performance in defect inspection networks significantly. ", + "bbox": [ + 174, + 310, + 825, + 435 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 459, + 344, + 476 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "GANs have shown their power in many computer vision tasks such as image synthesis (Luciˇ c et al., ´ 2019), style translation (Johnson et al., 2016), super-resolution (Ledig et al., 2017), image impainting (Pathak et al., 2016) and many other applications. To quantify the performance of GANs, visual quality and the diversity of generated images are considered as two of the most important criteria. Recent models address these requirements either by dedicated loss functions (Mao et al., 2019b; Yang et al., 2019) or architectural design (Brock et al., 2019). StyleGAN v2 (Karras et al., 2020b), the latest state-of-the-art model in image synthesis, introduces stochastic variation in image generating process by adding per-pixel noise after each convolution. However, it is non-trivial to adapt the model to transform given input images due to the design of the generator. ", + "bbox": [ + 173, + 492, + 825, + 618 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In contrast, image-to-image translation methods (Isola et al., 2017) provide a way to recover the connection between inputs and the generated images while encouraging diversity. For example, Zhu et al. (2017b) and Huang et al. (2018) impose consistent mappings in latent space to achieve the goal. Some approaches (Ma et al., 2019; Park et al., 2019) use reference images as guidance to generate diverse outputs. Mokady et al. (2020) further extends the translation task from styles to contents. It learns to identify a specific content in a given input (e.g., a specific pair of glasses) and transfer it to the target image. However, aforementioned methods only consider the translation between two domains and their extension to multiple domains is non-trivial. ", + "bbox": [ + 174, + 625, + 825, + 736 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Surface defect detection is one of the important tasks in real-world industrial manufacturing. It aims at identifying and classifying defects with the help of machine vision. Traditional methods (Ngan et al., 2011) build models upon hand-crafted feature extractors, which are unstable and outperformed by deep learning based models. However, the performance and generalization ability of deep learning approaches are restricted due to limited number of defective samples in real-world scenarios. Data augmentation aims to enrich the training dataset by introducing different kinds of invariance for the model to capture. Several recent works (Niu et al., 2020; Zhang et al., 2021) have proposed to adopt GANs as a data augmentation method to generate realistic defective samples. Among them, Defect-GAN (Zhang et al., 2021) tries to capture the stochastic variation within defects by mimicking the defacement and restoration processes. However, it still learns a deterministic mapping between inputs and outputs while DT-GAN achieves multi-modality by varying styles. Moreover, our method can generate realistic defects with sophisticated patterns copied from real-world defective samples. ", + "bbox": [ + 173, + 743, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/83af523dcfeb8e486bbe482fe2fcb6e531192ec8f0ab542aeb5d211eff025025.jpg", + "image_caption": [ + "Figure 2: Overview of all modules in DT-GAN. " + ], + "image_footnote": [], + "bbox": [ + 173, + 98, + 821, + 272 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHODOLOGY ", + "text_level": 1, + "bbox": [ + 174, + 316, + 341, + 333 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our primary aim is to perform unpaired image-to-image translation across multiple foreground domains within a single model. In our use case, the foreground domains refer to the defect types, which means we want to achieve translations between different types of defects while the background remains unaffected. We assume that there is always an adequate amount of normal samples (e.g., non-defective) available, while anomaly samples are rare and hard to acquire. ", + "bbox": [ + 174, + 351, + 825, + 421 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 PROPOSED FRAMEWORK ", + "text_level": 1, + "bbox": [ + 174, + 444, + 387, + 458 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our framework builds on StarGAN v2, a multimodal image-to-image translation model. Given an input image $\\textbf { x } \\in { \\mathcal { X } }$ and an arbitrary domain $y \\in \\mathcal { V }$ , StarGAN v2 generates a domain specific style code in a learned style space and outputs an image that is stylized to fit the domain of $y$ . Its network architecture consists of four modules: a generator, a mapping network, a style encoder and a discriminator. We modify and extend all four modules (see Figure 2) and describe the key differences in details as below. ", + "bbox": [ + 174, + 470, + 825, + 554 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Style-Content Separation. Given a latent code $\\mathbf { z }$ and a domain $y$ , the mapping network $M$ (Figure 2(b)) generates a style code $\\mathbf { s } = M _ { y } ( \\mathbf { z } )$ and a domain specific content $\\mathbf { c } = M _ { y } ( \\mathbf { z } )$ in different branches. It is worth mentioning that $M _ { y }$ here denotes an output of $M$ corresponding to the domain $y$ . This feature allows our method to separately model the structural appearance (i.e. content) and its artistic looks (i.e. style), which is essential because applying different styles to the same content enriches the diversity of outputs. By randomly sampling $\\mathbf { z }$ from a standard normal distribution and $y$ from all available foreground domains, $M$ is able to produce diverse style codes and domain specific contents. ", + "bbox": [ + 173, + 561, + 825, + 674 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The encoder $E$ (Figure 2(c)) extracts the style code $\\mathbf { s } = E _ { y } ( \\mathbf { x } )$ and the domain specific content $\\mathbf { c } = E _ { y } ( \\mathbf { x } )$ from an given image $\\mathbf { x }$ , which reflect the characteristics of reference images instead of randomly sampled noise. ", + "bbox": [ + 178, + 680, + 821, + 723 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Foreground/Background (FG/BG) Disentanglement. The generator $G$ (Figure 2(a)) translates an input image x into an output image $G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } )$ according to given domain specific style code es and content ec, which are provided either by the mapping network $M$ when generating from random noise or by the style-content encoder $E$ when transferring an existing content from a reference image. To achieve a FG/BG disentanglement, we split the channels of the three-dimensional feature map (i.e., $H \\times W \\times C$ ) at the bottle neck of $G$ into two parts. The model is then forced to encode the background into the first channels and the domain specific content cˆ into the latter channels by classification losses as discussed in Section 3.2. cˆ can then be replaced with content ec from the target domain. The adaptive instance normalization (AdaIN) (Huang & Belongie, 2017) is then used to inject es into ec during the decoding process while the background $B G _ { G } ( \\mathbf { x } )$ is decoded separately. StarGAN v2 learns $\\mathbf { F G }$ and BG together which leads to a conditional relationship between both. Our disentanglement and separate encoding break this conditioning and therefore enable our method to freely combine FG and BG as well as learn the full variation of FG content. Finally, $B G _ { G } ( \\mathbf { x } )$ and ec are concatenated together and then fused before output. ", + "bbox": [ + 173, + 729, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Multi-task discriminator with auxiliary classifiers. The discriminator $D$ (Figure 2(d)) is a multitask discriminator with two auxiliary classifiers: a foreground domain classifier and a background domain classifier. This feature strengthens the disentanglement of FG and BG by first ensuring the input image $\\mathbf { x }$ contains a domain specific content that can be recognized by the foreground domain classifier independent of the background. Later, each branch $D _ { y }$ in the multi-task discriminator $D$ is trained to determine if an image $\\mathbf { x }$ is a real image of its foreground domain or a fake image $G ( \\mathbf { x } , \\mathbf { s } , \\mathbf { c } )$ generated by $G$ . Apart from that, one extra branch $B G _ { \\mathrm { c l s } }$ is attached to decide whether the background information of the input images is well preserved. ", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Content Transfer. Mokady et al. (2020) introduced a concept that a model should be able to identity the difference between two domains when one of the domains contains a feature that the other does not have. We refer to this concept as ‘anchor’ and extend to multiple domains $( > 2 )$ by the FG/BG disentanglement, the multi-task discriminator and the foreground content classifier in $D$ . We treat domain Normal as the anchor domain i.e. set the domain specific content to zero, because a normal image has no domain specific content in our definition. As a result, we can now transfer contents between all combination of FG and BG domains (see Figure 10). ", + "bbox": [ + 173, + 222, + 825, + 320 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Compared to StarGAN v2, our method not only models style codes and contents separately but also disentangles the foreground and background of an image in a weakly-supervised manner. These features allow explicit control over output images by combining desired style codes and contents from one of the subnetworks with the input images. Therefore, it leads to higher variance regarding the location, structural pattern and artistic style of defects in the synthetic images of DT-GAN. ", + "bbox": [ + 174, + 325, + 825, + 396 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 TRAINING OBJECTIVES ", + "text_level": 1, + "bbox": [ + 176, + 412, + 379, + 428 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given an image $\\mathbf { x } \\in \\mathcal { X }$ , its original foreground domain $y \\in \\mathcal { V }$ and its background domain $p \\in \\mathcal { P }$ , the following objectives are used to train our framework. ", + "bbox": [ + 173, + 439, + 823, + 468 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Adversarial loss. In the training phase, a noise vector $\\mathbf { z } \\in { \\mathcal { Z } }$ and a target foreground domain $\\widetilde y \\in \\mathcal { V }$ are sampled randomly. Both of them are fed to $M$ , producing a target style code $\\widetilde { \\mathbf { s } }$ and a target content ec as follows: $\\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } = M _ { \\widetilde { y } } ( \\mathbf { z } )$ . Goal of the training is to ensure that $\\widetilde { \\mathbf { s } }$ and ec are sampled from the distribution over styles and contents of the target domain $\\widetilde { y }$ . The generator $G$ then combines an image $\\mathbf { x }$ with $\\widetilde { \\mathbf { s } }$ and $\\widetilde { \\mathbf c }$ and learns to generate an output image $G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } )$ that is indistinguishable from real images in the target domain $\\widetilde { y }$ . We encourage this behavior by using an adversarial loss same as in Choi et al. (2020) ", + "bbox": [ + 173, + 474, + 825, + 571 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/cd793fc06d67b12ed9b5a25f7cc4effb3e16a9336e8b3693d7bd6c236854a0d3.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { a d v } } = \\mathbb { E } _ { { \\mathbf { x } } , y } \\big [ \\log D _ { y } ( { \\mathbf { x } } ) \\big ] + \\mathbb { E } _ { { \\mathbf { x } } , \\widetilde { y } , { \\mathbf { z } } } [ \\log \\left( 1 - D _ { \\widetilde { y } } ( G ( { \\mathbf { x } } , \\widetilde { { \\mathbf { s } } } , \\widetilde { { \\mathbf { c } } } ) ) \\right) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 290, + 579, + 705, + 598 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $D _ { y }$ and $D _ { \\widetilde { y } }$ are the output branches of $D$ that correspond to the source domain $y$ and the target domain $\\widetilde { y }$ , respectively. ", + "bbox": [ + 173, + 606, + 823, + 636 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Style-content reconstruction loss. Similar to StarGAN v2, to enforce the generator $G$ takes the style code $\\widetilde { \\mathbf { s } }$ and the domain specific content ec into consideration during the generation process, we employ a style-content reconstruction loss ", + "bbox": [ + 173, + 640, + 825, + 684 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ae70d605b176b542cef1b090f291d2c692ecbbaaceb1c0f0b119e790a5f72661.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { s t y . c o n } } = \\mathbb { E } _ { \\mathbf { x } , \\widetilde { y } , \\mathbf { z } } \\big [ \\| \\widetilde { \\mathbf { s } } - S _ { E } ( G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } ) ) \\| _ { 1 } \\big ] + \\mathbb { E } _ { \\mathbf { x } , \\widetilde { y } , \\mathbf { z } } \\big [ \\| \\widetilde { \\mathbf { c } } - C _ { E } ( G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } ) ) \\| _ { 1 } \\big ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 241, + 690, + 756, + 710 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This objective urges the style-content encoder $E$ to recover $\\widetilde { \\mathbf { s } }$ and c from $G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } )$ . Here, the stylecontent encoder $E$ learns a mapping from an image to its style and content domains, which allows $G$ to synthesize an image with given s and c from reference images at test time. ", + "bbox": [ + 173, + 717, + 825, + 761 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Diversity loss. In order to further boost the diversity of output images from $G$ , we introduce a loss that encourages diversity as follows: for a pair of random latent codes $\\mathbf { z } _ { 1 }$ and $\\mathbf { z } _ { 2 }$ we compute $\\widetilde { \\mathbf { s } } _ { i } , \\widetilde { \\mathbf { c } } _ { i } = M _ { \\widetilde { y } } ( \\mathbf { z } _ { i } )$ for $i \\in \\{ 1 , 2 \\}$ and enforce a different outcome of the generator $G$ for differently mixed style and content input pairs: ", + "bbox": [ + 174, + 766, + 823, + 823 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/af2defdd3af95b358356d3f45b7f10b9e2ba3e2f9851f7366b219e39a10c234c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { d s } } = \\mathbb { E } _ { \\mathbf { x } , \\widetilde { y } , \\mathbf { z } _ { 1 } , \\mathbf { z } _ { 2 } } \\left[ \\| G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } _ { 1 } , \\widetilde { \\mathbf { c } } _ { 2 } ) - G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } _ { 2 } , \\widetilde { \\mathbf { c } } _ { 1 } ) \\| _ { 1 } \\right] } \\\\ & { \\quad \\quad + \\mathbb { E } _ { \\mathbf { x } , \\widetilde { y } , \\mathbf { z } _ { 1 } , \\mathbf { z } _ { 2 } } \\left[ \\| G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } _ { 1 } , \\widetilde { \\mathbf { c } } _ { 1 } ) - G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } _ { 2 } , \\widetilde { \\mathbf { c } } _ { 2 } ) \\| _ { 1 } \\right] } \\\\ & { \\quad \\quad + \\sum _ { m , n , o } \\left[ \\mathbb { E } _ { \\mathbf { x } , \\widetilde { y } , \\mathbf { z } _ { 1 } , \\mathbf { z } _ { 2 } } \\left[ \\| G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } _ { m } , \\widetilde { \\mathbf { c } } _ { n } ) - G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } _ { o } , \\widetilde { \\mathbf { c } } _ { o } ) \\| _ { 1 } \\right] \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 828, + 699, + 891 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $m , n \\in \\{ 1 , 2 | m \\neq n \\}$ and $o \\in \\{ 1 , 2 \\}$ . Driven by this term, the generator $G$ is forced to discover meaningful style features and contents that eventually lead to diversity in generated images. ", + "bbox": [ + 176, + 895, + 821, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We ignore the denominator ${ \\left\\| { \\bf z } _ { 1 } - { \\bf z } _ { 2 } \\right\\| } _ { 1 }$ of the original diversity loss (Mao et al., 2019a) for stable training as in StarGAN v2. ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Cycle consistency loss. To ensure that the generated image $G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } )$ preserves the domaininvariant properties of its input image $\\mathbf { x }$ , we impose the cycle consistency loss (Zhu et al., 2017a) ", + "bbox": [ + 171, + 138, + 821, + 167 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8a28d37d9ae5394b57315d9c8d4765defe93168ce7c89ec9750064fee390377d.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { c y c } } = \\mathbb { E } _ { { \\mathbf { x } } , y , \\widetilde { y } , { \\mathbf { z } } } \\big [ | | { \\mathbf { x } } - G ( G ( { \\mathbf { x } } , \\widetilde { { \\mathbf { s } } } , \\widetilde { { \\mathbf { c } } } ) , \\widehat { { \\mathbf { s } } } , \\widehat { { \\mathbf { c } } } ) | | _ { 1 } \\big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 346, + 172, + 648, + 191 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\hat { \\bf s } , \\hat { \\bf c } = E _ { y } ( { \\bf x } )$ is the extracted style code and domain specific content of the input image $\\mathbf { x }$ , and $y$ is the original domain of $\\mathbf { x }$ . By learning to reconstruct the input image $\\mathbf { x }$ with given style code ˆs and content cˆ, the generator $G$ is then further encouraged to disentangle the background, the domain specific content and the style code. ", + "bbox": [ + 173, + 196, + 825, + 253 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Content consistency loss. Besides the cycle consistency loss, we apply another constraint to enforce that the detached domain specific content from $G$ is consistent with the one retrieved from $E$ according to ", + "bbox": [ + 173, + 260, + 825, + 303 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/16799b09efe3daa473a44c7377c649313946b26779783a12f869bc3fcbfa421e.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { c o n . c y c } } = \\mathbb { E } _ { \\mathbf { x } , y , \\widetilde { y } , \\mathbf { z } } \\left[ \\left\\| F G _ { G } ( \\mathbf { x } ) - \\widehat { \\mathbf { c } } \\right\\| _ { 1 } \\right] + \\mathbb { E } _ { \\mathbf { x } , y , \\widetilde { y } , \\mathbf { z } } \\left[ \\left\\| F G _ { G } ( G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } ) ) - \\widetilde { \\mathbf { c } } \\right\\| _ { 1 } \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 308, + 753, + 327 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\hat { \\mathbf { c } } = E _ { y } ( \\mathbf { x } ) , \\widetilde { \\mathbf { c } } = E _ { \\widetilde { y } } ( \\mathbf { x } ) , F G _ { G } ( \\mathbf { x } )$ and $F G _ { G } ( G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } ) )$ are the pop-out domain specific content from input image $\\mathbf { x }$ and generated image $G ( \\mathbf { x } , \\widetilde { \\mathbf { s } } , \\widetilde { \\mathbf { c } } )$ , respectively. ", + "bbox": [ + 176, + 330, + 823, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Classification losses. We employ two classification losses: the first one is the foreground content classification loss ", + "bbox": [ + 171, + 367, + 823, + 395 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6712ae0ca7d1db4598667f0ee1d11db11e23257990d2e793428b4f8a10453059.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { F G . c l s } } = \\mathbb { E } _ { \\mathbf { x } _ { \\mathrm { r e a l } } , y } \\Big [ - \\log D _ { \\mathrm { F G . c l s } } \\big ( y | \\mathbf { x } _ { \\mathrm { r e a l } } \\big ) \\Big ] + \\mathbb { E } _ { \\mathbf { x } _ { \\mathrm { f a k e } } , \\widetilde { y } } \\Big [ - \\log D _ { \\mathrm { F G . c l s } } \\big ( \\widetilde { y } | \\mathbf { x } _ { \\mathrm { f a k e } } \\big ) \\Big ] \\ ,\n$$", + "text_format": "latex", + "bbox": [ + 248, + 401, + 746, + 421 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "which aims to ensure that the domain specific content is properly encoded and carries enough information from the target domain. The second one is the background classification loss ", + "bbox": [ + 174, + 425, + 823, + 454 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/115c772df39b6fd258d86dbe2a48df8fb2caeffe1cc1c194cbf32120e55fcba7.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { B G } . \\mathrm { c l s } } = \\mathbb { E } _ { \\mathbf { x } _ { \\mathrm { r e a l } } , p } \\big [ - \\log D _ { \\mathrm { B G } . \\mathrm { c l s } } ( p | \\mathbf { x } _ { \\mathrm { r e a l } } ) \\big ] + \\mathbb { E } _ { \\mathbf { x } _ { \\mathrm { f a k e } } , p } \\big [ - \\log D _ { \\mathrm { B G } . \\mathrm { c l s } } ( p | \\mathbf { x } _ { \\mathrm { f a k e } } ) \\big ] \\ ,\n$$", + "text_format": "latex", + "bbox": [ + 246, + 458, + 748, + 478 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $p$ is the corresponding background type of $\\mathbf { x } _ { \\mathrm { r e a l } }$ and $\\mathbf { x } _ { \\mathrm { f a k e } }$ . With the help of this objective, the generator $G$ learns to preserve the domain-invariant characteristics of its input image $\\mathbf { x }$ while dissociating the foreground domain specific part. ", + "bbox": [ + 174, + 483, + 825, + 526 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Full objective. Our full objective functions can be summarized as ", + "bbox": [ + 174, + 532, + 607, + 547 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/05fe2b4c8716195856bd4bb70ae9fafb48b7dd38578c2cce55917eb22af3c280.jpg", + "text": "$$\n\\begin{array} { r l } { \\underset { G , F , E } { \\operatorname* { m i n } } \\underset { D } { \\operatorname* { m a x } } } & { \\mathcal { L } _ { \\mathrm { a d v } } + \\lambda _ { \\mathrm { s t y } . \\mathrm { c o n } } \\mathcal { L } _ { \\mathrm { s t y } . \\mathrm { c o n } } - \\lambda _ { \\mathrm { d s } } \\mathcal { L } _ { \\mathrm { d s } } + \\lambda _ { \\mathrm { c y c } } \\mathcal { L } _ { \\mathrm { c y c } } + } \\\\ & { \\lambda _ { \\mathrm { c o n . c y c } } \\mathcal { L } _ { \\mathrm { c o n . c y c } } + \\lambda _ { \\mathrm { F G . c l s } } \\mathcal { L } _ { \\mathrm { F G . c l s } } + \\lambda _ { \\mathrm { B G . c l s } } \\mathcal { L } _ { \\mathrm { B G . c l s } } \\ , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 551, + 707, + 595 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\lambda _ { \\mathrm { s t y } }$ , $\\lambda _ { \\mathrm { d s } }$ , $\\lambda _ { \\mathrm { c y c } }$ , $\\lambda _ { \\mathrm { c o n \\mathrm { { - } c y c } } }$ , $\\lambda _ { \\mathrm { F G \\mathrm { - } c l s } }$ and $\\lambda _ { \\mathrm { B G \\mathrm { { - } c l s } } }$ are the hyperparameters for each term. ", + "bbox": [ + 173, + 598, + 743, + 616 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 633, + 326, + 650 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We evaluated the images generated by DT-GAN through a series of experiments both quantitatively and qualitatively. Finally, we demonstrate the benefits of our generated images when being used as data augmentation for a defect classification task on limited data. ", + "bbox": [ + 176, + 665, + 823, + 707 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Dataset. All experiments were performed on a real industrial dataset: a Surface Defect Inspection (SDI) dataset that contains three different kinds of products from production lines and samples from each product are classified into three mutually exclusive classes: Normal, Scratch and Spot. All of the images are grayscale. Detailed statistics of the dataset are summarized in Appendix A. Note that only the training set was used in GAN training, the test set was left untouched for final evaluation in classifier training. For a fair comparison, all images were resized to $1 2 8 \\times 1 2 8$ resolution for both GAN training and classifier training, which was also the highest resolution used in the baselines for image generation. For comparison, we also conducted experiments on the widely used MVTec Anomaly Detection dataset (Bergmann et al., 2019) in Appendix E.4. ", + "bbox": [ + 173, + 713, + 825, + 839 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 DEFECT GENERATION ", + "text_level": 1, + "bbox": [ + 176, + 856, + 367, + 869 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Baselines. As discussed in Section 3, DT-GAN can either use the mapping network to randomly generate styles and defects, or it can use the style-content encoder to extract both from reference images. We refer to these cases as ‘latent-guided’ and ‘reference-guided’, respectively. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since the two ways of guidance are fundamentally different, we evaluated them against two sets of baselines: Our reference-guided image generation was compared to Mokady et al. (2020) and StarGAN v2, because both of them can perform a reference-guided translation. Note that Mokady et al. (2020) can only translate between two domains while StarGAN v2 and DT-GAN can achieve multi-domain translation within a single model. Images generated through the latent-guided part of DT-GAN were compared to state-of-the-art GANs in image synthesis: BigGAN (Brock et al., 2019) and StyleGAN v2 (Karras et al., 2020b). We set BigGAN to condition on defect types during training while StyleGAN v2 was trained unconditionally. All baselines were trained from scratch with the public implementations provided by the authors1. ", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1.1 QUANTITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 246, + 428, + 260 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Metrics. We employed the commonly used frechet inception distance (FID) (Heusel et al., 2017) to evaluate both the visual quality and the diversity of the generated images. We also report the kernel inception distance (KID) (Binkowski et al., 2018) which is a more stable metric for small sets of images like our SDI dataset. Lower FID and KID scores indicate better performance. ", + "bbox": [ + 174, + 270, + 825, + 327 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Both scores are shown in Table 1. We observe that methods like BigGAN and StyleGAN v2, which perform defect synthesis purely based on latent codes, generally provide unsatisfactory results on the SDI dataset, presumably due to the small number of defective samples that were available. These methods then struggle to capture the complex and irregular patterns of defects. We also experimented with augmentation methods for GAN training (Karras et al., 2020a; Zhao et al., 2020) but did not find a consistent improvement (see Appendix E.2). We thus only report the best scores. ", + "bbox": [ + 174, + 333, + 825, + 417 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Reference-guided synthesis methods like Mokady et al. (2020) and StarGAN v2 seem to generate more realistic images. The scores of StarGAN v2 on a single product are omitted here because generating images with specified background is not possible due to its network design—the product type changes in output images, which we refer to as ‘identity-shift’. As seen in Table 1, our method achieves better scores in all cases. We believe this is due to the fact that our method allows free combination of foreground defects and backgrounds, making the generated images more diverse even with a small number of training samples. ", + "bbox": [ + 174, + 424, + 825, + 522 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/7998b6b2f82acd4e8a5d2d8cfa152b29cacad12cfc7d986a268c2db781bbb119.jpg", + "table_caption": [ + "Table 1: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID and KID. Note that the reported values are not comparable between columns, because they were calculated on different training sets. " + ], + "table_footnote": [], + "table_body": "
MethodFID↓KID↓
ABCAllABCAll
Mokady (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v21137.70-110.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
", + "bbox": [ + 176, + 585, + 815, + 699 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1.2 QUALITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 719, + 418, + 734 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We present a qualitative comparison with the baseline methods in latent-guided image synthesis in Figure 3. To make a fair comparison, we trained StyleGAN v2 and BigGAN on each product separately to have control on background products. Note however, that images from DT-GAN were always obtained from a single model. We can see that some generated samples from StyleGAN v2 do not contain clear defects, and samples from BigGAN present abnormal grid patterns. Both methods do not take images as inputs but generate synthetic images according to a given latent code which contains information for both FG and BG. This conditioning leads to limited diversity in the output images. On the other hand, StarGAN v2 performs translation based on input images but suffers from the same entanglement issue. Thus, it fails to preserve the background, which results in artifacts or identity-shift in its outputs. Our network architecture that disentangles foreground and background seems to mitigate these issues. See Appendix E.4 for more images. ", + "bbox": [ + 174, + 743, + 825, + 897 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/e8844ad801507fcb58625110b5ddd1e08b278ccc393d4b6551f6e75e4f93f6ca.jpg", + "image_caption": [ + "Figure 3: Qualitative comparison of latent-guided image synthesis results. In each subfigure: on the left, defective images are fully generated from random noise. On the right, random defects are synthesized onto given normal samples. Note that BigGAN\\* denotes it was trained with DiffAug. " + ], + "image_footnote": [], + "bbox": [ + 176, + 103, + 825, + 267 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/c8d4f138dd7a663cb4fbcdd6d95ca7b5d88bc4505014c289d8d78e678a6ef857.jpg", + "image_caption": [ + "Figure 4: Qualitative comparison of reference-guided image synthesis results on the SDI dataset. Each method transforms the given source images into target foreground domains (e.g., Scratches) with the styles and contents extracted from the reference images. " + ], + "image_footnote": [], + "bbox": [ + 183, + 319, + 816, + 484 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Also for reference-guided image synthesis, where we used different background and foreground reference images as illustrated in Figure 4, only our method produces high quality images with preserved background from the source and transferred foreground defect from the reference. ", + "bbox": [ + 174, + 560, + 825, + 603 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Ablation study. We visually demonstrate the effect of each component we added to DT-GAN compared to StarGAN v2 in Figure 5, using the examples of both latent- and reference-guided image synthesis from Normal to Scratches. The quantitative evaluation can be found in Appendix E.3. ", + "bbox": [ + 174, + 609, + 825, + 651 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Column (a) corresponds to StarGAN v2 and highlights the drawback of entangled FG/BG again (i.e. the identity-shift in the background). We first tackle this problem by modeling the style code and foreground content explicitly and feeding them separately to the generator. This leads to a better preservation of the background structure in column (b) for the reference-guided subnetwork, but not for the latent-guided synthesis on the bottom of Figure 5. Thus, we add a foreground classifier in the discriminator in (c) to ensure the output image contains the desired foreground content (scratch). Similarly, we introduce a background classifier to the discriminator in column (d). Note that the additional product type labels can be acquired automatically from production lines. ", + "bbox": [ + 173, + 659, + 825, + 771 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For column (e), we add the separate decoders for foreground and background in the generator which are fused only in the end. This enhances the preservation of background characteristics like lighting even more. Imposing an additional penalty for foreground content extracted from a normal sample as described in Section 3.1 leads to another visual improvement of the foreground edges for reference-guided synthesis in column (f). Finally, inspired by StyleGAN, we incorporate adaptive noise injection to the mapping network, which significantly boosts the performance of our latentguided image synthesis as shown in column (g). ", + "bbox": [ + 174, + 777, + 825, + 875 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Styling. We visually demonstrate the effect of style codes in our method by randomly sampling those and combining them with fixed reference background and foreground images in Figure 6, where a variety of artistic styles can be seen on the output columns. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/91696fc85e55662030167a35ef4aae436fa0857c2b3a62e62f0e14e5967749b8.jpg", + "image_caption": [ + "Figure 5: Ablation study. (a) The baseline StarGAN v2. (b) $^ +$ Style-Content branches. (c) $^ +$ Foreground classifier. (d) $^ +$ Background classifier. (e) $^ +$ Separately decoding foreground and background in $G$ . (f) $^ +$ Anchor foreground domain (e.g. Normal). (g) $^ +$ Noise injection in Mapping Network. " + ], + "image_footnote": [], + "bbox": [ + 215, + 102, + 781, + 310 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/10b4554eb02cdbab48f0a3b29e5fed2318aee10f044aacf5703500d8bb21eef0.jpg", + "image_caption": [ + "Figure 6: Visual effect of randomly sampled style codes on fixed pairs of reference background (Source) and foreground (Content) images. " + ], + "image_footnote": [], + "bbox": [ + 183, + 375, + 815, + 539 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 DT-GAN FOR DATA AUGMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 602, + 472, + 616 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also evaluated our method as a data augmentation method for defect classification on the SDI dataset. We defined one task ‘general’, where the classifier was trained on images from all products at once, while task ‘single product’ only used the subset of images for one product. ", + "bbox": [ + 174, + 631, + 825, + 672 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Besides, we incrementally varied the amount of real Normal data available for classifier training: 4500, 6600, 12000 and 18600. In the case of defective images, all of them were always used due to the small amount unless otherwise specified. As backbone we used a ResNet-50 (He et al., 2016a) with ImageNet pretrained weights. For experiments with synthetic data, we attached an auxiliary domain classifier to the network through a Gradient Reversal Layer (Ganin & Lempitsky, 2015). ", + "bbox": [ + 174, + 680, + 825, + 751 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/928c69ed8d652dba082f99aeef3e7de87cc055491f1db2d2d419fae8e599b312.jpg", + "table_caption": [ + "Table 2: Quantitative comparison of the baseline methods on defect classification task at the scale of 12000 images/class. The reported values are the achieved error rates $( \\% )$ over five runs. " + ], + "table_footnote": [], + "table_body": "
MethodResNet-50EfficientNet-b4
No-Aug21.64±1.2412.06±0.64
Trad-Aug12.58±0.819.33±0.73
Mokady (2020)11.11±1.1913.26±1.13
StarGAN v213.07±1.3012.25±0.79
StyleGAN v211.55±1.7911.68±0.76
BigGAN+DiffAug11.45±0.6112.06±0.50
Ours9.9±0.699.14±1.02
", + "bbox": [ + 266, + 805, + 725, + 925 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Since the SDI dataset is highly imbalanced, we oversampled the minority classes (Ling et al., 1998) unless the data was balanced through synthetic images. Additionally, we always applied traditional data augmentation techniques like random horizontal flips, jittering and lighting (Shorten & Khoshgoftaar, 2019) except where noted. All following results were evaluated by the achieved error rates over five runs with different random seeds. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Effectiveness of synthetic data. We first compare classifier performance for no augmentation (NoAug), traditional data augmentation (Trad-Aug), and a combination of traditional augmentation with synthetic images for GAN methods including DT-GAN. We also introduce a stronger backbone, EfficientNet-b4 (Tan & Le, 2019), to demonstrate that our results are not confined to a specific network. Table 2 shows that our method is the only one that improves performance for both backbones, presumably due to the combination of high visual image quality and diversity in our samples. ", + "bbox": [ + 173, + 180, + 825, + 265 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/8d583e7701fa68183a5b7e4a5236d7a737bd6b648588956f2a2416af51c4e12b.jpg", + "table_caption": [ + "Table 3: Experimental results on using different amount of synthetic images generated by DT-GAN to train classifiers. The left-most column stands for number of samples per class to be classified. The training set of the baselines is balanced by oversampling while ours is by synthetic images. " + ], + "table_footnote": [], + "table_body": "
Dataset Size20AAll
Trad-AugOursTrad-AugOurs
450015.55±0.6314.28±1.2512.75±0.6111.04±0.76
660016.69±0.7614.41±3.1213.07±1.5710.60±0.48
1200016.95±1.0214.22±1.5312.05±0.819.90±0.69
1860016.12±2.1915.36±0.8612.37±0.3210.21±0.96
", + "bbox": [ + 282, + 327, + 712, + 428 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Impact of dataset size. Motivated by the limited availability of data in real-world production scenarios, we therefore evaluated DT-GAN for data augmentation on a subset of the full SDI dataset (All), which only contains 20 defective samples in product A for each defect type (20A). In this case, DT-GAN was also trained on the reduced subset. As shown in Table 3, there is a clear improvement when synthetic images from DT-GAN are used as data augmentation, even for the extremely limited data subset. Further results on single product classifiers can be found in Appendix E.1. ", + "bbox": [ + 173, + 438, + 825, + 522 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/974a838e3b729a382b7808545ffbfcb351041081afc0d6165f506ec8cd4fa186.jpg", + "table_caption": [ + "Table 4: Cross-domain effect on single product classifiers trained with reference-guided synthetic images at the scale of 12000 images/class. " + ], + "table_footnote": [], + "table_body": "
Trad-AugvAVBvCvABC
A13.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
B6.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
C16.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
", + "bbox": [ + 253, + 568, + 741, + 637 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Cross-domain effect. We hypothesized that limited data can be counteracted by transferring defects across multiple background products, if there are at least some defects that occur on multiple products (See Appendix E.1 for further discussion). We tested this approach by comparing the performance of classifiers trained on synthetic images with defects from a specific source (vA, vB, vC) to classifiers trained on images with defects from all products (vABC). As we can see in Table 4, the best performances are reached by the models that take over defects from other products. We interpret this as support for our hypothesis and its practical usefulness. ", + "bbox": [ + 173, + 647, + 825, + 746 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 766, + 318, + 782 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose a novel method, DT-GAN, which allows diverse defect synthesis both by generating from randomly sampled noise and by following the guidance of given reference images. Due to explicit style-content separation and FG/BG disentanglement, DT-GAN achieves higher image fidelity, better variance in defects and full control over background and foreground while being sample-efficient. We demonstrated the feasibility and benefits of DT-GAN on a real industrial defect classification task and the results show our method provides consistent gains even with limited data and boosts the performance of classifiers compared to state-of-the-art image synthesis methods. For future investigation, we aim to represent defects more explicitly (e.g., localization) to improve the explainability of the model and also enhance the model transferability to unseen products. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REPRODUCIBILITY STATEMENT ", + "text_level": 1, + "bbox": [ + 174, + 103, + 395, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We aim for full reproducibility by publishing the source code and dataset with the final version of the paper. Besides, we provide descriptions of the training details in Appendix B, the evaluation setup in Appendix C and the network architecture in Appendix D. ", + "bbox": [ + 174, + 127, + 823, + 170 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES \nPaul Bergmann, Michael Fauser, David Sattlegger, and Carsten Steger. Mvtec ad — a comprehensive real-world dataset for unsupervised anomaly detection. 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", + "bbox": [ + 173, + 398, + 823, + 428 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A THE SURFACE DEFECT INSPECTION DATASET ", + "text_level": 1, + "bbox": [ + 174, + 102, + 589, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The Surface Defect Inspection (SDI) dataset consists of 20,414 images at $1 2 8 \\times 1 2 8$ resolution. It contains three background domains—product A, product $\\mathbf { B }$ and product C, each can be further classified into three foreground domains—Normal, Scratches and Spots. Figure 7 shows example images of the SDI dataset. To be noticed that the dataset is highly imbalanced not only between normal and defective samples but also between different products as shown in Table 5. This sets a more challenging task when training deep neural networks like GANs and downstream classifiers. ", + "bbox": [ + 174, + 133, + 825, + 231 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For each foreground and background domains, we randomly select 50 images for a joint validation/test set, which is then further split into separate sets in the ratio of 3:7, and use all remaining images as training sets for GAN and classifier training. We present the distribution of the training set when training DT-GAN in Table 6. Note that the normal samples used in GAN training are only a subset of all available samples in Normal and we keep the rest of them for generating defective samples at test time. For classifier training, we show the statistics in Table 7, where the number of normal samples involved in classifier training increase incrementally. The validation set is used to select the best model during classifier training while the test set is left untouched until the final evaluation. Both of the validation and test set are inaccessible by DT-GAN. ", + "bbox": [ + 174, + 237, + 825, + 363 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/c5ad4cce383f42b999dbec5baa09abd0eec003f8449b17cca61eb4e7218407fe.jpg", + "table_caption": [ + "Table 5: Distribution of the full SDI dataset. " + ], + "table_footnote": [], + "table_body": "
Overview
ABC
Normal625062506250
Scratches340167121
Spots108670258
", + "bbox": [ + 192, + 410, + 485, + 496 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/d3b1a834deec1f06f6598262b6c7958b3d090165d454d73f8326527e6462444b.jpg", + "table_caption": [ + "Table 6: The training set for DT-GAN and the baseline image synthesis methods. " + ], + "table_footnote": [], + "table_body": "
Overview
ABC
Normal700700700
Scratches29011771
Spots58620208
", + "bbox": [ + 514, + 416, + 808, + 503 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/ecde65a45fe9d34b0e307da325180a667544009c522094149b40f88f28ea42b7.jpg", + "table_caption": [ + "Table 7: The training, validation and test set for classifier training, where $N$ increases incrementally—1500, 2200, 4000 and 6200. " + ], + "table_footnote": [], + "table_body": "
TrainValidationTest
ABCABCABC
NormalNNN121815383235
Scratches29011771141615363435
Spots58620208141615363435
", + "bbox": [ + 173, + 564, + 833, + 652 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B TRAINING DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 680, + 372, + 696 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "DT-GAN. We follow the training scheme as described in StarGAN v2 with minor modifications. To fit the model on a single Nvidia GTX TITAN X, the batch size is reduced to four while the model is still trained for 100,000 iterations. The training time is about three and a half days on the dedicated GPU with the modified network architecture2 and loss functions mentioned in Section 3 in PyTorch (Paszke et al., 2017). We set $\\lambda _ { \\mathrm { s t y } } = 1$ , $\\lambda _ { \\mathrm { d s } } = 1$ , $\\lambda _ { \\mathrm { c y c } } = 1$ , $\\lambda _ { \\mathrm { c o n . c y c } } = 1$ , $\\lambda _ { \\mathrm { c l s } } = 1$ and $\\lambda _ { \\mathrm { B G . c l s } } = 1$ for the SDI dataset. All other design choices remain the same as in StarGAN v2. ", + "bbox": [ + 173, + 712, + 825, + 795 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Classifiers. We train all the classifiers that use ResNet-50 as backbone for 100 epochs with the SGD optimizer (Ruder, 2016) and batch size 256. The initial learning rate is 0.001, momentum is 0.9 and weight decay is 1e-4. A learning rate scheduler is set to reduce the learning rate by factor of 0.1 when the validation loss stops decreasing for 5 epochs. The same setting also applies to EfficientNet-b4, except the batch size is reduced to 128. Although DT-GAN can synthesize realistic defective samples, we notice that there still exists a domain gap between the generated samples and the real samples. To explore the full potential of the generated samples, we attach an auxiliary source classifier to distinguish between synthetic and real samples. Then, this classifier is connected to the backbone (e.g. ResNet-50) through a Gradient Reversal Layer. With the help of the Gradient Reversal Layer, the backbone is forced to extract the shared features between synthetic and real samples, which ensures all training samples are effectively learned. ", + "bbox": [ + 173, + 801, + 825, + 900 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/2b5df2b8d264369f1ee04ca1e3d7919ee1d3d69e7370aa9deec145bb6941e15e.jpg", + "image_caption": [ + "Figure 7: Overview of the SDI dataset. " + ], + "image_footnote": [], + "bbox": [ + 173, + 83, + 826, + 664 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 776, + 825, + 833 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We design a two-layer perceptron that connects to the average pooling layer in ResNet-50 as shown in Figure 8. Note that the usual fully connected layer after the average pooling in ResNet-50 remains the same and is not affected by the extra branch we added. Inspired by Chen et al. (2018), a threelayer perceptron is used for EfficientNet-b4 instead as shown in Figure 9. Its layers are initialized with a random normal distribution, where the standard deviation is set to 0.01 for the first two layers and 0.05 for the output layer. The biases for all layers are set to 0. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/5876fcb93af3f0aad7fc6536e733bff52799290b671bdeef4fd81bf21ab172fb.jpg", + "image_caption": [ + "Figure 8: ResNet-50 with GRL. " + ], + "image_footnote": [], + "bbox": [ + 212, + 102, + 460, + 167 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/79d423c36937100d387ec0e803940f1adbc96236a1a0b915320ec3741d5f7095.jpg", + "image_caption": [ + "Figure 9: EfficientNet-b4 with GRL. " + ], + "image_footnote": [], + "bbox": [ + 500, + 102, + 826, + 167 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C EVALUATION SETUP ", + "text_level": 1, + "bbox": [ + 176, + 220, + 377, + 236 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Generated samples from DT-GAN. DT-GAN requires images as input for generating synthetic data. At test time, we translated each Normal image in the SDI dataset into four defective images: two with Scratches and two with Spots. The translations were performed by two subnetworks: by the mapping network $M$ using random noise (‘latent-guided’) and by the style-content encoder $E$ using a reference image (‘reference-guided’). We first randomly sampled one latent code for each defective foreground domain. Similarly, we also randomly sampled one reference image from the training set for each defective foreground domain. The corresponding style codes and defect contents were then produced by the two subnetworks respectively and fed to the generator for target image generation. ", + "bbox": [ + 173, + 251, + 825, + 376 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We conducted classification experiments separately on images generated from the two subnetworks and a mixture set of both (i.e. $50 \\%$ from each subnetwork). Experiments show consistent gains of using synthetic images generated from DT-GAN (Table 8). We observe that the latent-guided synthetic images in general perform better than the reference-guided one, while the mixture set provides more stable results with regard to the standard deviation. Presumably the mixture set benefits from the combination of samples from reference-guided synthesis, which are well aligned with the original defect distribution, and the samples from latent-guided synthesis, i.e. from random noise, which adds novel but plausible defects to the dataset. In the main text, we report the results of the mixture set for all experiments, including the quantitative evaluation of DT-GAN. ", + "bbox": [ + 174, + 382, + 825, + 508 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/c1d4e75efc50b8836e9f56d1642ae073df8bae94b456b2fba3105e400e5d6572.jpg", + "table_caption": [ + "Table 8: Classification results with regard to the synthetic images generated from the two subnetworks and the mixture set. " + ], + "table_footnote": [], + "table_body": "
Dataset SizeAll
Trad-AugLatentReferenceMix
450012.75±0.6110.72±0.9611.48±0.8811.04±0.76
660013.07±1.5710.34±1.8611.55±1.6410.60±0.48
1200012.05±0.819.90±1.2610.40±0.999.90±0.69
1860012.37±0.3211.04±1.2612.12±0.7510.21±0.96
", + "bbox": [ + 285, + 558, + 709, + 659 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Frechet inception distance (FID) and Kernel inception distance (KID). ´ We used the feature vectors from the last average pooling layer of the ImageNet pretrained Inception-V3 to calculate both scores. For each test image from the Normal domain, we translated it into a synthetic defective image of each defect domain. The style codes and contents for the translation were acquired in two ways: by randomly sampling from the standard normal distribution and by randomly sampling a reference image from the train set of a defect domain. To calculate the FID and KID score, we generated 4000 defective samples per product per defect domain for each way of guidance, and formed the mixture set by randomly sampling 2000 images per product per defect domain from each way. The reported FID and KID scores were then computed between the defective images in the training set and the mixture set of synthetic defective images. The same procedure was applied when computing scores on single product subsets of the SDI dataset. For example, for product A, we calculated the scores between the defective image of product A in the training set and the mixture set of synthetic defective images of product A. ", + "bbox": [ + 173, + 678, + 825, + 858 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D NETWORK ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 878, + 436, + 895 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In this section, we provide the architectural details of all four modules in DT-GAN. ", + "bbox": [ + 173, + 909, + 717, + 924 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/1c48ba4f6a9c5f47866cc9bb38143dabaa253479bdaae65b20f932c892e1f632.jpg", + "table_caption": [ + "Table 9: Generator architecture. " + ], + "table_footnote": [], + "table_body": "
(a)Encoder
LayerResampleNormOutput Shape
Image x-128 × 128×3
Conv 1×1-128 ×128 ×128
ResBlkAvgPoolIN64× 64×256
ResBlkAvgPoolIN32 × 32 × 512
ResBlkAvgPoolIN16 × 16 × 512
ResBlkIN16 ×16× 512
ResBlkIN16 ×16 × 512
(b) Background Decoder(c) Foreground Decoder
LayerResampleNorm Output ShapeLayerResample NormOutput Shape
Input-16 × 16 × 448Input ResBlk16 × 16 × 64
ResBlkIN 16 ×16× 448 16 ×16× 512ResBlkAdaIN16 ×16× 64
ResBlkIN=AdaIN16 × 16 × 256
ResBlk=IN16 ×16 × 512ResBlkAdaIN16 ×16× 256
ResBlkUpsampleIN32 × 32×512ResBlk Upsample ResBlkAdaIN32 × 32 × 256
ResBlk ResBlkUpsample UpsampleIN IN64 × 64× 256 128 × 128× 448UpsampleAdaIN64×64×128
ResBlkUpsampleAdaIN128 ×128 × 64
(d) Fusion
LayerResampleNormOutput Shape
Input128 × 128 × (448 + 64)
Conv 1×1128 × 128× 3
", + "bbox": [ + 176, + 132, + 818, + 497 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/1098199c42bf1a6e4062025eaaa671f3e1525ccdf2528cdf76e0b9528b15881a.jpg", + "table_caption": [ + "Table 10: Mapping network architecture. " + ], + "table_footnote": [], + "table_body": "
LayerActivationOutput Shape
Latent z=16
LinearReLU512
LinearReLU512
LinearReLU512
LinearReLU512
(b) Style Code(c) Content
LayerActivation Output ShapeLayerResample ActivationNoise Output Shape
Input=512Input=512
LinearReLU512Reshape--1×1×512
LinearReLU512ResBlkUpsampleINTrue2×2×512
LinearReLU512ResBlkUpsampleINTrue4×4×512
Linear164ResBlkUpsampleINTrue8×8×256
ResBlkUpsampleINTrue16 ×16×128
Conv 1×1=INTrue16 × 16× 64
", + "bbox": [ + 173, + 554, + 825, + 804 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Generator (Table 9). For the SDI dataset, the encoder part of the generator consists of three downsampling blocks and two intermediate blocks (Table 9 (a)), all of them are pre-activation residual units (He et al., 2016b). Then the encoded feature map is split channel-wise into background (Table 9 (b)) and foreground (Table 9 (c)). Both of them are then carried through separate decoders. We use the instance normalization (IN) and the adaptive instance normalization (AdaIN) as indicated. The style code is injected into all AdaIN layers to modulate the affine transformations. Note that ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/0490c17ed37be089529218dd050adf14532c06146d2b840c7e6b071380cd65e8.jpg", + "table_caption": [ + "Table 11: Style-content encoder and discriminator architectures. " + ], + "table_footnote": [], + "table_body": "
(a)SharedLayers
LayerResampleNormOutput Shape
Input x128 × 128 ×3
Conv 1×11128 × 128 × 64
ResBlkAvgPool64 × 64× 256
ResBlkAvgPool32 × 32×512
ResBlkAvgPool16 ×16 × 512
(b) Style Code /Discriminator and BG Classifier(c) Content /FG Classifier
LayerResample NormOutput ShapeLayerResample NormOutput Shape
Input-16 ×16× 512Input16 ×16× 512
ResBlkAvgPool8×8×512LReLU16 ×16 × 512
ResBlkAvgPool4×4×512Conv 1×1*K16×16×64*K
LReLU4×4×512
Conv 4×41×1×512
LReLU1×1×512
Reshape512
Linear *KD*K
", + "bbox": [ + 171, + 131, + 834, + 404 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "AdaIN is only used in the foreground decoder. The outputs of both decoders are only fused in the end (Table 9 (d)). ", + "bbox": [ + 174, + 428, + 823, + 457 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Mapping Network (Table 10). The mapping network consists of four shared linear layers (Table 10 (a)) and two separate branches: one for generating style codes (Table 10(b)) and one for contents (Table 10(c)). Each of them is further divided into $K$ output branches, where $K$ denotes the number of domains. The dimension of the input, the output style code and the output content is set to 16, 64, and $1 6 \\times 1 6 \\times 6 4$ , respectively. The latent code is sampled from the standard normal distribution. Note that we apply per-pixel noise after each convolution in the content branch, which we have observed to increase the diversity of generated defects significantly (cf. Figure 5 (g)). ", + "bbox": [ + 173, + 463, + 825, + 560 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Style-Content Encoder (Table 11). The style-content encoder consists of a CNN (Table 11 (a)) with two branches (Table 11 (b) and (c)) as in the mapping network. Each branch has $K$ outputs, where $K$ is the number of domains. Three pre-activation residual blocks are shared among two branches, followed by a specific structure for each branch. The output dimension $D$ in Table 11 is set to 64, which denotes the dimension of the style code. ", + "bbox": [ + 174, + 568, + 825, + 637 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Discriminator (Table 11). The discriminator is a multi-task discriminator with two auxiliary classifiers for the foreground content and the background. The structure is almost identical to the stylecontent encoder, except $D$ is set to 1 for real/fake classification. The background classifier acts in parallel to final linear layer in Table 11 (b) and provides the logits for background classification. The foreground classifier instead acts on top of the output in Table 11 (c) and four more pre-activation residual layers are applied to encode the content into logits for foreground content classification. ", + "bbox": [ + 173, + 643, + 825, + 728 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 747, + 398, + 763 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E.1 ADDITIONAL RESULTS ON THE SDI DATASET ", + "text_level": 1, + "bbox": [ + 176, + 779, + 527, + 792 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We provide additional reference-guided image synthesis results on the SDI dataset in Figure 10. We demonstrate all the possible transfers among all foreground domains. Both style codes and contents are extracted from the reference images. To be noted that DT-GAN can append and remove foreground defects not only onto Normal samples but also to defective samples. For example, in the fifth column of Figure 10, the original scratch in the source image is removed and only the defects from the reference images are presented in the output images. ", + "bbox": [ + 174, + 804, + 825, + 888 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Besides, we present additional evaluations showing the effectiveness of our synthetic data according to Table 3. As seen in Table 12, the synthetic images from DT-GAN also boost the performance in single product classifiers, where the classifiers were trained on the subset of images for one product (A, B, C) instead of the full dataset (ABC). ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/ba0e2234e87b25b63cc95ac4e5644f8da2a00a1e319667a7cf9fa0a32eed56c7.jpg", + "image_caption": [ + "Figure 10: Reference-guided image synthesis results on the SDI dataset. The first row and the first column are the real images sampled from the dataset, while the rest are synthetic images generated by the proposed DT-GAN. Our model provides translations between different foreground domains (Normal, Scratches and Spots) with styles and contents extracted from reference images while the backgrounds from source images are well preserved. " + ], + "image_footnote": [], + "bbox": [ + 178, + 98, + 823, + 609 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 734, + 823, + 763 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "As discussed in Section 4.2, we assumed that the data-insufficiency problem can be mitigated by transferring defects across multiple background products. To examine if this assumption holds, we compared the performance of classifiers trained on synthetic images with defects from a specific source (vA, vB, vC) to classifiers trained on images with defects from all products (vABC). The results on the cross-domain effect with regard to different sizes of the training set are shown in Table 13. We again notice that using our synthetic data is beneficial. Moreover, in most cases the performance is further improved by exploiting cross-domain information (i.e. by transferring defects from other products). We interpret this as support for our assumption and the practical usefulness of our method in the real-world scenario. The case of cross-domain image synthesis when the desired combination is not presented in the training set is covered in the study on the MVTec Anomaly Detection dataset (Bergmann et al., 2019) (see Appendix E.4). ", + "bbox": [ + 173, + 771, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/20974be996d9fce37f644ecdf9151a7e5df8554b4465f8cbe6a1c0ef5f25b6a6.jpg", + "table_caption": [ + "Table 12: Quantitative results for DT-GAN as a data augmentation method to train general and single product classifiers. The left-most column indicates the number of samples per class, including all images from the training set plus increasing amounts of synthetic images. In the first row, 20A refers to the case of 20 real defective samples for product A, while All refers to the full training set. " + ], + "table_footnote": [], + "table_body": "
Dataset Size20A
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450035.09±2.62 27.64±3.127.8±1.485.6±1.6715.24±1.90 13.14±1.7015.55±0.63 14.28±1.25
660039.64±2.28 27.64±1.658.8±1.646.2±1.6415.81±1.73112.38±1.6516.69±0.76 14.41±3.12
1200034.18±4.39 28.55±7.325.8±0.455.6±1.1416.19±1.17 10.86±1.28 16.95±1.02 14.22±1.53
1860039.45±7.06 32.55±5.047.2±0.845.2±1.1014.86±0.85 13.14±2.06 16.12±2.19 15.36±0.86
Dataset SizeAll
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450016.00±1.04110.18±1.75 8.79±0.455.60±1.5117.13±6.62 14.09±2.2712.75±0.61 11.04±0.76
660014.90±1.3810.54±1.22 7.60±1.516.80±3.1115.23±2.33 11.42±013.07±1.57 10.60±0.48
1200013.81±2.366.72±1.65 6.80±1.644.60±016.57±3.20 13.90±2.5712.05±0.819.90±0.69
1860013.63±2.22 10.54±2.45 6.80±1.794.99±1.87 15.62±0.85 11.61±1.24 12.37±0.32 10.21±0.96
", + "bbox": [ + 178, + 169, + 816, + 407 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/f17f93e8ebcc379ed4eada67e2de92805226db4f931acda640080a5aad9fc349.jpg", + "table_caption": [ + "Table 13: Cross-domain effect on single product classifiers trained with reference-guided synthetic images at all scales. Note that here A, B and C stand for 3 products in the SDI dataset while vA, vB, vC and vABC indicate the defects are copied from which reference set. " + ], + "table_footnote": [], + "table_body": "
Dataset SizeA
Trad-AugvAvBvCvABC
450016.00±1.0412.90±2.6113.08±1.6514.90±2.4615.27±3.49
660014.90±1.3813.99±1.8911.26±1.0414.36±4.0416.00±2.85
1200013.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
1860013.63±2.2212.72±5.2214.36±3.8314.18±5.0513.81±8.56
DatasetB
SizeTrad-AugvAvBvCVABC
45008.79±0.457.80±2.155.60±1.1410.19±0.846.79±1.30
66007.60±1.516.80±1.657.80±1.108.00±2.346.00±1.41
120006.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
186006.80±1.796.19±1.784.40±1.146.60±1.955.99±1.58
DatasetC
SizeTrad-AugvAVBvCvABC
450017.14±4.6214.85±0.5216.76±2.5813.90±1.9812.00±1.59
660015.23±2.3313.14±1.2413.90±2.2914.28±1.3412.57±1.57
1200016.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
1860015.62±0.8513.71±1.7315.99±6.7512.57±3.2612.95±2.98
", + "bbox": [ + 240, + 478, + 753, + 781 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.2 ADDITIONAL FID AND KID RESULTS ON THE SDI DATASET ", + "text_level": 1, + "bbox": [ + 174, + 810, + 630, + 827 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We provide additional results in the case of training GANs with augmentation methods in Table 14. Augmentation methods like ADA (Karras et al., 2020a) or DiffAug (Zhao et al., 2020) are proposed to adapt GAN training to limited data. We applied these augmentation methods to StyleGAN v2 and BigGAN, because these state-of-art image synthesis methods are not optimized for small dataset. However, incorporating the augmentation methods in training GANs on the SDI dataset is not always beneficial. The performance of StyleGAN v2 is largely degraded when using ADA, potentially due to the conflict between augmentation methods and the decentralized location of defects—in the SDI dataset, defects can occur anywhere on the surface. This is in contrast to datasets that were used to evaluate the aforementioned augmentation methods in GANs, where the objects are centralized (e.g., ImageNet (Deng et al., 2009), Cifar (Krizhevsky & Hinton, 2009)) and their attributes (e.g., beard, eye glasses in CelebA (Liu et al., 2015)) only occur in specific images parts. ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/45a92e9be03f105fca9e42ed49982648d41734c531d7e3f15ce1a34f30d32186.jpg", + "table_caption": [ + "Table 14: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID and KID. Note that the reported values are not comparable between columns, because they are calculated on different training sets. The scores of StarGAN v2 on single products are omitted because generating images with specified background is not possible due to its network design. " + ], + "table_footnote": [], + "table_body": "
MethodFID↓KID↓
ABCAllABCAll
Mokady et al. (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v2--137.701-10.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
StyleGAN v2 + ADA149.6642.75135.6976.160.1380.0190.1910.055
BigGAN235.66192.89193.61151.430.2480.1990.2760.115
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
", + "bbox": [ + 173, + 252, + 821, + 390 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.3 ABLATION STUDY WITH REGARD TO FID AND KID SCORES ", + "text_level": 1, + "bbox": [ + 174, + 417, + 629, + 433 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We report the FID and KID scores of the ablation study in Table 15. We notice that both subnetworks show positive correlation to each modification except for structural change as in (a) and (e) . Among the two subnetworks, the reference-guided subnetwork outperforms the latent-guided one in the beginning, which is due to the fact that transferring existing contents is easier than generating them from random noise. This effect is also observed in Figure 5. However, the performance of the latentguided subnetwork improves significantly after applying per-pixel noise injection. The subnetwork can now output non-deterministic foreground contents even for a fixed input vector which results in better visual quality and higher diversity of generated defects. In the main text, the scores of the mixture set are reported. ", + "bbox": [ + 173, + 443, + 825, + 569 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/d91c27f754a46e96b8e6645e98fa2909f79399eb5f16398ddc585747f2f70c05.jpg", + "table_caption": [ + "Table 15: Ablation study with regard to FID and KID scores. " + ], + "table_footnote": [], + "table_body": "
FID↓KID↓
LatentReferenceMixLatentReferenceMix
(a) Baseline StarGAN v237.7337.9937.700.0130.0130.013
(b)+ Style-Content branches43.9032.6133.360.0170.0110.011
(c)+Foreground classifier37.1432.3427.690.0140.0110.008
(d) + Background classifier34.1232.5030.230.0110.0110.010
(e)+ Separately decoding foreground and background in G48.5238.1134.790.0170.0150.011
(f) + Anchor foreground domain (e.g. No rmal)43.6637.4532.150.0190.0150.011
33.050.0090.011
(g)+ Noise injection in Mapping Network34.4229.730.009
", + "bbox": [ + 173, + 607, + 836, + 757 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.4 ADDITIONAL RESULTS ON THE MVTEC ANOMALY DETECTION DATASET ", + "text_level": 1, + "bbox": [ + 174, + 786, + 720, + 801 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The MVTec Anomaly Detection dataset (Bergmann et al., 2019) contains 15 different object and texture categories for anomaly detection. The dataset is formed of non-defective image for training and both non-defective and defective images with various kinds of defects for testing. The pixel-level annotations of all defective images are also provided. It is worth noting that the MVTec Anomaly Detection dataset is relatively small scale in number of images, where the number of training images is ranging from 60 to 391. Moreover, the number of defective images for each defect category in the test set is varying only from 8 to 30, which is relatively limited considering the sophisticated pattern of defects. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We conducted image synthesis experiments on a subset of MVTec Anomaly Detection dataset, where we selected four texture categories: Carpet, Leather, Wood and Tile for our targeted scenario i.e. surface defects. Furthermore, we aggregated some of the original defect types defined in the MVTec Anomaly Detection dataset into scratches and spots according to their visual appearance. We then simply added the subset of the MVTec Anomaly Detection dataset to the training set together with the SDI dataset for training DT-GAN. Details of the resulting dataset are shown in Table 17. Note that the small scale of available data posts a major challenge for training generative models. ", + "bbox": [ + 174, + 103, + 825, + 214 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Quantitative Evaluation. We present additional quantitative results on the subset of the MVTec Anomaly Detection dataset in Table 16, following the same evaluation setup as described in Appendix C. As shown in Table 16, our method achieves the best scores in Carpet and Wood, which supports our claim that DT-GAN generates synthetic images with higher fidelity and more diverse defect. However, we also observe that StyleGAN v2 seems to outperform our method in Leather and Tile. ", + "bbox": [ + 174, + 222, + 825, + 305 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Please note that FID and KID are not optimized to evaluate such a small dataset, there the results should only be interpreted together with the qualitative results. ", + "bbox": [ + 176, + 313, + 821, + 340 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Note the we again omit the FID and KID of StarGAN v2 because it is not cable of generating images for a specified product due to the ‘identity-shift’, which is also explained in detail in the qualitative evaluation. ", + "bbox": [ + 174, + 347, + 825, + 390 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/48579259fe7630f714f2c1e1b5470350c7a4af04e5a533325a61185f9ce9ab51.jpg", + "table_caption": [ + "Table 16: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID and KID. Note that the reported values are not comparable between columns, because they were calculated on different training sets. " + ], + "table_footnote": [], + "table_body": "
MethodFID↓KID↓
CarpetLeatherTileWoodCarpetLeatherTileWood
Mokady (2020)41.8760.26275.1281.710.040.030.290.04
StarGAN v211=1
StyleGAN v251.3751.60225.96140.010.050.030.230.12
BigGAN + DiffAug34.47101.70391.54113.320.030.070.420.07
Ours22.7986.13321.3575.830.010.070.360.03
", + "bbox": [ + 194, + 453, + 799, + 566 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Qualitative Evaluation. For qualitative results, we again discuss the ‘latent-guided’ and ‘referenceguided’ synthesis separately. ", + "bbox": [ + 176, + 589, + 821, + 618 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We present the ‘latent-guided’ image synthesis results of StyleGAN v2 in Figure 11 and Figure 12 and BigGAN in Figure 13 and Figure 14. The results are acquired by training one model for each product and then generating 16 images from randomly sampled latent codes from each of them. As pointed out in Section 4.1.2, both methods can not adapt well on small dataset. They suffer from model collapsing and show signs of overfitting by generating images similar to the training data. For example, StyleGAN v2 generates images either with no clear defect or identical to the training set (e.g., Leather in Figure 11 and Product B in Figure 12). The overfitting we observe here also explains the better FID and KID scores in Table 16. For Tile, we can see clear signs of mode collapse in the generated Tile images of StyleGAN v2. Similarly, BigGAN produces images with single mode and abnormal patterns (e.g., grid structure and gray edges). Unlike StyleGAN v2 and BigGAN, StarGAN v2 and our method both require images as input (i.e. Source). Therefore, we randomly sampled two Normal images and applied eight defects which are generated from randomly sampled latent codes to each of them. As seen in Figure 15 and Figure 16, StarGAN v2 fails to preserve the background from the given input images due to the highly entangled FG and BG. Also it fails to generates legit and diverse defects without separately modeling the style and the content. In contrast to aforementioned methods, our DT-GAN produces images with higher fidelity and more diversity in defect patterns as shown in Figure 17 and Figure 18. We believe this again prove the importance of style-content separation and FG/BG disentanglement, which we introduce in Section 3.1. ", + "bbox": [ + 174, + 625, + 825, + 888 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For ‘reference-guided’ image synthesis, the results of Mokady et al. (2020) are shown in Figure 19 and Figure 20 while the results of StarGAN v2 are in Figure 21 and Figure 22. We can observe a clear shift in color in all the outputs from Mokady et al. (2020). Moreover, Mokady et al. (2020) can only transfer content between two domains. In order to perform translation from a non-defective sample to a defective one, we trained a model for each type of defect and for each product. This sums up to be 13 models (Scratches and Spots for 6 categories and Scratches only for Tile). The results from the intended use within one background domain can be found on the diagonal and are marked in red in both Figure 19 and Figure 20. We still show the images that we feed in images from other background domains. As expected, the model then fails to preserve the background of given source images and introduce artifacts to the outputs. Similarly, StarGAN v2 does not preserve the background from the input images. Without style-content separation and FG/BG disentanglement, we observe that StarGAN v2 encodes the background characteristics together with the foreground content of the reference images, which results in identity-shits in its output images. Moreover, the output images either show no clear defect or contain abnormal patterns which sabotage the fidelity. On the contrary, our method can faithfully transfer the foreground content of reference images across given background of different products as shown in Figure 23 and Figure 24, which demonstrate the effectiveness of the style-content separation and FG/BG disentanglement we introduced in Section 3.1. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 104, + 825, + 325 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "It is also worth noting that our method can perform cross-domain image synthesis even the desired combination is not presented in the training set. We demonstrate this on product Tile, which only has images with Scratches but no Spots. As shown in Figure 18 and Figure 24, DT-GAN can generated spots one given Tile images. However, this kind of transformation is most useful when the desired combination is reasonable for the downstream applications. ", + "bbox": [ + 174, + 333, + 823, + 402 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Limitation and Future Work. We have demonstrated the feasibility of the proposed DT-GAN by incorporating more products from the MVTec Anomaly Detection dataset in our training procedure. Intensive experiments have shown that the generated images from DT-GAN yielded better results compared to the baseline image synthesis methods. However, we noticed that despite the diverse patterns of the generated defects, DT-GAN tends to apply the styles learned from the SDI dataset also to the samples from the MVTec Anomaly Detection dataset. For example, we can observe some ”halo” effects in Leather and Wood in Figure 18 and some of the generated scratches in Figure 17 and Figure 23 are rather weakly pronounced. We hypothesize this can be counteracted by explicitly localizing the defect and enforcing the model to learn conditional relationships between ‘styles’ and different backgrounds. We aim to address these issues in future work. ", + "bbox": [ + 174, + 410, + 825, + 547 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/e872d4c36a2361439e0ef7091ef852bb00bbd7ba065cebeee40346b44c21a5cb.jpg", + "table_caption": [ + "Table 17: Overview of our formation of the MVTec Anomaly Detection sub-dataset. The first column represents the original defect types in the MVTec Anomaly Detection dataset while the first row stands for the defect types in our targeted scenario. We list the ID of samples we took from the MVTec Anomaly Detection dataset and show the number of samples in row Sum. " + ], + "table_footnote": [], + "table_body": "
(a) Carpet
ScratchesSpots
Color011,012,014,016, 017000,003,004,007,015,018
Thread000-018
Hole-000 - 016
Sum2423
(b) Leather
ScratchesSpots
Color001,003,005,007,009,011,013,015,018000,002,006,008,010,012,014
Cut000 -018 000 - 006,009 - 016-
Fold Glue000 - 002,005-009,011-015,018
Poke003,009,010,016,017000-017
Sum39
48
(c) Tile
ScratchesSpots
Crack000 - 016二
Sum170
(d) Wood
ScratchesSpots
Color003,005
Scratch001-006,008- 010,013 -016,018- 020000 - 016
Hole=000 - 004,006-009
Combined008001,002,009
Sum1912
", + "bbox": [ + 171, + 324, + 828, + 761 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Randomly Sampled Defects (Scratches) ", + "text_level": 1, + "bbox": [ + 400, + 143, + 619, + 155 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/567f75bf12cbd5a5119634c75240da3d88366422eb7484cba51a63409149e476.jpg", + "image_caption": [ + "Figure 11: Latent-guided image synthesis results of StyleGAN v2 on the SDI dataset and the MVTec AD dataset. We train a model for each product and generate 16 Scratches images from randomly sampled latent codes. " + ], + "image_footnote": [], + "bbox": [ + 243, + 156, + 758, + 830 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/3f55519405e1df2af7deb5998aea2cd1baa9a5f4eca9816aac99ce93d6637aef.jpg", + "image_caption": [ + "Figure 12: Latent-guided image synthesis results of StyleGAN v2 on the SDI dataset and the MVTec AD dataset. We train a model for each product and generate 16 Spots images from randomly sampled latent codes. " + ], + "image_footnote": [], + "bbox": [ + 243, + 200, + 758, + 782 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/9af2881df4cd89bcd7f83e40ecac133c4486b383aca6a5f80167e218d4861e0f.jpg", + "image_caption": [ + "Figure 13: Latent-guided image synthesis results of BigGAN with DiffAug on the SDI dataset and the MVTec AD dataset. We train a model for each product and generate 16 Scratches images from randomly sampled latent codes. " + ], + "image_footnote": [], + "bbox": [ + 243, + 155, + 758, + 833 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/6502d31fa4830920177c7a1201405a5d38fd61293a22fa5cc40cc805b4888a23.jpg", + "image_caption": [ + "Figure 14: Latent-guided image synthesis results of BigGAN with DiffAug on the SDI dataset and the MVTec AD dataset. We train a model for each product and generate 16 Spots images from randomly sampled latent codes. " + ], + "image_footnote": [], + "bbox": [ + 243, + 200, + 758, + 782 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/753dedb2a88181dd1a02122d6d4012b9e01f9fd382ccd9580c8645207ed06d8d.jpg", + "image_caption": [ + "Figure 15: Latent-guided image synthesis results of StarGAN v2 on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Scratches. Note that without the style-content separation and the FG/BG disentanglement, StarGAN v2 not only fails to preserve the background from the given Source image but also fail to generates legit defects. " + ], + "image_footnote": [], + "bbox": [ + 210, + 142, + 790, + 814 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "images/49e2bc6b1e4d4de05a60e2ef2fb4992d1dc00bdd8bcc62cfa38819c364a1a72e.jpg", + "image_caption": [ + "Figure 16: Latent-guided image synthesis results of StarGAN v2 on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Spots. Note that without the style-content separation and the FG/BG disentanglement, StarGAN v2 not only fails to preserve the background from the given Source image but also fail to generates legit defects. " + ], + "image_footnote": [], + "bbox": [ + 212, + 140, + 790, + 816 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/d7665bc9189f97191cb363ff43243d45d9c0eed07ddb8d132a1899df30a87129.jpg", + "image_caption": [ + "Figure 17: Latent-guided image synthesis results of DT-GAN on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Scratches. Note the our model takes input Source images as background and only synthesizes the foreground defects from randomly sampled latent code compared to StyleGAN v2 and BigGAN. " + ], + "image_footnote": [], + "bbox": [ + 212, + 142, + 790, + 815 + ], + "page_idx": 29 + }, + { + "type": "image", + "img_path": "images/6dc33723b61a8a29ea99523303f759e2ec80cca613f5db714539c271f27d80a8.jpg", + "image_caption": [ + "Figure 18: Latent-guided image synthesis results of DT-GAN on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Spots. Note the our model takes input Source images as background and only synthesizes the foreground defects from randomly sampled latent code compared to StyleGAN v2 and BigGAN. " + ], + "image_footnote": [], + "bbox": [ + 212, + 140, + 790, + 816 + ], + "page_idx": 30 + }, + { + "type": "image", + "img_path": "images/6fbc28db81f825e8f35749ff8c9692af1a0d34358c15a1ea8e335765a76b1ab6.jpg", + "image_caption": [ + "Figure 19: Reference-guided image synthesis results of Mokady et al. (2020) on the SDI dataset and the MVTec AD dataset. We train a model for each product and each defect type. Then we translate Normal images to Scratches by taking the Source as background and applying the foreground defect from Reference to it. " + ], + "image_footnote": [], + "bbox": [ + 215, + 263, + 789, + 700 + ], + "page_idx": 31 + }, + { + "type": "image", + "img_path": "images/52103c7943c67c62f890f5864b5e34df9520df52349661b703bc363d3ec05466.jpg", + "image_caption": [ + "Figure 20: Reference-guided image synthesis results of Mokady et al. (2020) on the SDI dataset and the MVTec AD dataset. We train a model for each product and each defect type. Then we translate Normal images to Spots by taking the Source as background and applying the foreground defect from Reference to it. " + ], + "image_footnote": [], + "bbox": [ + 215, + 263, + 790, + 699 + ], + "page_idx": 32 + }, + { + "type": "image", + "img_path": "images/cb73ee815eca7466279848e34aba2a7625dd3b27655d8dcff88d57a806fdbab6.jpg", + "image_caption": [ + "Figure 21: Reference-guided image synthesis results StarGAN v2 on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Scratches by taking the Source as background and applying the foreground defect from Reference to it. Note that without the style-content separation and the FG/BG disentanglement, StarGAN v2 not only fails to preserve the background from the given Source image but also fail to generates legit defects. " + ], + "image_footnote": [], + "bbox": [ + 215, + 246, + 789, + 685 + ], + "page_idx": 33 + }, + { + "type": "image", + "img_path": "images/c0757dd5081906d67150a0b1e7071d90d4070ef9a33fedd3004d5d77e8301ddd.jpg", + "image_caption": [ + "Figure 22: Reference-guided image synthesis results of StarGAN v2 on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Spots by taking the Source as background and applying the foreground defect from Reference to it. Note that without the style-content separation and the FG/BG disentanglement, StarGAN v2 not only fails to preserve the background from the given Source image but also fail to generates legit defects. " + ], + "image_footnote": [], + "bbox": [ + 215, + 247, + 790, + 685 + ], + "page_idx": 34 + }, + { + "type": "image", + "img_path": "images/8f5416bea6a57fe53694b7b81ca32fab188b10b5307dd5743801f95cfe3dc61b.jpg", + "image_caption": [ + "Figure 23: Reference-guided image synthesis results DT-GAN on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Scratches by taking the Source as background and applying the foreground defect from Reference to it. " + ], + "image_footnote": [], + "bbox": [ + 215, + 260, + 790, + 700 + ], + "page_idx": 35 + }, + { + "type": "image", + "img_path": "images/038273ee81067405ed8edb30707628ce21e36f0f73845fa8004a02f19865a69d.jpg", + "image_caption": [ + "Figure 24: Reference-guided image synthesis results of DT-GAN on the SDI dataset and the MVTec AD dataset. The model is trained on a joint set of aforementioned datasets and performs translation from Normal to Spots by taking the Source as background and applying the foreground defect from Reference to it. " + ], + "image_footnote": [], + "bbox": [ + 215, + 262, + 790, + 700 + ], + "page_idx": 36 + } +] \ No newline at end of file diff --git a/parse/dev/2hMEdc35xZ6/2hMEdc35xZ6_middle.json b/parse/dev/2hMEdc35xZ6/2hMEdc35xZ6_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..b2b97b63bd158719aa2c400cd9b8e71e44c33829 --- /dev/null +++ b/parse/dev/2hMEdc35xZ6/2hMEdc35xZ6_middle.json @@ -0,0 +1,54962 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "DEFECT TRANSFER GAN: DIVERSE DEFECT SYNTHE-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 101, + 331, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 331, + 117 + ], + "score": 1.0, + "content": "SIS FOR DATA AUGMENTATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 187, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 212, + 468, + 376 + ], + "lines": [ + { + "bbox": [ + 140, + 210, + 469, + 226 + ], + "spans": [ + { + "bbox": [ + 140, + 210, + 469, + 226 + ], + "score": 1.0, + "content": "Large amounts of data are a common requirement for many deep learning ap-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 223, + 470, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 223, + 470, + 236 + ], + "score": 1.0, + "content": "proaches. 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Also, we show consistent gains of using our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 366, + 357, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 366, + 357, + 378 + ], + "score": 1.0, + "content": "method to enlarge training sets in classification tasks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 206, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "Automated Visual Inspection (AVI) is vital for quality control in modern production lines. Despite", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "the fact that AVI has been studied for decades, it remains a challenging task with many open research", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "questions await to be answered. One of the main challenges in data-driven AVI is the acquisition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "of suitable training data. This is for two reasons: First, collecting a vast amount of labelled data is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "usually labor-intensive and time-consuming. In many cases, even experts are required to identify", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "where and what to look for. However, the acquired label information is task-specific and cannot", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "be reused or transferred to a new task in most cases. Thus, the tedious labelling process must be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "repeated for each new product, even if its defect is similar to other products in people’s eyes. Second,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "in real-world scenarios such as highly optimized production lines, a more severe problem emerges:", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "data imbalance. Only very few defective parts are produced by design. Moreover, the acquired", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "anomaly images from a single product are lacking diversity and may not capture the full defect", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 495, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 495, + 558 + ], + "score": 1.0, + "content": "distribution. Training a robust deep neural network model in such conditions is very challenging.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 504, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "Since collecting sufficient real-world defective samples is impractical, algorithms to synthesize re-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "quired images became a focus in research. Image synthesis through Generative Adversarial Net-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "works (GANs) (Goodfellow et al., 2014) has shown promising performance in recent years. But it", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "also requires large amounts of balanced data which are not available in most industrial use cases, in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "particular for irregular defect patterns and large variation. Therefore, GANs tend to overfit to the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 383, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 383, + 628 + ], + "score": 1.0, + "content": "training examples when trained with little data (Karras et al., 2020a).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "In this work, we tackle these issues by exploiting cross-domain information: we first define two", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "sets of domains—foreground domains and background domains. The foreground domain describes", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "a set of images that contains a specific foreground content to be grouped into a distinctive category,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "and each content has a different style. The background domain instead is considered as a group of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "images that shares similar structural appearance over the whole image. For example, we can set", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "foreground domains as defect types and background domains as product types while the styles of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "defects indicate their artistic looks such as light or heavy strokes. Building upon StarGAN v2 (Choi", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "et al., 2020), the concept underlying this work is to transfer and generate foreground contents with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 434, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 434, + 733 + ], + "score": 1.0, + "content": "a variety of styles across different background domains, as illustrated in Figure 1.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "DEFECT TRANSFER GAN: DIVERSE DEFECT SYNTHE-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 101, + 331, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 331, + 117 + ], + "score": 1.0, + "content": "SIS FOR DATA AUGMENTATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 112, + 136, + 245, + 159 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 187, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 212, + 468, + 376 + ], + "lines": [ + { + "bbox": [ + 140, + 210, + 469, + 226 + ], + "spans": [ + { + "bbox": [ + 140, + 210, + 469, + 226 + ], + "score": 1.0, + "content": "Large amounts of data are a common requirement for many deep learning ap-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 223, + 470, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 223, + 470, + 236 + ], + "score": 1.0, + "content": "proaches. However, data is not always equally available at large scale for all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 470, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 470, + 246 + ], + "score": 1.0, + "content": "classes. For example, on highly optimized production lines, defective samples are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 244, + 470, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 470, + 257 + ], + "score": 1.0, + "content": "hardly acquired while non-defective samples come almost for free. The defects", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "score": 1.0, + "content": "however often seem to resemble each other, e.g., scratches on different products", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "score": 1.0, + "content": "may only differ in few characteristics. In this work, we propose to make use of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 470, + 290 + ], + "score": 1.0, + "content": "the shared characteristics by transferring a stylized defect-specific content from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 470, + 300 + ], + "score": 1.0, + "content": "one type of background product to another. Moreover, the stochastic variations of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 470, + 312 + ], + "score": 1.0, + "content": "the shared characteristics are captured, which also allows generating novel defects", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 470, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 470, + 323 + ], + "score": 1.0, + "content": "from random noise. These synthetic defective samples enlarge the dataset and in-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 333 + ], + "score": 1.0, + "content": "crease the diversity of defects on the target product. Experiments demonstrate that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 333, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 142, + 333, + 469, + 344 + ], + "score": 1.0, + "content": "our model is able to disentangle the defect-specific content from the background", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 357 + ], + "score": 1.0, + "content": "of an image without pixel-level labels. We present convincing results on images", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "score": 1.0, + "content": "from real industrial production lines. Also, we show consistent gains of using our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 366, + 357, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 366, + 357, + 378 + ], + "score": 1.0, + "content": "method to enlarge training sets in classification tasks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 140, + 210, + 470, + 378 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 206, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "Automated Visual Inspection (AVI) is vital for quality control in modern production lines. Despite", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "the fact that AVI has been studied for decades, it remains a challenging task with many open research", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "questions await to be answered. One of the main challenges in data-driven AVI is the acquisition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "of suitable training data. This is for two reasons: First, collecting a vast amount of labelled data is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "usually labor-intensive and time-consuming. In many cases, even experts are required to identify", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "where and what to look for. However, the acquired label information is task-specific and cannot", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "be reused or transferred to a new task in most cases. Thus, the tedious labelling process must be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "repeated for each new product, even if its defect is similar to other products in people’s eyes. Second,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "in real-world scenarios such as highly optimized production lines, a more severe problem emerges:", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "data imbalance. Only very few defective parts are produced by design. Moreover, the acquired", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "anomaly images from a single product are lacking diversity and may not capture the full defect", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 495, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 495, + 558 + ], + "score": 1.0, + "content": "distribution. Training a robust deep neural network model in such conditions is very challenging.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 424, + 506, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 504, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "Since collecting sufficient real-world defective samples is impractical, algorithms to synthesize re-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "quired images became a focus in research. Image synthesis through Generative Adversarial Net-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "works (GANs) (Goodfellow et al., 2014) has shown promising performance in recent years. But it", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "also requires large amounts of balanced data which are not available in most industrial use cases, in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "particular for irregular defect patterns and large variation. Therefore, GANs tend to overfit to the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 383, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 383, + 628 + ], + "score": 1.0, + "content": "training examples when trained with little data (Karras et al., 2020a).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 560, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "In this work, we tackle these issues by exploiting cross-domain information: we first define two", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "sets of domains—foreground domains and background domains. The foreground domain describes", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "a set of images that contains a specific foreground content to be grouped into a distinctive category,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "and each content has a different style. The background domain instead is considered as a group of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "images that shares similar structural appearance over the whole image. For example, we can set", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "foreground domains as defect types and background domains as product types while the styles of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "defects indicate their artistic looks such as light or heavy strokes. Building upon StarGAN v2 (Choi", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "et al., 2020), the concept underlying this work is to transfer and generate foreground contents with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 434, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 434, + 733 + ], + "score": 1.0, + "content": "a variety of styles across different background domains, as illustrated in Figure 1.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 633, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 158, + 82, + 450, + 188 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 158, + 82, + 450, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 158, + 82, + 450, + 188 + ], + "spans": [ + { + "bbox": [ + 158, + 82, + 450, + 188 + ], + "score": 0.972, + "type": "image", + "image_path": "7375134a93e4d6c487812a69c420b54b7f33ccdc499b7470357d6f8e1d3d2531.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 158, + 82, + 450, + 117.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 158, + 117.33333333333334, + 450, + 152.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 158, + 152.66666666666669, + 450, + 188.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 201, + 505, + 224 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "Figure 1: The underlying concept of DT-GAN is to transfer and generate foreground contents with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 212, + 469, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 469, + 225 + ], + "score": 1.0, + "content": "a variety of artistic styles (e.g., light / heavy strokes) across different background domains.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "The contributions of this work are three-fold: First, we introduce Defect Transfer GAN (DT-GAN),", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 258, + 504, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 504, + 270 + ], + "score": 1.0, + "content": "a model that learns transferring existing foreground content and generating novel contents onto dif-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 269, + 504, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 504, + 281 + ], + "score": 1.0, + "content": "ferent backgrounds at the same time. In the real-world scenario, it allows defect inspection networks", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 280, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 504, + 291 + ], + "score": 1.0, + "content": "to learn from a variety of synthetic defective images by composing the foreground defects together", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "with various non-defective images from different products. 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These defective images", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 334, + 464, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 464, + 348 + ], + "score": 1.0, + "content": "generated by DT-GAN boost the performance in defect inspection networks significantly.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 364, + 211, + 377 + ], + "lines": [ + { + "bbox": [ + 104, + 362, + 213, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 213, + 379 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "GANs have shown their power in many computer vision tasks such as image synthesis (Luciˇ c et al., ´", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "2019), style translation (Johnson et al., 2016), super-resolution (Ledig et al., 2017), image impaint-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "ing (Pathak et al., 2016) and many other applications. To quantify the performance of GANs, visual", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 424, + 504, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 504, + 437 + ], + "score": 1.0, + "content": "quality and the diversity of generated images are considered as two of the most important criteria.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "Recent models address these requirements either by dedicated loss functions (Mao et al., 2019b;", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "Yang et al., 2019) or architectural design (Brock et al., 2019). StyleGAN v2 (Karras et al., 2020b),", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "the latest state-of-the-art model in image synthesis, introduces stochastic variation in image gener-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "ating process by adding per-pixel noise after each convolution. However, it is non-trivial to adapt", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 479, + 415, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 415, + 491 + ], + "score": 1.0, + "content": "the model to transform given input images due to the design of the generator.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "In contrast, image-to-image translation methods (Isola et al., 2017) provide a way to recover the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "connection between inputs and the generated images while encouraging diversity. For example, Zhu", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "et al. (2017b) and Huang et al. (2018) impose consistent mappings in latent space to achieve the goal.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Some approaches (Ma et al., 2019; Park et al., 2019) use reference images as guidance to generate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "diverse outputs. Mokady et al. (2020) further extends the translation task from styles to contents. It", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 507, + 563 + ], + "score": 1.0, + "content": "learns to identify a specific content in a given input (e.g., a specific pair of glasses) and transfer it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "to the target image. 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Traditional methods", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "(Ngan et al., 2011) build models upon hand-crafted feature extractors, which are unstable and out-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "performed by deep learning based models. 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These defective images", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 334, + 464, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 464, + 348 + ], + "score": 1.0, + "content": "generated by DT-GAN boost the performance in defect inspection networks significantly.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 246, + 505, + 348 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 364, + 211, + 377 + ], + "lines": [ + { + "bbox": [ + 104, + 362, + 213, + 379 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 213, + 379 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "GANs have shown their power in many computer vision tasks such as image synthesis (Luciˇ c et al., ´", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "2019), style translation (Johnson et al., 2016), super-resolution (Ledig et al., 2017), image impaint-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "ing (Pathak et al., 2016) and many other applications. To quantify the performance of GANs, visual", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 424, + 504, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 504, + 437 + ], + "score": 1.0, + "content": "quality and the diversity of generated images are considered as two of the most important criteria.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "Recent models address these requirements either by dedicated loss functions (Mao et al., 2019b;", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "Yang et al., 2019) or architectural design (Brock et al., 2019). StyleGAN v2 (Karras et al., 2020b),", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "the latest state-of-the-art model in image synthesis, introduces stochastic variation in image gener-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "ating process by adding per-pixel noise after each convolution. 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For example, Zhu", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "et al. (2017b) and Huang et al. (2018) impose consistent mappings in latent space to achieve the goal.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Some approaches (Ma et al., 2019; Park et al., 2019) use reference images as guidance to generate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "diverse outputs. Mokady et al. (2020) further extends the translation task from styles to contents. It", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 507, + 563 + ], + "score": 1.0, + "content": "learns to identify a specific content in a given input (e.g., a specific pair of glasses) and transfer it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "to the target image. 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It", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "aims at identifying and classifying defects with the help of machine vision. Traditional methods", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "(Ngan et al., 2011) build models upon hand-crafted feature extractors, which are unstable and out-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "performed by deep learning based models. 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We modify and extend all four modules (see Figure 2) and describe the key", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 428, + 231, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 231, + 441 + ], + "score": 1.0, + "content": "differences in details as below.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 303, + 459 + ], + "score": 1.0, + "content": "Style-Content Separation. Given a latent code", + "type": "text" + }, + { + "bbox": [ + 303, + 448, + 310, + 456 + ], + "score": 0.59, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 444, + 369, + 459 + ], + "score": 1.0, + "content": "and a domain", + "type": "text" + }, + { + "bbox": [ + 369, + 448, + 376, + 457 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 444, + 469, + 459 + ], + "score": 1.0, + "content": ", the mapping network", + "type": "text" + }, + { + "bbox": [ + 469, + 446, + 481, + 456 + ], + "score": 0.72, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "(Fig-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 235, + 470 + ], + "score": 1.0, + "content": "ure 2(b)) generates a style code", + "type": "text" + }, + { + "bbox": [ + 236, + 456, + 283, + 468 + ], + "score": 0.94, + "content": "\\mathbf { s } = M _ { y } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 456, + 408, + 470 + ], + "score": 1.0, + "content": "and a domain specific content", + "type": "text" + }, + { + "bbox": [ + 408, + 457, + 456, + 469 + ], + "score": 0.93, + "content": "\\mathbf { c } = M _ { y } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "in different", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 255, + 480 + ], + "score": 1.0, + "content": "branches. It is worth mentioning that", + "type": "text" + }, + { + "bbox": [ + 256, + 468, + 271, + 480 + ], + "score": 0.9, + "content": "M _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 467, + 375, + 480 + ], + "score": 1.0, + "content": "here denotes an output of", + "type": "text" + }, + { + "bbox": [ + 376, + 468, + 388, + 478 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "corresponding to the domain", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 113, + 490 + ], + "score": 0.68, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 478, + 506, + 492 + ], + "score": 1.0, + "content": ". 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To", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "achieve a FG/BG disentanglement, we split the channels of the three-dimensional feature map", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 128, + 645 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 129, + 633, + 186, + 644 + ], + "score": 0.91, + "content": "H \\times W \\times C", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 632, + 278, + 645 + ], + "score": 1.0, + "content": ") at the bottle neck of", + "type": "text" + }, + { + "bbox": [ + 278, + 633, + 287, + 643 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "into two parts. The model is then forced to encode", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "the background into the first channels and the domain specific content cˆ into the latter channels by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "classification losses as discussed in Section 3.2. cˆ can then be replaced with content ec from the target", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "domain. 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Its", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "network architecture consists of four modules: a generator, a mapping network, a style encoder", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "and a discriminator. We modify and extend all four modules (see Figure 2) and describe the key", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 428, + 231, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 231, + 441 + ], + "score": 1.0, + "content": "differences in details as below.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 372, + 506, + 441 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 303, + 459 + ], + "score": 1.0, + "content": "Style-Content Separation. Given a latent code", + "type": "text" + }, + { + "bbox": [ + 303, + 448, + 310, + 456 + ], + "score": 0.59, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 444, + 369, + 459 + ], + "score": 1.0, + "content": "and a domain", + "type": "text" + }, + { + "bbox": [ + 369, + 448, + 376, + 457 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 444, + 469, + 459 + ], + "score": 1.0, + "content": ", the mapping network", + "type": "text" + }, + { + "bbox": [ + 469, + 446, + 481, + 456 + ], + "score": 0.72, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "(Fig-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 235, + 470 + ], + "score": 1.0, + "content": "ure 2(b)) generates a style code", + "type": "text" + }, + { + "bbox": [ + 236, + 456, + 283, + 468 + ], + "score": 0.94, + "content": "\\mathbf { s } = M _ { y } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 456, + 408, + 470 + ], + "score": 1.0, + "content": "and a domain specific content", + "type": "text" + }, + { + "bbox": [ + 408, + 457, + 456, + 469 + ], + "score": 0.93, + "content": "\\mathbf { c } = M _ { y } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "in different", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 255, + 480 + ], + "score": 1.0, + "content": "branches. 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We employ two classification losses: the first one is the foreground content", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 301, + 180, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 180, + 314 + ], + "score": 1.0, + "content": "classification loss", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 318, + 457, + 334 + ], + "lines": [ + { + "bbox": [ + 152, + 318, + 457, + 334 + ], + "spans": [ + { + "bbox": [ + 152, + 318, + 457, + 334 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\mathrm { F G . c l s } } = \\mathbb { E } _ { \\mathbf { x } _ { \\mathrm { r e a l } } , y } \\Big [ - \\log D _ { \\mathrm { F G . c l s } } \\big ( y | \\mathbf { x } _ { \\mathrm { r e a l } } \\big ) \\Big ] + \\mathbb { E } _ { \\mathbf { x } _ { \\mathrm { f a k e } } , \\widetilde { y } } \\Big [ - \\log D _ { \\mathrm { F G . c l s } } \\big ( \\widetilde { y } | \\mathbf { x } _ { \\mathrm { f a k e } } \\big ) \\Big ] \\ ,", + "type": "interline_equation", + "image_path": "6712ae0ca7d1db4598667f0ee1d11db11e23257990d2e793428b4f8a10453059.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 152, + 318, + 457, + 334 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 504, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 504, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 504, + 350 + ], + "score": 1.0, + "content": "which aims to ensure that the domain specific content is properly encoded and carries enough infor-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 349, + 444, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 444, + 360 + ], + "score": 1.0, + "content": "mation from the target domain. 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Finally, we demonstrate the benefits of our generated images when being used as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 549, + 368, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 368, + 561 + ], + "score": 1.0, + "content": "data augmentation for a defect classification task on limited data.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 577 + ], + "score": 1.0, + "content": "Dataset. 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We employ two classification losses: the first one is the foreground content", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 301, + 180, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 180, + 314 + ], + "score": 1.0, + "content": "classification loss", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 290, + 505, + 314 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 318, + 457, + 334 + ], + "lines": [ + { + "bbox": [ + 152, + 318, + 457, + 334 + ], + "spans": [ + { + "bbox": [ + 152, + 318, + 457, + 334 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\mathrm { F G . c l s } } = \\mathbb { E } _ { \\mathbf { x } _ { \\mathrm { r e a l } } , y } \\Big [ - \\log D _ { \\mathrm { F G . c l s } } \\big ( y | \\mathbf { x } _ { \\mathrm { r e a l } } \\big ) \\Big ] + \\mathbb { E } _ { \\mathbf { x } _ { \\mathrm { f a k e } } , \\widetilde { y } } \\Big [ - \\log D _ { \\mathrm { F G . c l s } } \\big ( \\widetilde { y } | \\mathbf { x } _ { \\mathrm { f a k e } } \\big ) \\Big ] \\ ,", + "type": "interline_equation", + "image_path": "6712ae0ca7d1db4598667f0ee1d11db11e23257990d2e793428b4f8a10453059.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 152, + 318, + 457, + 334 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 504, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 504, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 504, + 350 + ], + "score": 1.0, + "content": "which aims to ensure that the domain specific content is properly encoded and carries enough infor-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 349, + 444, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 444, + 360 + ], + "score": 1.0, + "content": "mation from the target domain. 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Our full objective functions can be summarized as", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 421, + 373, + 435 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 437, + 433, + 472 + ], + "lines": [ + { + "bbox": [ + 178, + 437, + 433, + 472 + ], + "spans": [ + { + "bbox": [ + 178, + 437, + 433, + 472 + ], + "score": 0.87, + "content": "\\begin{array} { r l } { \\underset { G , F , E } { \\operatorname* { m i n } } \\underset { D } { \\operatorname* { m a x } } } & { \\mathcal { L } _ { \\mathrm { a d v } } + \\lambda _ { \\mathrm { s t y } . \\mathrm { c o n } } \\mathcal { L } _ { \\mathrm { s t y } . \\mathrm { c o n } } - \\lambda _ { \\mathrm { d s } } \\mathcal { L } _ { \\mathrm { d s } } + \\lambda _ { \\mathrm { c y c } } \\mathcal { L } _ { \\mathrm { c y c } } + } \\\\ & { \\lambda _ { \\mathrm { c o n . c y c } } \\mathcal { L } _ { \\mathrm { c o n . c y c } } + \\lambda _ { \\mathrm { F G . c l s } } \\mathcal { L } _ { \\mathrm { F G . c l s } } + \\lambda _ { \\mathrm { B G . c l s } } \\mathcal { L } _ { \\mathrm { B G . c l s } } \\ , } \\end{array}", + "type": "interline_equation", + "image_path": "05fe2b4c8716195856bd4bb70ae9fafb48b7dd38578c2cce55917eb22af3c280.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 178, + 437, + 433, + 448.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 178, + 448.6666666666667, + 433, + 460.33333333333337 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 178, + 460.33333333333337, + 433, + 472.00000000000006 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 455, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 458, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 133, + 489 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 475, + 149, + 487 + ], + "score": 0.32, + "content": "\\lambda _ { \\mathrm { s t y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 473, + 153, + 489 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 153, + 475, + 166, + 487 + ], + "score": 0.35, + "content": "\\lambda _ { \\mathrm { d s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 473, + 170, + 489 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 170, + 475, + 187, + 488 + ], + "score": 0.49, + "content": "\\lambda _ { \\mathrm { c y c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 473, + 193, + 489 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 194, + 475, + 223, + 487 + ], + "score": 0.73, + "content": "\\lambda _ { \\mathrm { c o n \\mathrm { { - } c y c } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 473, + 227, + 489 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 227, + 475, + 254, + 487 + ], + "score": 0.85, + "content": "\\lambda _ { \\mathrm { F G \\mathrm { - } c l s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 473, + 272, + 489 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 272, + 475, + 299, + 487 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { B G \\mathrm { { - } c l s } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 473, + 458, + 489 + ], + "score": 1.0, + "content": "are the hyperparameters for each term.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 473, + 458, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 502, + 200, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 201, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 201, + 516 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 108, + 527, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 540 + ], + "score": 1.0, + "content": "We evaluated the images generated by DT-GAN through a series of experiments both quantitatively", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "and qualitatively. Finally, we demonstrate the benefits of our generated images when being used as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 549, + 368, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 368, + 561 + ], + "score": 1.0, + "content": "data augmentation for a defect classification task on limited data.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 525, + 505, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 577 + ], + "score": 1.0, + "content": "Dataset. All experiments were performed on a real industrial dataset: a Surface Defect Inspection", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 577, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 504, + 589 + ], + "score": 1.0, + "content": "(SDI) dataset that contains three different kinds of products from production lines and samples from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "each product are classified into three mutually exclusive classes: Normal, Scratch and Spot.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "All of the images are grayscale. Detailed statistics of the dataset are summarized in Appendix", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "A. Note that only the training set was used in GAN training, the test set was left untouched for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 621, + 504, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 459, + 634 + ], + "score": 1.0, + "content": "final evaluation in classifier training. For a fair comparison, all images were resized to", + "type": "text" + }, + { + "bbox": [ + 460, + 621, + 504, + 631 + ], + "score": 0.89, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "resolution for both GAN training and classifier training, which was also the highest resolution used", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "in the baselines for image generation. For comparison, we also conducted experiments on the widely", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 653, + 439, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 439, + 666 + ], + "score": 1.0, + "content": "used MVTec Anomaly Detection dataset (Bergmann et al., 2019) in Appendix E.4.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 566, + 506, + 666 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 225, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 227, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 227, + 691 + ], + "score": 1.0, + "content": "4.1 DEFECT GENERATION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Baselines. As discussed in Section 3, DT-GAN can either use the mapping network to randomly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "generate styles and defects, or it can use the style-content encoder to extract both from reference", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 454, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 454, + 734 + ], + "score": 1.0, + "content": "images. We refer to these cases as ‘latent-guided’ and ‘reference-guided’, respectively.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Since the two ways of guidance are fundamentally different, we evaluated them against two sets", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "of baselines: Our reference-guided image generation was compared to Mokady et al. (2020) and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "score": 1.0, + "content": "StarGAN v2, because both of them can perform a reference-guided translation. Note that Mokady", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "et al. (2020) can only translate between two domains while StarGAN v2 and DT-GAN can achieve", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "multi-domain translation within a single model. Images generated through the latent-guided part", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "of DT-GAN were compared to state-of-the-art GANs in image synthesis: BigGAN (Brock et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "2019) and StyleGAN v2 (Karras et al., 2020b). We set BigGAN to condition on defect types during", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "training while StyleGAN v2 was trained unconditionally. All baselines were trained from scratch", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 340, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 340, + 182 + ], + "score": 1.0, + "content": "with the public implementations provided by the authors1.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 195, + 262, + 206 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 264, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 264, + 208 + ], + "score": 1.0, + "content": "4.1.1 QUANTITATIVE EVALUATION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "Metrics. We employed the commonly used frechet inception distance (FID) (Heusel et al., 2017) to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "evaluate both the visual quality and the diversity of the generated images. We also report the kernel", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "inception distance (KID) (Binkowski et al., 2018) which is a more stable metric for small sets of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 448, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 448, + 261 + ], + "score": 1.0, + "content": "images like our SDI dataset. Lower FID and KID scores indicate better performance.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 265, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 505, + 276 + ], + "score": 1.0, + "content": "Both scores are shown in Table 1. We observe that methods like BigGAN and StyleGAN v2, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "perform defect synthesis purely based on latent codes, generally provide unsatisfactory results on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "the SDI dataset, presumably due to the small number of defective samples that were available.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "These methods then struggle to capture the complex and irregular patterns of defects. We also", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "experimented with augmentation methods for GAN training (Karras et al., 2020a; Zhao et al., 2020)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 318, + 502, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 502, + 332 + ], + "score": 1.0, + "content": "but did not find a consistent improvement (see Appendix E.2). We thus only report the best scores.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "Reference-guided synthesis methods like Mokady et al. (2020) and StarGAN v2 seem to generate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "more realistic images. The scores of StarGAN v2 on a single product are omitted here because", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "generating images with specified background is not possible due to its network design—the product", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "type changes in output images, which we refer to as ‘identity-shift’. As seen in Table 1, our method", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "achieves better scores in all cases. We believe this is due to the fact that our method allows free", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "combination of foreground defects and backgrounds, making the generated images more diverse", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 293, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 293, + 415 + ], + "score": 1.0, + "content": "even with a small number of training samples.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "table", + "bbox": [ + 108, + 464, + 499, + 554 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 430, + 502, + 464 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 430, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 504, + 443 + ], + "score": 1.0, + "content": "Table 1: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 441, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 504, + 453 + ], + "score": 1.0, + "content": "and KID. Note that the reported values are not comparable between columns, because they were", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 452, + 252, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 252, + 465 + ], + "score": 1.0, + "content": "calculated on different training sets.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "table_body", + "bbox": [ + 108, + 464, + 499, + 554 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 464, + 499, + 554 + ], + "spans": [ + { + "bbox": [ + 108, + 464, + 499, + 554 + ], + "score": 0.979, + "html": "
MethodFID↓KID↓
ABCAllABCAll
Mokady (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v21137.70-110.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
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To make a fair comparison, we trained StyleGAN v2 and BigGAN on each product", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 611, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 625 + ], + "score": 1.0, + "content": "separately to have control on background products. Note however, that images from DT-GAN were", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "always obtained from a single model. We can see that some generated samples from StyleGAN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "v2 do not contain clear defects, and samples from BigGAN present abnormal grid patterns. Both", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "methods do not take images as inputs but generate synthetic images according to a given latent code", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "which contains information for both FG and BG. This conditioning leads to limited diversity in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 679 + ], + "score": 1.0, + "content": "the output images. On the other hand, StarGAN v2 performs translation based on input images but", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "suffers from the same entanglement issue. Thus, it fails to preserve the background, which results in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "artifacts or identity-shift in its outputs. Our network architecture that disentangles foreground and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 425, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 425, + 713 + ], + "score": 1.0, + "content": "background seems to mitigate these issues. See Appendix E.4 for more images.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 381, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 381, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 381, + 733 + ], + "score": 1.0, + "content": "1We could not obtain the code of Defect-GAN to reproduce their results.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Since the two ways of guidance are fundamentally different, we evaluated them against two sets", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "of baselines: Our reference-guided image generation was compared to Mokady et al. (2020) and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "score": 1.0, + "content": "StarGAN v2, because both of them can perform a reference-guided translation. Note that Mokady", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "et al. (2020) can only translate between two domains while StarGAN v2 and DT-GAN can achieve", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "multi-domain translation within a single model. Images generated through the latent-guided part", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "of DT-GAN were compared to state-of-the-art GANs in image synthesis: BigGAN (Brock et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "2019) and StyleGAN v2 (Karras et al., 2020b). We set BigGAN to condition on defect types during", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "training while StyleGAN v2 was trained unconditionally. All baselines were trained from scratch", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 340, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 340, + 182 + ], + "score": 1.0, + "content": "with the public implementations provided by the authors1.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 506, + 182 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 195, + 262, + 206 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 264, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 264, + 208 + ], + "score": 1.0, + "content": "4.1.1 QUANTITATIVE EVALUATION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "Metrics. We employed the commonly used frechet inception distance (FID) (Heusel et al., 2017) to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "evaluate both the visual quality and the diversity of the generated images. We also report the kernel", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "inception distance (KID) (Binkowski et al., 2018) which is a more stable metric for small sets of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 448, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 448, + 261 + ], + "score": 1.0, + "content": "images like our SDI dataset. Lower FID and KID scores indicate better performance.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 214, + 506, + 261 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 265, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 505, + 276 + ], + "score": 1.0, + "content": "Both scores are shown in Table 1. We observe that methods like BigGAN and StyleGAN v2, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "perform defect synthesis purely based on latent codes, generally provide unsatisfactory results on", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "the SDI dataset, presumably due to the small number of defective samples that were available.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "These methods then struggle to capture the complex and irregular patterns of defects. We also", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "experimented with augmentation methods for GAN training (Karras et al., 2020a; Zhao et al., 2020)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 318, + 502, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 502, + 332 + ], + "score": 1.0, + "content": "but did not find a consistent improvement (see Appendix E.2). We thus only report the best scores.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 265, + 505, + 332 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "Reference-guided synthesis methods like Mokady et al. (2020) and StarGAN v2 seem to generate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "more realistic images. The scores of StarGAN v2 on a single product are omitted here because", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "generating images with specified background is not possible due to its network design—the product", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "type changes in output images, which we refer to as ‘identity-shift’. As seen in Table 1, our method", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "achieves better scores in all cases. We believe this is due to the fact that our method allows free", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "combination of foreground defects and backgrounds, making the generated images more diverse", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 293, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 293, + 415 + ], + "score": 1.0, + "content": "even with a small number of training samples.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 335, + 506, + 415 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 464, + 499, + 554 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 430, + 502, + 464 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 430, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 504, + 443 + ], + "score": 1.0, + "content": "Table 1: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 441, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 504, + 453 + ], + "score": 1.0, + "content": "and KID. Note that the reported values are not comparable between columns, because they were", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 452, + 252, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 252, + 465 + ], + "score": 1.0, + "content": "calculated on different training sets.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "table_body", + "bbox": [ + 108, + 464, + 499, + 554 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 464, + 499, + 554 + ], + "spans": [ + { + "bbox": [ + 108, + 464, + 499, + 554 + ], + "score": 0.979, + "html": "
MethodFID↓KID↓
ABCAllABCAll
Mokady (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v21137.70-110.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
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To make a fair comparison, we trained StyleGAN v2 and BigGAN on each product", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 611, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 625 + ], + "score": 1.0, + "content": "separately to have control on background products. Note however, that images from DT-GAN were", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "always obtained from a single model. We can see that some generated samples from StyleGAN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "v2 do not contain clear defects, and samples from BigGAN present abnormal grid patterns. Both", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "methods do not take images as inputs but generate synthetic images according to a given latent code", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "which contains information for both FG and BG. This conditioning leads to limited diversity in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 679 + ], + "score": 1.0, + "content": "the output images. On the other hand, StarGAN v2 performs translation based on input images but", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "suffers from the same entanglement issue. Thus, it fails to preserve the background, which results in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "artifacts or identity-shift in its outputs. Our network architecture that disentangles foreground and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 425, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 425, + 713 + ], + "score": 1.0, + "content": "background seems to mitigate these issues. 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In each subfigure: on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "the left, defective images are fully generated from random noise. On the right, random defects are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 236, + 500, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 500, + 250 + ], + "score": 1.0, + "content": "synthesized onto given normal samples. 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Each method transforms the given source images into target foreground domains (e.g.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 407, + 427, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 427, + 421 + ], + "score": 1.0, + "content": "Scratches) with the styles and contents extracted from the reference images.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "Also for reference-guided image synthesis, where we used different background and foreground", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "reference images as illustrated in Figure 4, only our method produces high quality images with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 466, + 477, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 477, + 479 + ], + "score": 1.0, + "content": "preserved background from the source and transferred foreground defect from the reference.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "Ablation study. We visually demonstrate the effect of each component we added to DT-GAN", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "compared to StarGAN v2 in Figure 5, using the examples of both latent- and reference-guided image", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "synthesis from Normal to Scratches. The quantitative evaluation can be found in Appendix E.3.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "Column (a) corresponds to StarGAN v2 and highlights the drawback of entangled FG/BG again", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "(i.e. the identity-shift in the background). We first tackle this problem by modeling the style code", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "and foreground content explicitly and feeding them separately to the generator. This leads to a better", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "preservation of the background structure in column (b) for the reference-guided subnetwork, but not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "for the latent-guided synthesis on the bottom of Figure 5. Thus, we add a foreground classifier in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "the discriminator in (c) to ensure the output image contains the desired foreground content (scratch).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "Similarly, we introduce a background classifier to the discriminator in column (d). Note that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 599, + 441, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 441, + 612 + ], + "score": 1.0, + "content": "additional product type labels can be acquired automatically from production lines.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "For column (e), we add the separate decoders for foreground and background in the generator which", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "are fused only in the end. 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In each subfigure: on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "the left, defective images are fully generated from random noise. On the right, random defects are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 236, + 500, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 500, + 250 + ], + "score": 1.0, + "content": "synthesized onto given normal samples. 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MethodResNet-50EfficientNet-b4
No-Aug21.64±1.2412.06±0.64
Trad-Aug12.58±0.819.33±0.73
Mokady (2020)11.11±1.1913.26±1.13
StarGAN v213.07±1.3012.25±0.79
StyleGAN v211.55±1.7911.68±0.76
BigGAN+DiffAug11.45±0.6112.06±0.50
Ours9.9±0.699.14±1.02
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MethodResNet-50EfficientNet-b4
No-Aug21.64±1.2412.06±0.64
Trad-Aug12.58±0.819.33±0.73
Mokady (2020)11.11±1.1913.26±1.13
StarGAN v213.07±1.3012.25±0.79
StyleGAN v211.55±1.7911.68±0.76
BigGAN+DiffAug11.45±0.6112.06±0.50
Ours9.9±0.699.14±1.02
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All following results were evaluated by the achieved error rates", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 279, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 279, + 138 + ], + "score": 1.0, + "content": "over five runs with different random seeds.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "Effectiveness of synthetic data. 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We also introduce a stronger back-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "bone, EfficientNet-b4 (Tan & Le, 2019), to demonstrate that our results are not confined to a specific", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "network. Table 2 shows that our method is the only one that improves performance for both back-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 504, + 210 + ], + "score": 1.0, + "content": "bones, presumably due to the combination of high visual image quality and diversity in our samples.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "table", + "bbox": [ + 173, + 259, + 436, + 339 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 226, + 504, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "Table 3: Experimental results on using different amount of synthetic images generated by DT-GAN", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "to train classifiers. The left-most column stands for number of samples per class to be classified.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 489, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 489, + 262 + ], + "score": 1.0, + "content": "The training set of the baselines is balanced by oversampling while ours is by synthetic images.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "table_body", + "bbox": [ + 173, + 259, + 436, + 339 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 173, + 259, + 436, + 339 + ], + "spans": [ + { + "bbox": [ + 173, + 259, + 436, + 339 + ], + "score": 0.977, + "html": "
Dataset Size20AAll
Trad-AugOursTrad-AugOurs
450015.55±0.6314.28±1.2512.75±0.6111.04±0.76
660016.69±0.7614.41±3.1213.07±1.5710.60±0.48
1200016.95±1.0214.22±1.5312.05±0.819.90±0.69
1860016.12±2.1915.36±0.8612.37±0.3210.21±0.96
", + "type": "table", + "image_path": "8d583e7701fa68183a5b7e4a5236d7a737bd6b648588956f2a2416af51c4e12b.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 173, + 259, + 436, + 285.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 173, + 285.6666666666667, + 436, + 312.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 173, + 312.33333333333337, + 436, + 339.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "Impact of dataset size. Motivated by the limited availability of data in real-world production sce-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "narios, we therefore evaluated DT-GAN for data augmentation on a subset of the full SDI dataset", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 368, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 104, + 368, + 506, + 383 + ], + "score": 1.0, + "content": "(All), which only contains 20 defective samples in product A for each defect type (20A). In this case,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "DT-GAN was also trained on the reduced subset. As shown in Table 3, there is a clear improvement", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "when synthetic images from DT-GAN are used as data augmentation, even for the extremely limited", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 455, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 455, + 415 + ], + "score": 1.0, + "content": "data subset. Further results on single product classifiers can be found in Appendix E.1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "table", + "bbox": [ + 155, + 450, + 454, + 505 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 428, + 504, + 450 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "Table 4: Cross-domain effect on single product classifiers trained with reference-guided synthetic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 438, + 277, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 277, + 451 + ], + "score": 1.0, + "content": "images at the scale of 12000 images/class.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "table_body", + "bbox": [ + 155, + 450, + 454, + 505 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 155, + 450, + 454, + 505 + ], + "spans": [ + { + "bbox": [ + 155, + 450, + 454, + 505 + ], + "score": 0.977, + "html": "
Trad-AugvAVBvCvABC
A13.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
B6.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
C16.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
", + "type": "table", + "image_path": "974a838e3b729a382b7808545ffbfcb351041081afc0d6165f506ec8cd4fa186.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 155, + 450, + 454, + 468.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 155, + 468.3333333333333, + 454, + 486.66666666666663 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 155, + 486.66666666666663, + 454, + 504.99999999999994 + ], + "spans": [], + "index": 27 + } + ] + } + ], + "index": 24.75 + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 512, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 527 + ], + "score": 1.0, + "content": "Cross-domain effect. We hypothesized that limited data can be counteracted by transferring de-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 525, + 504, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 504, + 536 + ], + "score": 1.0, + "content": "fects across multiple background products, if there are at least some defects that occur on multiple", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "products (See Appendix E.1 for further discussion). We tested this approach by comparing the per-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 547, + 504, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 504, + 559 + ], + "score": 1.0, + "content": "formance of classifiers trained on synthetic images with defects from a specific source (vA, vB, vC)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "to classifiers trained on images with defects from all products (vABC). As we can see in Table 4,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "the best performances are reached by the models that take over defects from other products. We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 390, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 390, + 592 + ], + "score": 1.0, + "content": "interpret this as support for our hypothesis and its practical usefulness.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 108, + 607, + 195, + 620 + ], + "lines": [ + { + "bbox": [ + 104, + 604, + 198, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 198, + 624 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "score": 1.0, + "content": "We propose a novel method, DT-GAN, which allows diverse defect synthesis both by generating", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "from randomly sampled noise and by following the guidance of given reference images. Due to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "explicit style-content separation and FG/BG disentanglement, DT-GAN achieves higher image", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "score": 1.0, + "content": "fidelity, better variance in defects and full control over background and foreground while being", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "sample-efficient. We demonstrated the feasibility and benefits of DT-GAN on a real industrial defect", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "classification task and the results show our method provides consistent gains even with limited data", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "and boosts the performance of classifiers compared to state-of-the-art image synthesis methods. For", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "future investigation, we aim to represent defects more explicitly (e.g., localization) to improve the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 467, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 467, + 734 + ], + "score": 1.0, + "content": "explainability of the model and also enhance the model transferability to unseen products.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Since the SDI dataset is highly imbalanced, we oversampled the minority classes (Ling et al., 1998)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "unless the data was balanced through synthetic images. Additionally, we always applied traditional", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 504, + 117 + ], + "score": 1.0, + "content": "data augmentation techniques like random horizontal flips, jittering and lighting (Shorten & Khosh-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "goftaar, 2019) except where noted. All following results were evaluated by the achieved error rates", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 279, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 279, + 138 + ], + "score": 1.0, + "content": "over five runs with different random seeds.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 81, + 506, + 138 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "Effectiveness of synthetic data. We first compare classifier performance for no augmentation (No-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "Aug), traditional data augmentation (Trad-Aug), and a combination of traditional augmentation", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "score": 1.0, + "content": "with synthetic images for GAN methods including DT-GAN. We also introduce a stronger back-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "bone, EfficientNet-b4 (Tan & Le, 2019), to demonstrate that our results are not confined to a specific", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "network. Table 2 shows that our method is the only one that improves performance for both back-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 504, + 210 + ], + "score": 1.0, + "content": "bones, presumably due to the combination of high visual image quality and diversity in our samples.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 142, + 505, + 210 + ] + }, + { + "type": "table", + "bbox": [ + 173, + 259, + 436, + 339 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 226, + 504, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "Table 3: Experimental results on using different amount of synthetic images generated by DT-GAN", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "to train classifiers. The left-most column stands for number of samples per class to be classified.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 489, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 489, + 262 + ], + "score": 1.0, + "content": "The training set of the baselines is balanced by oversampling while ours is by synthetic images.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "table_body", + "bbox": [ + 173, + 259, + 436, + 339 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 173, + 259, + 436, + 339 + ], + "spans": [ + { + "bbox": [ + 173, + 259, + 436, + 339 + ], + "score": 0.977, + "html": "
Dataset Size20AAll
Trad-AugOursTrad-AugOurs
450015.55±0.6314.28±1.2512.75±0.6111.04±0.76
660016.69±0.7614.41±3.1213.07±1.5710.60±0.48
1200016.95±1.0214.22±1.5312.05±0.819.90±0.69
1860016.12±2.1915.36±0.8612.37±0.3210.21±0.96
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Trad-AugvAVBvCvABC
A13.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
B6.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
C16.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
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We hypothesized that limited data can be counteracted by transferring de-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 525, + 504, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 504, + 536 + ], + "score": 1.0, + "content": "fects across multiple background products, if there are at least some defects that occur on multiple", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "products (See Appendix E.1 for further discussion). We tested this approach by comparing the per-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 547, + 504, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 504, + 559 + ], + "score": 1.0, + "content": "formance of classifiers trained on synthetic images with defects from a specific source (vA, vB, vC)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "to classifiers trained on images with defects from all products (vABC). As we can see in Table 4,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "the best performances are reached by the models that take over defects from other products. We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 390, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 390, + 592 + ], + "score": 1.0, + "content": "interpret this as support for our hypothesis and its practical usefulness.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 512, + 505, + 592 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 607, + 195, + 620 + ], + "lines": [ + { + "bbox": [ + 104, + 604, + 198, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 198, + 624 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "score": 1.0, + "content": "We propose a novel method, DT-GAN, which allows diverse defect synthesis both by generating", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "from randomly sampled noise and by following the guidance of given reference images. 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For", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "future investigation, we aim to represent defects more explicitly (e.g., localization) to improve the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 467, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 467, + 734 + ], + "score": 1.0, + "content": "explainability of the model and also enhance the model transferability to unseen products.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 631, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 242, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 243, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 243, + 94 + ], + "score": 1.0, + "content": "REPRODUCIBILITY STATEMENT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 101, + 504, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 113 + ], + "score": 1.0, + "content": "We aim for full reproducibility by publishing the source code and dataset with the final version of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "the paper. 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To explore the full potential of the generated samples, we attach an auxiliary", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 115, + 721, + 493, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 718, + 494, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 718, + 494, + 734 + ], + "score": 1.0, + "content": "2We based our implementation on source code from StarGAN v2: https://github.com/clovaai/stargan-v2", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 361, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 362, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 362, + 95 + ], + "score": 1.0, + "content": "A THE SURFACE DEFECT INSPECTION DATASET", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 412, + 118 + ], + "score": 1.0, + "content": "The Surface Defect Inspection (SDI) dataset consists of 20,414 images at", + "type": "text" + }, + { + "bbox": [ + 413, + 106, + 458, + 117 + ], + "score": 0.89, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "resolution.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 349, + 129 + ], + "score": 1.0, + "content": "It contains three background domains—product A, product", + "type": "text" + }, + { + "bbox": [ + 349, + 118, + 358, + 127 + ], + "score": 0.3, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "and product C, each can be further", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "classified into three foreground domains—Normal, Scratches and Spots. Figure 7 shows", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 138, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 104, + 138, + 505, + 153 + ], + "score": 1.0, + "content": "example images of the SDI dataset. To be noticed that the dataset is highly imbalanced not only", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 504, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 504, + 161 + ], + "score": 1.0, + "content": "between normal and defective samples but also between different products as shown in Table 5.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "score": 1.0, + "content": "This sets a more challenging task when training deep neural networks like GANs and downstream", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 151, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 151, + 184 + ], + "score": 1.0, + "content": "classifiers.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 106, + 506, + 184 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 188, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "For each foreground and background domains, we randomly select 50 images for a joint valida-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 213 + ], + "score": 1.0, + "content": "tion/test set, which is then further split into separate sets in the ratio of 3:7, and use all remaining", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 225 + ], + "score": 1.0, + "content": "images as training sets for GAN and classifier training. 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A learning rate scheduler is set to reduce the learning rate by factor", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "of 0.1 when the validation loss stops decreasing for 5 epochs. The same setting also applies to", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "EfficientNet-b4, except the batch size is reduced to 128. 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Then, this classifier is connected", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "to the backbone (e.g. ResNet-50) through a Gradient Reversal Layer. With the help of the Gradient", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "Reversal Layer, the backbone is forced to extract the shared features between synthetic and real", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 649, + 377, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 377, + 661 + ], + "score": 1.0, + "content": "samples, which ensures all training samples are effectively learned.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We design a two-layer perceptron that connects to the average pooling layer in ResNet-50 as shown", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "in Figure 8. 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Its layers are initialized", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "with a random normal distribution, where the standard deviation is set to 0.01 for the first two layers", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 721, + 371, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 371, + 732 + ], + "score": 1.0, + "content": "and 0.05 for the output layer. 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DT-GAN requires images as input for generating synthetic", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "data. At test time, we translated each Normal image in the SDI dataset into four defective images:", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 504, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 504, + 234 + ], + "score": 1.0, + "content": "two with Scratches and two with Spots. The translations were performed by two subnetworks:", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 207, + 245 + ], + "score": 1.0, + "content": "by the mapping network", + "type": "text" + }, + { + "bbox": [ + 208, + 233, + 219, + 243 + ], + "score": 0.69, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "using random noise (‘latent-guided’) and by the style-content encoder", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 116, + 253 + ], + "score": 0.78, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "using a reference image (‘reference-guided’). We first randomly sampled one latent code for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "each defective foreground domain. Similarly, we also randomly sampled one reference image from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "the training set for each defective foreground domain. The corresponding style codes and defect", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 275, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 506, + 290 + ], + "score": 1.0, + "content": "contents were then produced by the two subnetworks respectively and fed to the generator for target", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 180, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 180, + 300 + ], + "score": 1.0, + "content": "image generation.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "We conducted classification experiments separately on images generated from the two subnetworks", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 235, + 327 + ], + "score": 1.0, + "content": "and a mixture set of both (i.e.", + "type": "text" + }, + { + "bbox": [ + 235, + 315, + 255, + 326 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 316, + 505, + 327 + ], + "score": 1.0, + "content": "from each subnetwork). Experiments show consistent gains", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "of using synthetic images generated from DT-GAN (Table 8). We observe that the latent-guided", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "score": 1.0, + "content": "synthetic images in general perform better than the reference-guided one, while the mixture set", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "provides more stable results with regard to the standard deviation. Presumably the mixture set", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "benefits from the combination of samples from reference-guided synthesis, which are well aligned", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "with the original defect distribution, and the samples from latent-guided synthesis, i.e. from random", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "noise, which adds novel but plausible defects to the dataset. In the main text, we report the results", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 392, + 460, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 460, + 404 + ], + "score": 1.0, + "content": "of the mixture set for all experiments, including the quantitative evaluation of DT-GAN.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "table", + "bbox": [ + 175, + 442, + 434, + 522 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 104, + 421, + 502, + 442 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Table 8: Classification results with regard to the synthetic images generated from the two subnet-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 432, + 214, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 214, + 444 + ], + "score": 1.0, + "content": "works and the mixture set.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "table_body", + "bbox": [ + 175, + 442, + 434, + 522 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 175, + 442, + 434, + 522 + ], + "spans": [ + { + "bbox": [ + 175, + 442, + 434, + 522 + ], + "score": 0.981, + "html": "
Dataset SizeAll
Trad-AugLatentReferenceMix
450012.75±0.6110.72±0.9611.48±0.8811.04±0.76
660013.07±1.5710.34±1.8611.55±1.6410.60±0.48
1200012.05±0.819.90±1.2610.40±0.999.90±0.69
1860012.37±0.3211.04±1.2612.12±0.7510.21±0.96
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For each test image from the Normal domain, we translated it into a synthetic defective", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "image of each defect domain. The style codes and contents for the translation were acquired in two", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "ways: by randomly sampling from the standard normal distribution and by randomly sampling a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "reference image from the train set of a defect domain. 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For example, for product A,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "we calculated the scores between the defective image of product A in the training set and the mixture", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 669, + 295, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 295, + 682 + ], + "score": 1.0, + "content": "set of synthetic defective images of product A.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 108, + 696, + 267, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 695, + 268, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 695, + 268, + 710 + ], + "score": 1.0, + "content": "D NETWORK ARCHITECTURE", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 439, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 719, + 441, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 441, + 733 + ], + "score": 1.0, + "content": "In this section, we provide the architectural details of all four modules in DT-GAN.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 130, + 81, + 282, + 133 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 130, + 81, + 282, + 133 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 130, + 81, + 282, + 133 + ], + "spans": [ + { + "bbox": [ + 130, + 81, + 282, + 133 + ], + "score": 0.957, + "type": "image", + "image_path": "5876fcb93af3f0aad7fc6536e733bff52799290b671bdeef4fd81bf21ab172fb.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 130, + 81, + 282, + 107.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 130, + 107.0, + 282, + 133.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 142, + 146, + 271, + 158 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 141, + 145, + 271, + 159 + ], + "spans": [ + { + "bbox": [ + 141, + 145, + 271, + 159 + ], + "score": 1.0, + "content": "Figure 8: ResNet-50 with GRL.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + } + ], + "index": 3.75 + }, + { + "type": "image", + "bbox": [ + 306, + 81, + 506, + 133 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 306, + 81, + 506, + 133 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 306, + 81, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 306, + 81, + 506, + 133 + ], + "score": 0.95, + "type": "image", + "image_path": "79d423c36937100d387ec0e803940f1adbc96236a1a0b915320ec3741d5f7095.jpg" + } + ] + } + ], + "index": 3.0, + "virtual_lines": [ + { + "bbox": [ + 306, + 81, + 506, + 94.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 306, + 94.0, + 506, + 107.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 306, + 107.0, + 506, + 120.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 306, + 120.0, + 506, + 133.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 333, + 146, + 480, + 158 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 332, + 145, + 480, + 159 + ], + "spans": [ + { + "bbox": [ + 332, + 145, + 480, + 159 + ], + "score": 1.0, + "content": "Figure 9: EfficientNet-b4 with GRL.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + } + ], + "index": 5.0 + }, + { + "type": "title", + "bbox": [ + 108, + 175, + 231, + 187 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 232, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 232, + 189 + ], + "score": 1.0, + "content": "C EVALUATION SETUP", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "Generated samples from DT-GAN. DT-GAN requires images as input for generating synthetic", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "data. At test time, we translated each Normal image in the SDI dataset into four defective images:", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 504, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 504, + 234 + ], + "score": 1.0, + "content": "two with Scratches and two with Spots. The translations were performed by two subnetworks:", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 207, + 245 + ], + "score": 1.0, + "content": "by the mapping network", + "type": "text" + }, + { + "bbox": [ + 208, + 233, + 219, + 243 + ], + "score": 0.69, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "using random noise (‘latent-guided’) and by the style-content encoder", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 116, + 253 + ], + "score": 0.78, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "using a reference image (‘reference-guided’). We first randomly sampled one latent code for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "each defective foreground domain. Similarly, we also randomly sampled one reference image from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "the training set for each defective foreground domain. The corresponding style codes and defect", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 275, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 506, + 290 + ], + "score": 1.0, + "content": "contents were then produced by the two subnetworks respectively and fed to the generator for target", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 180, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 180, + 300 + ], + "score": 1.0, + "content": "image generation.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 199, + 506, + 300 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "We conducted classification experiments separately on images generated from the two subnetworks", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 235, + 327 + ], + "score": 1.0, + "content": "and a mixture set of both (i.e.", + "type": "text" + }, + { + "bbox": [ + 235, + 315, + 255, + 326 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 316, + 505, + 327 + ], + "score": 1.0, + "content": "from each subnetwork). Experiments show consistent gains", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "of using synthetic images generated from DT-GAN (Table 8). We observe that the latent-guided", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "score": 1.0, + "content": "synthetic images in general perform better than the reference-guided one, while the mixture set", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "provides more stable results with regard to the standard deviation. Presumably the mixture set", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "benefits from the combination of samples from reference-guided synthesis, which are well aligned", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "with the original defect distribution, and the samples from latent-guided synthesis, i.e. from random", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "noise, which adds novel but plausible defects to the dataset. In the main text, we report the results", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 392, + 460, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 460, + 404 + ], + "score": 1.0, + "content": "of the mixture set for all experiments, including the quantitative evaluation of DT-GAN.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 303, + 506, + 404 + ] + }, + { + "type": "table", + "bbox": [ + 175, + 442, + 434, + 522 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 104, + 421, + 502, + 442 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Table 8: Classification results with regard to the synthetic images generated from the two subnet-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 432, + 214, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 214, + 444 + ], + "score": 1.0, + "content": "works and the mixture set.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "table_body", + "bbox": [ + 175, + 442, + 434, + 522 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 175, + 442, + 434, + 522 + ], + "spans": [ + { + "bbox": [ + 175, + 442, + 434, + 522 + ], + "score": 0.981, + "html": "
Dataset SizeAll
Trad-AugLatentReferenceMix
450012.75±0.6110.72±0.9611.48±0.8811.04±0.76
660013.07±1.5710.34±1.8611.55±1.6410.60±0.48
1200012.05±0.819.90±1.2610.40±0.999.90±0.69
1860012.37±0.3211.04±1.2612.12±0.7510.21±0.96
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For each test image from the Normal domain, we translated it into a synthetic defective", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "image of each defect domain. The style codes and contents for the translation were acquired in two", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "ways: by randomly sampling from the standard normal distribution and by randomly sampling a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "reference image from the train set of a defect domain. To calculate the FID and KID score, we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "generated 4000 defective samples per product per defect domain for each way of guidance, and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "formed the mixture set by randomly sampling 2000 images per product per defect domain from", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 625, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 506, + 638 + ], + "score": 1.0, + "content": "each way. The reported FID and KID scores were then computed between the defective images in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "the training set and the mixture set of synthetic defective images. The same procedure was applied", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "when computing scores on single product subsets of the SDI dataset. 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(a)Encoder
LayerResampleNormOutput Shape
Image x-128 × 128×3
Conv 1×1-128 ×128 ×128
ResBlkAvgPoolIN64× 64×256
ResBlkAvgPoolIN32 × 32 × 512
ResBlkAvgPoolIN16 × 16 × 512
ResBlkIN16 ×16× 512
ResBlkIN16 ×16 × 512
(b) Background Decoder(c) Foreground Decoder
LayerResampleNorm Output ShapeLayerResample NormOutput Shape
Input-16 × 16 × 448Input ResBlk16 × 16 × 64
ResBlkIN 16 ×16× 448 16 ×16× 512ResBlkAdaIN16 ×16× 64
ResBlkIN=AdaIN16 × 16 × 256
ResBlk=IN16 ×16 × 512ResBlkAdaIN16 ×16× 256
ResBlkUpsampleIN32 × 32×512ResBlk Upsample ResBlkAdaIN32 × 32 × 256
ResBlk ResBlkUpsample UpsampleIN IN64 × 64× 256 128 × 128× 448UpsampleAdaIN64×64×128
ResBlkUpsampleAdaIN128 ×128 × 64
(d) Fusion
LayerResampleNormOutput Shape
Input128 × 128 × (448 + 64)
Conv 1×1128 × 128× 3
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LayerActivationOutput Shape
Latent z=16
LinearReLU512
LinearReLU512
LinearReLU512
LinearReLU512
(b) Style Code(c) Content
LayerActivation Output ShapeLayerResample ActivationNoise Output Shape
Input=512Input=512
LinearReLU512Reshape--1×1×512
LinearReLU512ResBlkUpsampleINTrue2×2×512
LinearReLU512ResBlkUpsampleINTrue4×4×512
Linear164ResBlkUpsampleINTrue8×8×256
ResBlkUpsampleINTrue16 ×16×128
Conv 1×1=INTrue16 × 16× 64
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(a)Encoder
LayerResampleNormOutput Shape
Image x-128 × 128×3
Conv 1×1-128 ×128 ×128
ResBlkAvgPoolIN64× 64×256
ResBlkAvgPoolIN32 × 32 × 512
ResBlkAvgPoolIN16 × 16 × 512
ResBlkIN16 ×16× 512
ResBlkIN16 ×16 × 512
(b) Background Decoder(c) Foreground Decoder
LayerResampleNorm Output ShapeLayerResample NormOutput Shape
Input-16 × 16 × 448Input ResBlk16 × 16 × 64
ResBlkIN 16 ×16× 448 16 ×16× 512ResBlkAdaIN16 ×16× 64
ResBlkIN=AdaIN16 × 16 × 256
ResBlk=IN16 ×16 × 512ResBlkAdaIN16 ×16× 256
ResBlkUpsampleIN32 × 32×512ResBlk Upsample ResBlkAdaIN32 × 32 × 256
ResBlk ResBlkUpsample UpsampleIN IN64 × 64× 256 128 × 128× 448UpsampleAdaIN64×64×128
ResBlkUpsampleAdaIN128 ×128 × 64
(d) Fusion
LayerResampleNormOutput Shape
Input128 × 128 × (448 + 64)
Conv 1×1128 × 128× 3
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LayerActivationOutput Shape
Latent z=16
LinearReLU512
LinearReLU512
LinearReLU512
LinearReLU512
(b) Style Code(c) Content
LayerActivation Output ShapeLayerResample ActivationNoise Output Shape
Input=512Input=512
LinearReLU512Reshape--1×1×512
LinearReLU512ResBlkUpsampleINTrue2×2×512
LinearReLU512ResBlkUpsampleINTrue4×4×512
Linear164ResBlkUpsampleINTrue8×8×256
ResBlkUpsampleINTrue16 ×16×128
Conv 1×1=INTrue16 × 16× 64
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(a)SharedLayers
LayerResampleNormOutput Shape
Input x128 × 128 ×3
Conv 1×11128 × 128 × 64
ResBlkAvgPool64 × 64× 256
ResBlkAvgPool32 × 32×512
ResBlkAvgPool16 ×16 × 512
(b) Style Code /Discriminator and BG Classifier(c) Content /FG Classifier
LayerResample NormOutput ShapeLayerResample NormOutput Shape
Input-16 ×16× 512Input16 ×16× 512
ResBlkAvgPool8×8×512LReLU16 ×16 × 512
ResBlkAvgPool4×4×512Conv 1×1*K16×16×64*K
LReLU4×4×512
Conv 4×41×1×512
LReLU1×1×512
Reshape512
Linear *KD*K
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The mapping network consists of four shared linear layers (Table 10", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "(a)) and two separate branches: one for generating style codes (Table 10(b)) and one for contents", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 389, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 307, + 400 + ], + "score": 1.0, + "content": "(Table 10(c)). 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The dimension of the input, the output style code and the output content is set to 16, 64,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 124, + 423 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 411, + 181, + 422 + ], + "score": 0.92, + "content": "1 6 \\times 1 6 \\times 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 411, + 505, + 423 + ], + "score": 1.0, + "content": ", respectively. 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The output dimension", + "type": "text" + }, + { + "bbox": [ + 397, + 483, + 406, + 492 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "in Table 11 is set to 64,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 493, + 294, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 294, + 505 + ], + "score": 1.0, + "content": "which denotes the dimension of the style code.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 504, + 522 + ], + "score": 1.0, + "content": "Discriminator (Table 11). 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(a)SharedLayers
LayerResampleNormOutput Shape
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Conv 1×11128 × 128 × 64
ResBlkAvgPool64 × 64× 256
ResBlkAvgPool32 × 32×512
ResBlkAvgPool16 ×16 × 512
(b) Style Code /Discriminator and BG Classifier(c) Content /FG Classifier
LayerResample NormOutput ShapeLayerResample NormOutput Shape
Input-16 ×16× 512Input16 ×16× 512
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LReLU4×4×512
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LReLU1×1×512
Reshape512
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To examine if this assumption holds, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "compared the performance of classifiers trained on synthetic images with defects from a specific", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "source (vA, vB, vC) to classifiers trained on images with defects from all products (vABC). The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "results on the cross-domain effect with regard to different sizes of the training set are shown in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "Table 13. We again notice that using our synthetic data is beneficial. 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The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "results on the cross-domain effect with regard to different sizes of the training set are shown in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "Table 13. We again notice that using our synthetic data is beneficial. 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Dataset Size20A
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450035.09±2.62 27.64±3.127.8±1.485.6±1.6715.24±1.90 13.14±1.7015.55±0.63 14.28±1.25
660039.64±2.28 27.64±1.658.8±1.646.2±1.6415.81±1.73112.38±1.6516.69±0.76 14.41±3.12
1200034.18±4.39 28.55±7.325.8±0.455.6±1.1416.19±1.17 10.86±1.28 16.95±1.02 14.22±1.53
1860039.45±7.06 32.55±5.047.2±0.845.2±1.1014.86±0.85 13.14±2.06 16.12±2.19 15.36±0.86
Dataset SizeAll
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450016.00±1.04110.18±1.75 8.79±0.455.60±1.5117.13±6.62 14.09±2.2712.75±0.61 11.04±0.76
660014.90±1.3810.54±1.22 7.60±1.516.80±3.1115.23±2.33 11.42±013.07±1.57 10.60±0.48
1200013.81±2.366.72±1.65 6.80±1.644.60±016.57±3.20 13.90±2.5712.05±0.819.90±0.69
1860013.63±2.22 10.54±2.45 6.80±1.794.99±1.87 15.62±0.85 11.61±1.24 12.37±0.32 10.21±0.96
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Dataset SizeA
Trad-AugvAvBvCvABC
450016.00±1.0412.90±2.6113.08±1.6514.90±2.4615.27±3.49
660014.90±1.3813.99±1.8911.26±1.0414.36±4.0416.00±2.85
1200013.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
1860013.63±2.2212.72±5.2214.36±3.8314.18±5.0513.81±8.56
DatasetB
SizeTrad-AugvAvBvCVABC
45008.79±0.457.80±2.155.60±1.1410.19±0.846.79±1.30
66007.60±1.516.80±1.657.80±1.108.00±2.346.00±1.41
120006.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
186006.80±1.796.19±1.784.40±1.146.60±1.955.99±1.58
DatasetC
SizeTrad-AugvAVBvCvABC
450017.14±4.6214.85±0.5216.76±2.5813.90±1.9812.00±1.59
660015.23±2.3313.14±1.2413.90±2.2914.28±1.3412.57±1.57
1200016.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
1860015.62±0.8513.71±1.7315.99±6.7512.57±3.2612.95±2.98
", + "type": "table", + "image_path": "f17f93e8ebcc379ed4eada67e2de92805226db4f931acda640080a5aad9fc349.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 147, + 379, + 461, + 459.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 147, + 459.0, + 461, + 539.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 147, + 539.0, + 461, + 619.0 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 642, + 386, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 388, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 388, + 656 + ], + "score": 1.0, + "content": "E.2 ADDITIONAL FID AND KID RESULTS ON THE SDI DATASET", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We provide additional results in the case of training GANs with augmentation methods in Table 14.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "Augmentation methods like ADA (Karras et al., 2020a) or DiffAug (Zhao et al., 2020) are proposed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "to adapt GAN training to limited data. We applied these augmentation methods to StyleGAN v2 and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "BigGAN, because these state-of-art image synthesis methods are not optimized for small dataset.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "However, incorporating the augmentation methods in training GANs on the SDI dataset is not always", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "beneficial. The performance of StyleGAN v2 is largely degraded when using ADA, potentially due", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 134, + 500, + 323 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 89, + 505, + 133 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "score": 1.0, + "content": "Table 12: Quantitative results for DT-GAN as a data augmentation method to train general and single", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 505, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 505, + 112 + ], + "score": 1.0, + "content": "product classifiers. The left-most column indicates the number of samples per class, including all", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "images from the training set plus increasing amounts of synthetic images. In the first row, 20A refers", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 479, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 479, + 134 + ], + "score": 1.0, + "content": "to the case of 20 real defective samples for product A, while All refers to the full training set.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 109, + 134, + 500, + 323 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 134, + 500, + 323 + ], + "spans": [ + { + "bbox": [ + 109, + 134, + 500, + 323 + ], + "score": 0.979, + "html": "
Dataset Size20A
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450035.09±2.62 27.64±3.127.8±1.485.6±1.6715.24±1.90 13.14±1.7015.55±0.63 14.28±1.25
660039.64±2.28 27.64±1.658.8±1.646.2±1.6415.81±1.73112.38±1.6516.69±0.76 14.41±3.12
1200034.18±4.39 28.55±7.325.8±0.455.6±1.1416.19±1.17 10.86±1.28 16.95±1.02 14.22±1.53
1860039.45±7.06 32.55±5.047.2±0.845.2±1.1014.86±0.85 13.14±2.06 16.12±2.19 15.36±0.86
Dataset SizeAll
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450016.00±1.04110.18±1.75 8.79±0.455.60±1.5117.13±6.62 14.09±2.2712.75±0.61 11.04±0.76
660014.90±1.3810.54±1.22 7.60±1.516.80±3.1115.23±2.33 11.42±013.07±1.57 10.60±0.48
1200013.81±2.366.72±1.65 6.80±1.644.60±016.57±3.20 13.90±2.5712.05±0.819.90±0.69
1860013.63±2.22 10.54±2.45 6.80±1.794.99±1.87 15.62±0.85 11.61±1.24 12.37±0.32 10.21±0.96
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Dataset SizeA
Trad-AugvAvBvCvABC
450016.00±1.0412.90±2.6113.08±1.6514.90±2.4615.27±3.49
660014.90±1.3813.99±1.8911.26±1.0414.36±4.0416.00±2.85
1200013.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
1860013.63±2.2212.72±5.2214.36±3.8314.18±5.0513.81±8.56
DatasetB
SizeTrad-AugvAvBvCVABC
45008.79±0.457.80±2.155.60±1.1410.19±0.846.79±1.30
66007.60±1.516.80±1.657.80±1.108.00±2.346.00±1.41
120006.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
186006.80±1.796.19±1.784.40±1.146.60±1.955.99±1.58
DatasetC
SizeTrad-AugvAVBvCvABC
450017.14±4.6214.85±0.5216.76±2.5813.90±1.9812.00±1.59
660015.23±2.3313.14±1.2413.90±2.2914.28±1.3412.57±1.57
1200016.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
1860015.62±0.8513.71±1.7315.99±6.7512.57±3.2612.95±2.98
", + "type": "table", + "image_path": "f17f93e8ebcc379ed4eada67e2de92805226db4f931acda640080a5aad9fc349.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 147, + 379, + 461, + 459.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 147, + 459.0, + 461, + 539.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 147, + 539.0, + 461, + 619.0 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 642, + 386, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 388, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 388, + 656 + ], + "score": 1.0, + "content": "E.2 ADDITIONAL FID AND KID RESULTS ON THE SDI DATASET", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We provide additional results in the case of training GANs with augmentation methods in Table 14.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "Augmentation methods like ADA (Karras et al., 2020a) or DiffAug (Zhao et al., 2020) are proposed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "to adapt GAN training to limited data. We applied these augmentation methods to StyleGAN v2 and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "BigGAN, because these state-of-art image synthesis methods are not optimized for small dataset.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "However, incorporating the augmentation methods in training GANs on the SDI dataset is not always", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "beneficial. The performance of StyleGAN v2 is largely degraded when using ADA, potentially due", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "to the conflict between augmentation methods and the decentralized location of defects—in the SDI", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "dataset, defects can occur anywhere on the surface. This is in contrast to datasets that were used", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "to evaluate the aforementioned augmentation methods in GANs, where the objects are centralized", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 507, + 131 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 507, + 131 + ], + "score": 1.0, + "content": "(e.g., ImageNet (Deng et al., 2009), Cifar (Krizhevsky & Hinton, 2009)) and their attributes (e.g.,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 439, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 439, + 140 + ], + "score": 1.0, + "content": "beard, eye glasses in CelebA (Liu et al., 2015)) only occur in specific images parts.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "to the conflict between augmentation methods and the decentralized location of defects—in the SDI", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "dataset, defects can occur anywhere on the surface. This is in contrast to datasets that were used", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "to evaluate the aforementioned augmentation methods in GANs, where the objects are centralized", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 507, + 131 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 507, + 131 + ], + "score": 1.0, + "content": "(e.g., ImageNet (Deng et al., 2009), Cifar (Krizhevsky & Hinton, 2009)) and their attributes (e.g.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 439, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 439, + 140 + ], + "score": 1.0, + "content": "beard, eye glasses in CelebA (Liu et al., 2015)) only occur in specific images parts.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table", + "bbox": [ + 106, + 200, + 503, + 309 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 155, + 505, + 200 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 155, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 506, + 169 + ], + "score": 1.0, + "content": "Table 14: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "and KID. Note that the reported values are not comparable between columns, because they are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "calculated on different training sets. The scores of StarGAN v2 on single products are omitted", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 188, + 488, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 488, + 201 + ], + "score": 1.0, + "content": "because generating images with specified background is not possible due to its network design.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 200, + 503, + 309 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 200, + 503, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 503, + 309 + ], + "score": 0.982, + "html": "
MethodFID↓KID↓
ABCAllABCAll
Mokady et al. (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v2--137.701-10.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
StyleGAN v2 + ADA149.6642.75135.6976.160.1380.0190.1910.055
BigGAN235.66192.89193.61151.430.2480.1990.2760.115
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
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FID↓KID↓
LatentReferenceMixLatentReferenceMix
(a) Baseline StarGAN v237.7337.9937.700.0130.0130.013
(b)+ Style-Content branches43.9032.6133.360.0170.0110.011
(c)+Foreground classifier37.1432.3427.690.0140.0110.008
(d) + Background classifier34.1232.5030.230.0110.0110.010
(e)+ Separately decoding foreground and background in G48.5238.1134.790.0170.0150.011
(f) + Anchor foreground domain (e.g. No rmal)43.6637.4532.150.0190.0150.011
33.050.0090.011
(g)+ Noise injection in Mapping Network34.4229.730.009
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Note that the reported values are not comparable between columns, because they are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "calculated on different training sets. The scores of StarGAN v2 on single products are omitted", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 188, + 488, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 488, + 201 + ], + "score": 1.0, + "content": "because generating images with specified background is not possible due to its network design.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 200, + 503, + 309 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 200, + 503, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 503, + 309 + ], + "score": 0.982, + "html": "
MethodFID↓KID↓
ABCAllABCAll
Mokady et al. (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v2--137.701-10.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
StyleGAN v2 + ADA149.6642.75135.6976.160.1380.0190.1910.055
BigGAN235.66192.89193.61151.430.2480.1990.2760.115
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
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FID↓KID↓
LatentReferenceMixLatentReferenceMix
(a) Baseline StarGAN v237.7337.9937.700.0130.0130.013
(b)+ Style-Content branches43.9032.6133.360.0170.0110.011
(c)+Foreground classifier37.1432.3427.690.0140.0110.008
(d) + Background classifier34.1232.5030.230.0110.0110.010
(e)+ Separately decoding foreground and background in G48.5238.1134.790.0170.0150.011
(f) + Anchor foreground domain (e.g. No rmal)43.6637.4532.150.0190.0150.011
33.050.0090.011
(g)+ Noise injection in Mapping Network34.4229.730.009
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The dataset is formed of non-defective image for training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "score": 1.0, + "content": "and both non-defective and defective images with various kinds of defects for testing. The pixel-level", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "annotations of all defective images are also provided. It is worth noting that the MVTec Anomaly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "Detection dataset is relatively small scale in number of images, where the number of training images", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "is ranging from 60 to 391. Moreover, the number of defective images for each defect category in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "test set is varying only from 8 to 30, which is relatively limited considering the sophisticated pattern", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 151, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 151, + 732 + ], + "score": 1.0, + "content": "of defects.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 644, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "We conducted image synthesis experiments on a subset of MVTec Anomaly Detection dataset,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "where we selected four texture categories: Carpet, Leather, Wood and Tile for our targeted sce-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "nario i.e. surface defects. Furthermore, we aggregated some of the original defect types defined in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the MVTec Anomaly Detection dataset into scratches and spots according to their visual ap-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "score": 1.0, + "content": "pearance. We then simply added the subset of the MVTec Anomaly Detection dataset to the training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "set together with the SDI dataset for training DT-GAN. Details of the resulting dataset are shown in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "Table 17. Note that the small scale of available data posts a major challenge for training generative", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 141, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 141, + 171 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "Quantitative Evaluation. We present additional quantitative results on the subset of the MVTec", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 201 + ], + "score": 1.0, + "content": "Anomaly Detection dataset in Table 16, following the same evaluation setup as described in Ap-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "pendix C. As shown in Table 16, our method achieves the best scores in Carpet and Wood, which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "supports our claim that DT-GAN generates synthetic images with higher fidelity and more diverse", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "defect. However, we also observe that StyleGAN v2 seems to outperform our method in Leather", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 145, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 145, + 243 + ], + "score": 1.0, + "content": "and Tile.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 108, + 248, + 503, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "Please note that FID and KID are not optimized to evaluate such a small dataset, there the results", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 358, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 358, + 271 + ], + "score": 1.0, + "content": "should only be interpreted together with the qualitative results.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "score": 1.0, + "content": "Note the we again omit the FID and KID of StarGAN v2 because it is not cable of generating images", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "for a specified product due to the ‘identity-shift’, which is also explained in detail in the qualitative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 153, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 153, + 310 + ], + "score": 1.0, + "content": "evaluation.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "table", + "bbox": [ + 119, + 359, + 489, + 449 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 326, + 504, + 358 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "Table 16: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "and KID. Note that the reported values are not comparable between columns, because they were", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 252, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 252, + 361 + ], + "score": 1.0, + "content": "calculated on different training sets.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "table_body", + "bbox": [ + 119, + 359, + 489, + 449 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 359, + 489, + 449 + ], + "spans": [ + { + "bbox": [ + 119, + 359, + 489, + 449 + ], + "score": 0.983, + "html": "
MethodFID↓KID↓
CarpetLeatherTileWoodCarpetLeatherTileWood
Mokady (2020)41.8760.26275.1281.710.040.030.290.04
StarGAN v211=1
StyleGAN v251.3751.60225.96140.010.050.030.230.12
BigGAN + DiffAug34.47101.70391.54113.320.030.070.420.07
Ours22.7986.13321.3575.830.010.070.360.03
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For qualitative results, we again discuss the ‘latent-guided’ and ‘reference-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 223, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 223, + 491 + ], + "score": 1.0, + "content": "guided’ synthesis separately.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "We present the ‘latent-guided’ image synthesis results of StyleGAN v2 in Figure 11 and Figure 12", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "and BigGAN in Figure 13 and Figure 14. The results are acquired by training one model for each", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "product and then generating 16 images from randomly sampled latent codes from each of them.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "As pointed out in Section 4.1.2, both methods can not adapt well on small dataset. They suffer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "score": 1.0, + "content": "from model collapsing and show signs of overfitting by generating images similar to the training", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "data. For example, StyleGAN v2 generates images either with no clear defect or identical to the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "training set (e.g., Leather in Figure 11 and Product B in Figure 12). The overfitting we observe", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "here also explains the better FID and KID scores in Table 16. For Tile, we can see clear signs of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 582, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 505, + 597 + ], + "score": 1.0, + "content": "mode collapse in the generated Tile images of StyleGAN v2. Similarly, BigGAN produces images", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 608 + ], + "score": 1.0, + "content": "with single mode and abnormal patterns (e.g., grid structure and gray edges). Unlike StyleGAN v2", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "and BigGAN, StarGAN v2 and our method both require images as input (i.e. Source). Therefore,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "we randomly sampled two Normal images and applied eight defects which are generated from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "randomly sampled latent codes to each of them. As seen in Figure 15 and Figure 16, StarGAN v2", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "fails to preserve the background from the given input images due to the highly entangled FG and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "BG. Also it fails to generates legit and diverse defects without separately modeling the style and the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 504, + 673 + ], + "score": 1.0, + "content": "content. In contrast to aforementioned methods, our DT-GAN produces images with higher fidelity", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "and more diversity in defect patterns as shown in Figure 17 and Figure 18. We believe this again", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "prove the importance of style-content separation and FG/BG disentanglement, which we introduce", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 693, + 167, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 167, + 703 + ], + "score": 1.0, + "content": "in Section 3.1.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "For ‘reference-guided’ image synthesis, the results of Mokady et al. (2020) are shown in Figure 19", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "and Figure 20 while the results of StarGAN v2 are in Figure 21 and Figure 22. We can observe a", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "We conducted image synthesis experiments on a subset of MVTec Anomaly Detection dataset,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "where we selected four texture categories: Carpet, Leather, Wood and Tile for our targeted sce-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "nario i.e. surface defects. Furthermore, we aggregated some of the original defect types defined in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the MVTec Anomaly Detection dataset into scratches and spots according to their visual ap-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 141 + ], + "score": 1.0, + "content": "pearance. We then simply added the subset of the MVTec Anomaly Detection dataset to the training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "set together with the SDI dataset for training DT-GAN. Details of the resulting dataset are shown in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "Table 17. Note that the small scale of available data posts a major challenge for training generative", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 141, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 141, + 171 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 506, + 171 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "Quantitative Evaluation. We present additional quantitative results on the subset of the MVTec", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 201 + ], + "score": 1.0, + "content": "Anomaly Detection dataset in Table 16, following the same evaluation setup as described in Ap-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "pendix C. As shown in Table 16, our method achieves the best scores in Carpet and Wood, which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "supports our claim that DT-GAN generates synthetic images with higher fidelity and more diverse", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "defect. However, we also observe that StyleGAN v2 seems to outperform our method in Leather", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 145, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 145, + 243 + ], + "score": 1.0, + "content": "and Tile.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 176, + 506, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 248, + 503, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "Please note that FID and KID are not optimized to evaluate such a small dataset, there the results", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 358, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 358, + 271 + ], + "score": 1.0, + "content": "should only be interpreted together with the qualitative results.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 106, + 247, + 505, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "score": 1.0, + "content": "Note the we again omit the FID and KID of StarGAN v2 because it is not cable of generating images", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "for a specified product due to the ‘identity-shift’, which is also explained in detail in the qualitative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 297, + 153, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 153, + 310 + ], + "score": 1.0, + "content": "evaluation.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 274, + 506, + 310 + ] + }, + { + "type": "table", + "bbox": [ + 119, + 359, + 489, + 449 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 326, + 504, + 358 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "Table 16: Quantitative comparison of DT-GAN with baseline image synthesis methods using FID", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "and KID. Note that the reported values are not comparable between columns, because they were", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 252, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 252, + 361 + ], + "score": 1.0, + "content": "calculated on different training sets.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "table_body", + "bbox": [ + 119, + 359, + 489, + 449 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 359, + 489, + 449 + ], + "spans": [ + { + "bbox": [ + 119, + 359, + 489, + 449 + ], + "score": 0.983, + "html": "
MethodFID↓KID↓
CarpetLeatherTileWoodCarpetLeatherTileWood
Mokady (2020)41.8760.26275.1281.710.040.030.290.04
StarGAN v211=1
StyleGAN v251.3751.60225.96140.010.050.030.230.12
BigGAN + DiffAug34.47101.70391.54113.320.030.070.420.07
Ours22.7986.13321.3575.830.010.070.360.03
", + "type": "table", + "image_path": "48579259fe7630f714f2c1e1b5470350c7a4af04e5a533325a61185f9ce9ab51.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 119, + 359, + 489, + 389.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 119, + 389.0, + 489, + 419.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 119, + 419.0, + 489, + 449.0 + ], + "spans": [], + "index": 24 + } + ] + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 108, + 467, + 503, + 490 + ], + "lines": [ + { + "bbox": [ + 107, + 467, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 107, + 467, + 504, + 480 + ], + "score": 1.0, + "content": "Qualitative Evaluation. For qualitative results, we again discuss the ‘latent-guided’ and ‘reference-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 479, + 223, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 223, + 491 + ], + "score": 1.0, + "content": "guided’ synthesis separately.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 106, + 467, + 504, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "We present the ‘latent-guided’ image synthesis results of StyleGAN v2 in Figure 11 and Figure 12", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "and BigGAN in Figure 13 and Figure 14. The results are acquired by training one model for each", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "product and then generating 16 images from randomly sampled latent codes from each of them.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "As pointed out in Section 4.1.2, both methods can not adapt well on small dataset. They suffer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "score": 1.0, + "content": "from model collapsing and show signs of overfitting by generating images similar to the training", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "data. For example, StyleGAN v2 generates images either with no clear defect or identical to the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "training set (e.g., Leather in Figure 11 and Product B in Figure 12). The overfitting we observe", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "here also explains the better FID and KID scores in Table 16. For Tile, we can see clear signs of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 582, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 505, + 597 + ], + "score": 1.0, + "content": "mode collapse in the generated Tile images of StyleGAN v2. Similarly, BigGAN produces images", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 608 + ], + "score": 1.0, + "content": "with single mode and abnormal patterns (e.g., grid structure and gray edges). Unlike StyleGAN v2", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "and BigGAN, StarGAN v2 and our method both require images as input (i.e. Source). Therefore,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "we randomly sampled two Normal images and applied eight defects which are generated from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "randomly sampled latent codes to each of them. As seen in Figure 15 and Figure 16, StarGAN v2", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "fails to preserve the background from the given input images due to the highly entangled FG and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "BG. Also it fails to generates legit and diverse defects without separately modeling the style and the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 504, + 673 + ], + "score": 1.0, + "content": "content. In contrast to aforementioned methods, our DT-GAN produces images with higher fidelity", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "and more diversity in defect patterns as shown in Figure 17 and Figure 18. We believe this again", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "prove the importance of style-content separation and FG/BG disentanglement, which we introduce", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 693, + 167, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 167, + 703 + ], + "score": 1.0, + "content": "in Section 3.1.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 495, + 506, + 703 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "For ‘reference-guided’ image synthesis, the results of Mokady et al. (2020) are shown in Figure 19", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "and Figure 20 while the results of StarGAN v2 are in Figure 21 and Figure 22. We can observe a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "clear shift in color in all the outputs from Mokady et al. (2020). Moreover, Mokady et al. (2020)", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "can only transfer content between two domains. In order to perform translation from a non-defective", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "sample to a defective one, we trained a model for each type of defect and for each product. This sums", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "up to be 13 models (Scratches and Spots for 6 categories and Scratches only for Tile). The", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 139 + ], + "score": 1.0, + "content": "results from the intended use within one background domain can be found on the diagonal and are", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 504, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 504, + 149 + ], + "score": 1.0, + "content": "marked in red in both Figure 19 and Figure 20. We still show the images that we feed in images from", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "other background domains. As expected, the model then fails to preserve the background of given", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "source images and introduce artifacts to the outputs. Similarly, StarGAN v2 does not preserve the", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "background from the input images. Without style-content separation and FG/BG disentanglement,", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "we observe that StarGAN v2 encodes the background characteristics together with the foreground", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 193, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 504, + 204 + ], + "score": 1.0, + "content": "content of the reference images, which results in identity-shits in its output images. Moreover, the", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 504, + 216 + ], + "score": 1.0, + "content": "output images either show no clear defect or contain abnormal patterns which sabotage the fidelity.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "On the contrary, our method can faithfully transfer the foreground content of reference images across", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "given background of different products as shown in Figure 23 and Figure 24, which demonstrate", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "the effectiveness of the style-content separation and FG/BG disentanglement we introduced in", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 247, + 156, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 156, + 258 + ], + "score": 1.0, + "content": "Section 3.1.", + "type": "text", + "cross_page": true + } + ], + "index": 15 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "clear shift in color in all the outputs from Mokady et al. (2020). Moreover, Mokady et al. (2020)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "can only transfer content between two domains. In order to perform translation from a non-defective", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "sample to a defective one, we trained a model for each type of defect and for each product. This sums", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "up to be 13 models (Scratches and Spots for 6 categories and Scratches only for Tile). The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 139 + ], + "score": 1.0, + "content": "results from the intended use within one background domain can be found on the diagonal and are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 504, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 504, + 149 + ], + "score": 1.0, + "content": "marked in red in both Figure 19 and Figure 20. We still show the images that we feed in images from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "other background domains. As expected, the model then fails to preserve the background of given", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "source images and introduce artifacts to the outputs. Similarly, StarGAN v2 does not preserve the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "background from the input images. 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MethodFID↓KID↓
ABCAllABCAll
Mokady (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v21137.70-110.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
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MethodResNet-50EfficientNet-b4
No-Aug21.64±1.2412.06±0.64
Trad-Aug12.58±0.819.33±0.73
Mokady (2020)11.11±1.1913.26±1.13
StarGAN v213.07±1.3012.25±0.79
StyleGAN v211.55±1.7911.68±0.76
BigGAN+DiffAug11.45±0.6112.06±0.50
Ours9.9±0.699.14±1.02
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A13.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
B6.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
C16.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
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Dataset Size20AAll
Trad-AugOursTrad-AugOurs
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1200016.95±1.0214.22±1.5312.05±0.819.90±0.69
1860016.12±2.1915.36±0.8612.37±0.3210.21±0.96
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(a)Encoder
LayerResampleNormOutput Shape
Image x-128 × 128×3
Conv 1×1-128 ×128 ×128
ResBlkAvgPoolIN64× 64×256
ResBlkAvgPoolIN32 × 32 × 512
ResBlkAvgPoolIN16 × 16 × 512
ResBlkIN16 ×16× 512
ResBlkIN16 ×16 × 512
(b) Background Decoder(c) Foreground Decoder
LayerResampleNorm Output ShapeLayerResample NormOutput Shape
Input-16 × 16 × 448Input ResBlk16 × 16 × 64
ResBlkIN 16 ×16× 448 16 ×16× 512ResBlkAdaIN16 ×16× 64
ResBlkIN=AdaIN16 × 16 × 256
ResBlk=IN16 ×16 × 512ResBlkAdaIN16 ×16× 256
ResBlkUpsampleIN32 × 32×512ResBlk Upsample ResBlkAdaIN32 × 32 × 256
ResBlk ResBlkUpsample UpsampleIN IN64 × 64× 256 128 × 128× 448UpsampleAdaIN64×64×128
ResBlkUpsampleAdaIN128 ×128 × 64
(d) Fusion
LayerResampleNormOutput Shape
Input128 × 128 × (448 + 64)
Conv 1×1128 × 128× 3
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LayerActivationOutput Shape
Latent z=16
LinearReLU512
LinearReLU512
LinearReLU512
LinearReLU512
(b) Style Code(c) Content
LayerActivation Output ShapeLayerResample ActivationNoise Output Shape
Input=512Input=512
LinearReLU512Reshape--1×1×512
LinearReLU512ResBlkUpsampleINTrue2×2×512
LinearReLU512ResBlkUpsampleINTrue4×4×512
Linear164ResBlkUpsampleINTrue8×8×256
ResBlkUpsampleINTrue16 ×16×128
Conv 1×1=INTrue16 × 16× 64
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(a)SharedLayers
LayerResampleNormOutput Shape
Input x128 × 128 ×3
Conv 1×11128 × 128 × 64
ResBlkAvgPool64 × 64× 256
ResBlkAvgPool32 × 32×512
ResBlkAvgPool16 ×16 × 512
(b) Style Code /Discriminator and BG Classifier(c) Content /FG Classifier
LayerResample NormOutput ShapeLayerResample NormOutput Shape
Input-16 ×16× 512Input16 ×16× 512
ResBlkAvgPool8×8×512LReLU16 ×16 × 512
ResBlkAvgPool4×4×512Conv 1×1*K16×16×64*K
LReLU4×4×512
Conv 4×41×1×512
LReLU1×1×512
Reshape512
Linear *KD*K
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Dataset SizeA
Trad-AugvAvBvCvABC
450016.00±1.0412.90±2.6113.08±1.6514.90±2.4615.27±3.49
660014.90±1.3813.99±1.8911.26±1.0414.36±4.0416.00±2.85
1200013.81±2.3611.81±2.6512.72±2.8711.99±1.6311.09±3.49
1860013.63±2.2212.72±5.2214.36±3.8314.18±5.0513.81±8.56
DatasetB
SizeTrad-AugvAvBvCVABC
45008.79±0.457.80±2.155.60±1.1410.19±0.846.79±1.30
66007.60±1.516.80±1.657.80±1.108.00±2.346.00±1.41
120006.80±1.646.40±1.346.60±1.526.59±1.345.60±1.34
186006.80±1.796.19±1.784.40±1.146.60±1.955.99±1.58
DatasetC
SizeTrad-AugvAVBvCvABC
450017.14±4.6214.85±0.5216.76±2.5813.90±1.9812.00±1.59
660015.23±2.3313.14±1.2413.90±2.2914.28±1.3412.57±1.57
1200016.57±3.2013.14±2.8111.23±0.8014.85±1.7311.42±0.96
1860015.62±0.8513.71±1.7315.99±6.7512.57±3.2612.95±2.98
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Dataset Size20A
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450035.09±2.62 27.64±3.127.8±1.485.6±1.6715.24±1.90 13.14±1.7015.55±0.63 14.28±1.25
660039.64±2.28 27.64±1.658.8±1.646.2±1.6415.81±1.73112.38±1.6516.69±0.76 14.41±3.12
1200034.18±4.39 28.55±7.325.8±0.455.6±1.1416.19±1.17 10.86±1.28 16.95±1.02 14.22±1.53
1860039.45±7.06 32.55±5.047.2±0.845.2±1.1014.86±0.85 13.14±2.06 16.12±2.19 15.36±0.86
Dataset SizeAll
ABCABC
Trad-AugOursTrad-AugOursTrad-AugOursTrad-AugOurs
450016.00±1.04110.18±1.75 8.79±0.455.60±1.5117.13±6.62 14.09±2.2712.75±0.61 11.04±0.76
660014.90±1.3810.54±1.22 7.60±1.516.80±3.1115.23±2.33 11.42±013.07±1.57 10.60±0.48
1200013.81±2.366.72±1.65 6.80±1.644.60±016.57±3.20 13.90±2.5712.05±0.819.90±0.69
1860013.63±2.22 10.54±2.45 6.80±1.794.99±1.87 15.62±0.85 11.61±1.24 12.37±0.32 10.21±0.96
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FID↓KID↓
LatentReferenceMixLatentReferenceMix
(a) Baseline StarGAN v237.7337.9937.700.0130.0130.013
(b)+ Style-Content branches43.9032.6133.360.0170.0110.011
(c)+Foreground classifier37.1432.3427.690.0140.0110.008
(d) + Background classifier34.1232.5030.230.0110.0110.010
(e)+ Separately decoding foreground and background in G48.5238.1134.790.0170.0150.011
(f) + Anchor foreground domain (e.g. No rmal)43.6637.4532.150.0190.0150.011
33.050.0090.011
(g)+ Noise injection in Mapping Network34.4229.730.009
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MethodFID↓KID↓
ABCAllABCAll
Mokady et al. (2020)68.6966.9036.2158.630.0500.0360.0300.036
StarGAN v2--137.701-10.013
StyleGAN v290.1052.95138.0935.340.0720.0270.1860.013
StyleGAN v2 + ADA149.6642.75135.6976.160.1380.0190.1910.055
BigGAN235.66192.89193.61151.430.2480.1990.2760.115
BigGAN + DiffAug218.74134.41270.89155.880.2200.1210.3780.099
Ours58.4336.4422.6829.730.0250.0130.0120.009
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MethodFID↓KID↓
CarpetLeatherTileWoodCarpetLeatherTileWood
Mokady (2020)41.8760.26275.1281.710.040.030.290.04
StarGAN v211=1
StyleGAN v251.3751.60225.96140.010.050.030.230.12
BigGAN + DiffAug34.47101.70391.54113.320.030.070.420.07
Ours22.7986.13321.3575.830.010.070.360.03
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Thread000-018
Hole-000 - 016
Sum2423
(b) Leather
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Poke003,009,010,016,017000-017
Sum39
48
(c) Tile
ScratchesSpots
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Sum170
(d) Wood
ScratchesSpots
Color003,005
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Combined008001,002,009
Sum1912
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Philipp Wu\* + +Alejandro Escontrela\* Danijar Hafner\* + +Ken Goldberg Pieter Abbeel + +University of California, Berkeley \*Equal contribution + +Abstract: To solve tasks in complex environments, robots need to learn from experience. Deep reinforcement learning is a common approach to robot learning but requires a large amount of trial and error to learn, limiting its deployment in the physical world. As a consequence, many advances in robot learning rely on simulators. On the other hand, learning inside of simulators fails to capture the complexity of the real world, is prone to simulator inaccuracies, and the resulting behaviors do not adapt to changes in the world. The Dreamer algorithm has recently shown great promise for learning from small amounts of interaction by planning within a learned world model, outperforming pure reinforcement learning in video games. Learning a world model to predict the outcomes of potential actions enables planning in imagination, reducing the amount of trial and error needed in the real environment. However, it is unknown whether Dreamer can facilitate faster learning on physical robots. In this paper, we apply Dreamer to 4 robots to learn online and directly in the real world, without any simulators. Dreamer trains a quadruped robot to roll off its back, stand up, and walk from scratch and without resets in only 1 hour. We then push the robot and find that Dreamer adapts within 10 minutes to withstand perturbations or quickly roll over and stand back up. On two different robotic arms, Dreamer learns to pick and place objects from camera images and sparse rewards, approaching human-level teleoperation performance. On a wheeled robot, Dreamer learns to navigate to a goal position purely from camera images, automatically resolving ambiguity about the robot orientation. Using the same hyperparameters across all experiments, we find that Dreamer is capable of online learning in the real world, which establishes a strong baseline. We release our infrastructure for future applications of world models to robot learning. Videos are available on the project website: https://danijar.com/daydreamer + +![](images/17f2d11eee9937e70f62a1993623ebccd221887d067e71919c350fa57662f4d3.jpg) +Figure 1: To study the applicability of Dreamer for sample-efficient robot learning, we apply the algorithm to learn robot locomotion, manipulation, and navigation tasks from scratch in the real world on 4 robots, without simulators. The tasks evaluate a diverse range of challenges, including continuous and discrete actions, dense and sparse rewards, proprioceptive and camera inputs, as well as sensor fusion of multiple input modalities. Learning successfully using the same hyperparameters across all experiments, Dreamer establishes a strong baseline for real world robot learning. + +# 1 Introduction + +Teaching robots to solve complex tasks in the real world is a foundational problem of robotics research. Deep reinforcement learning (RL) offers a popular approach to robot learning that enables robots to improve their behavior over time through trial and error. However, current algorithms require too much interaction with the environment to learn successful behaviors. Recently, modern world models have shown great promise for data efficient learning in simulated domains and video games (Hafner et al., 2019; 2020). Learning world models from past experience enables robots to imagine the future outcomes of potential actions, reducing the amount of trial and error in the real environment needed to learn. + +While learning accurate world models can be challenging, they offer compelling properties for robot learning. By predicting future outcomes, world models allow for planning and behavior learning given only small amounts of real world interaction (Gal et al., 2016; Ebert et al., 2018). Moreover, world models summarize general dynamics knowledge about the environment that, once learned, could be reused for a wide range of downstream tasks (Sekar et al., 2020). World models also learn representations that fuse multiple sensor modalities and integrate them into latent states, reducing the need for sophisticated state estimators. Finally, world models generalize well from available offline data (Yu et al., 2021), which further accelerates learning in the real world. + +![](images/e30f877426a1aa2686b70c08629a56889403c96105b017890f8ea57b7982c4a2.jpg) +Figure 2: Dreamer follows a simple pipeline for online learning on robot hardware without simulators. The current learned policy collects experience on the robot. This experience is added to the replay buffer. The world model is trained on replayed off-policy sequences through supervised learning. An actor critic algorithm optimizes a neural network policy from imagined rollouts in the latent space of the world model. We parallelize data collection and neural network learning. + +Despite the promises of world models, learning accurate world models for the real world is a open challenge. In this paper, we leverage recent advances of the Dreamer world model for training a variety of robots in the most straight-forward and fundamental problem setting: online reinforcement learning in the real world, without simulators or demonstrations. As shown in Figure 2, Dreamer learns a world model from a replay buffer of past experience, learns behaviors from rollouts imagined in the latent space of the world model, and continuously interacts with the environment to explore and improve its behaviors. Our aim is to push the limits of robot learning directly in the real world and offer a robust platform to enable future work that develops the benefits of world models for robot learning. The key contributions of this paper are summarized as follows: + +• Dreamer on Robots We apply Dreamer to 4 robots, demonstrating successful learning directly in the real world, without introducing new algorithms. The tasks cover a range of challenges, including different action spaces, sensory modalities, and reward structures. +• Walking in 1 Hour We teach a quadruped from scratch in the real world to roll off its back, stand up, and walk in only 1 hour. Afterwards, we find that the robot adapts to being pushed within 10 minutes, learning to withstand pushes or quickly roll over and get back on its feet. +• Visual Pick and Place We train robotic arms to pick and place objects from sparse rewards, which requires localizing objects from pixels and fusing images with proprioceptive inputs. The learned behavior outperforms model-free agents and approaches the performance of a human teleoperator using the same control interface as the robot. +• Open Source We publicly release the software infrastructure for all our experiments, which supports different action spaces and sensory modalities, offering a flexible platform for future research of world models for robot learning in the real world. + +![](images/bbd9aa6b3f541685e1ecf9dd1c4451b92904b361a6547ee2e39414769cb64de4.jpg) +Figure 3: Neural Network Training We leverage the Dreamer algorithm (Hafner et al., 2019; 2020) for fast robot learning in real world. Dreamer consists of two main neural network components, the world model and the policy. Left: The world model follows the structure of a deep Kalman filter that is trained on subsequences drawn from the replay buffer. The encoder fuses all sensory modalities into discrete codes. The decoder reconstructs the inputs from the codes, providing a rich learning signal and enabling human inspection of model predictions. A recurrent state-space model (RSSM) is trained to predict future codes given actions, without observing intermediate inputs. + +Right: The world model enables massively parallel policy optimization from imagined rollouts in the compact latent space using a large batch size, without having to reconstruct sensory inputs. Dreamer trains a policy network and value network from the imagined rollouts and a learned reward function. + +# 2 Approach + +We leverage the Dreamer algorithm (Hafner et al., 2019; 2020) for online learning on physical robots, without the need for simulators. Figure 2 shows an overview of the approach. Dreamer learns a world model from a replay buffer of past experiences, uses an actor critic algorithm to learn behaviors from trajectories predicted by the learned model, and deploys its behavior in the environment to continuously grow the replay buffer. We decouple learning updates from data collection to meet latency requirements and to enable fast training without waiting for the environment. In our implementation, a learner thread continuously trains the world model and actor critic behavior, while an actor thread in parallel computes actions for environment interaction. + +World Model Learning The world model is a deep neural network that learns to predict the environment dynamics, as shown in Figure 3 (left). Because sensory inputs can be large images, we predict future representations rather than future inputs. This reduces accumulating errors and enables massively parallel training with a large batch size. Thus, the world model can be thought of as a fast simulator of the environment that the robot learns autonomously, starting from a blank slate and continuously improving its model as it explores the real world. The world model is based on the Recurrent State-Space Model (RSSM; Hafner et al., 2018), which consists of four components: + +$$ +{ \begin{array} { r l r l } & { \operatorname { e n c } _ { \theta } { \big ( } s _ { t } \ { \big | } \ s _ { t - 1 } , a _ { t - 1 } , x _ { t } { \big ) } } & & { { \mathrm { D e c o d e r ~ N e t w o r k : } } \quad \operatorname* { d e c } _ { \theta } { \big ( } s _ { t } { \big ) } \approx x _ { t } } \\ & { \operatorname { d y n } _ { \theta } { \big ( } s _ { t } \ { \big | } \ s _ { t - 1 } , a _ { t - 1 } { \big ) } } & & { { \mathrm { R e w a r d ~ N e t w o r k : } } \quad \operatorname { r e w } _ { \theta } { \big ( } s _ { t + 1 } { \big ) } \approx r _ { t } } \end{array} } +$$ + +Physical robots are often equipped with multiple sensors of different modalities, such as proprioceptive joint readings, force sensors, and high-dimensional inputs such as RGB and depth camera images. The encoder network fuses all sensory inputs $x _ { t }$ together into the stochastic representations $z _ { t }$ . The dynamics model learns to predict the sequence of stochastic representations by using its recurrent state $h _ { t }$ . The decoder reconstructs the sensory inputs to provide a rich signal for learning representations and enables human inspection of model predictions. In our experiments, the robot has to discover task rewards by interacting with the real world, which the reward network learns to predict. Using manually specified rewards as a function of the decoded sensory inputs is also possible. We optimize all components of the world model jointly by stochastic backpropagation (Kingma and Welling, 2013; Rezende et al., 2014). + +Actor Critic Learning While the world model represents task-agnostic knowledge about the dynamics, the actor critic algorithm learns a behavior that is specific to the task at hand. As shown in Figure 3 (right), we learn behaviors from rollouts that are predicted in the latent space of the world model, without decoding observations. This enables massively parallel behavior learning with typical batch sizes of 16K on a single GPU. The actor critic algorithm consists of an actor network $\pi ( a _ { t } | s _ { t } )$ and a critic network $v ( s _ { t } )$ . + +The role of the actor network is to learn a distribution over successful actions $a _ { t }$ for each latent model state $s _ { t }$ that maximizes the sum of future predicted task rewards. The critic network learns to predict the sum of future task rewards through temporal difference learning (Sutton and Barto, 2018). This allows the algorithm to take into account rewards beyond the planning horizon of $H = 1 6$ steps to learn long-term strategies. Given a predicted trajectory of model states, the critic is trained to regress the return of the trajectory. We compute $\lambda$ -returns following Hafner et al. (2020; 2019): + +$$ +V _ { t } ^ { \lambda } \doteq r _ { t } + \gamma \Big ( ( 1 - \lambda ) v ( s _ { t + 1 } ) + \lambda V _ { t + 1 } ^ { \lambda } \Big ) , \quad V _ { H } ^ { \lambda } \doteq v ( s _ { H } ) . +$$ + +While the critic network is trained to regress the $\lambda$ -returns, the actor network is trained to maximize them. Different gradient estimators are available for computing the policy gradient for optimizing the actor, such as Reinforce (Williams, 1992) and the reparameterization trick (Kingma and Welling, 2013; Rezende et al., 2014) that directly backpropagates return gradients through the differentiable dynamics network (Henaff et al., 2019). Following Hafner et al. (2020), we choose reparameterization gradients for continuous control tasks and Reinforce gradients for tasks with discrete actions. In addition to maximizing returns, the actor is also incentivized to maintain high entropy to prevent collapse to a deterministic policy and maintain some amount of exploration throughout training: + +$$ +\begin{array} { r } { \mathcal { L } ( \pi ) \doteq - \operatorname { E } \bigl [ \sum _ { t = 1 } ^ { H } \ln \pi ( a _ { t } \mid s _ { t } ) \mathrm { s g } ( V _ { t } ^ { \lambda } - v ( s _ { t } ) ) + \eta \mathrm { H } \bigl [ \pi ( a _ { t } \mid s _ { t } ) \bigr ] \bigr ] } \end{array} +$$ + +We optimize the actor and critic using the Adam optimizer (Kingma and Ba, 2014). To compute the $\lambda$ -returns, we use a slowly updated copy of the critic network as common in the literature (Mnih et al., 2015; Lillicrap et al., 2015). The actor and critic gradients do not affect the world model, as this would lead to incorrect and overly optimistic model predictions. The hyperparameters are listed in Appendix D. + +# 3 Experiments + +We evaluate Dreamer on 4 robots, each with a different task, and compare its performance to appropriate algorithmic and human baselines. The experiments are representative of common robotic tasks, such as locomotion, manipulation, and navigation. The tasks pose a diverse range of challenges, including continuous and discrete actions, dense and sparse rewards, proprioceptive and image observations, and sensor fusion. The goal of the experiments is to evaluate whether the recent successes of learned world models enables sample-efficient robot learning directly in the real world. Specifically, we aim to answer the following research questions: + +• Does Dreamer enable robot learning directly in the real world, without simulators? • Does Dreamer succeed across various robot platforms, sensory modalities, and action spaces? • How does the data-efficiency of Dreamer compare to previous reinforcement learning algorithms? + +Implementation We build on the official implementation of DreamerV2 (Hafner et al., 2020). We develop an asynchronous actor and learner setup, which is essential in environments with high control rates, such as the quadruped, and also accelerates learning for slower environments, such as the robot arms. The actor thread computes online actions for the robot and sends trajectories of 128 time steps to the replay buffer. The learner thread samples data from the replay buffer, updates the world model, and optimizes the policy using imagination rollouts. Policy weights are synced from the learner to the actor every 20 seconds. We use an RSSM with 256 units to speed up the training computation. We use identical hyperparameters across all experiments, enabling off-the-shelf training on different robot embodiments. + +![](images/159d86a4fe017221206965fa98efc6ce35e16bebec7536f231b04a5fa470830b.jpg) +Figure 4: A1 Quadruped Walking Starting from lying on its back with the feet in the air, Dreamer learns to roll over, stand up, and walk in 1 hour of real world training time, without simulators or resets. In contrast, SAC only learns to roll over but neither to stand up nor to walk. For SAC, we also had to help the robot out of a dead-locked leg configuration during training. On the right we show training curves for both SAC and Dreamer. The maximum reward is 14. The filled circles indicate times where the robot fell on its back, requiring the learning of a robust strategy for getting back up. After 1 hour of training, we start pushing the robot and find that it adapts its behavior within 10 minutes to withstand light pushes and quickly roll back on its feet for hard pushes. The graph shows a single training run with the shaded area indicating one standard deviation within each time bin. + +Baselines We compare to a strong learning algorithm for each of our experimental setups. The A1 quadruped robot uses continuous actions and low-dimensional inputs, allowing us to compare to SAC (Haarnoja et al., 2018a;b), a popular algorithm for data-efficient continuous control. For the visual pick and place experiments on the XArm and UR5 robots, inputs are images and proprioceptive readings and actions are discrete, suggesting algorithms from the DQN (Mnih et al., 2015) line of work as baselines. We choose Rainbow (Hessel et al., 2018) as a powerful representative of this category, an algorithm that combines many improvements of DQN. To input the proprioceptive readings, we concatenate them as broadcasted planes to the RGB channels of the image, a common practice in the literature (Schrittwieser et al., 2019). For the UR5, we additionally compare against PPO (Schulman et al., 2017), with similar modifications for fusing image and proprioceptive readings. In addition, we compare against a human operator controlling the robot arm through the robot control interface. For the Sphero navigation task, inputs are images and actions are continuous. The state-ofthe-art baseline in this category is DrQv2 (Yarats et al., 2021), which uses image augmentation to increase sample-efficiency. + +# 3.1 A1 Quadruped Walking + +This high-dimensional continuous control task requires training a quadruped robot to roll over from its back, stand up, and walk forward at a fixed target velocity. Prior work in quadruped locomotion requires either extensive training in simulation under domain randomization, using recovery controllers to avoid unsafe states, or defining the action space as parameterized trajectory generators that restrict the space of motions (Rusu et al., 2016; Peng et al., 2018; Rudin et al., 2021; Lee et al., 2020; Yang et al., 2019). In contrast, we train in the end-to-end reinforcement learning setting directly on the robot, without simulators or resets. We use the Unitree A1 robot that consists of 12 direct drive motors. The motors are controlled at $2 0 \mathrm { H z }$ via continuous actions that represent motor angles that are realized by a PD controller on the hardware. Actions are filtered with a Butterworth filter to protect the motor from high-frequency actions. The input consists of motor angles, orientations, and angular velocities. Due to space constraints, we manually intervene when the robot has reached the end of the available training area, without modifying the joint configuration or orientation that the robot is in. + +![](images/69e7a0dc11e3a7ecd812dec526777f2a39e7aed63605587ef789903e7f57fb8c.jpg) +Figure 8: Within 10 minutes of perturbing the learned walking behavior, the robot adapts to withstanding pushes or quickly rolling over and back on its feet. + +The reward function is the sum of five terms. An upright reward is computed from the base frame up vector $\hat { z } ^ { T }$ , terms for matching the standing pose are computed from the joint angles of the hips, shoulders, and knees, and a forward velocity term is computed from the projected forward velocity $\boldsymbol { s } _ { v } \boldsymbol { x }$ and the total velocity $s _ { v }$ . Without the reward curriculum, the agent receives spurious reward values due to the velocity estimator’s dependence on foot-ground contact events. Each of the five terms is active while its preceding terms are satisfied to at least 0.7 and otherwise set to 0: + +![](images/69863294723843746383e47fe99dcd32744e499d1c526f51df4121c52ff99fe8.jpg) +Figure 5: UR5 Multi Object Visual Pick and Place This task requires learning to locate three ball objects from third-person camera images, grasp them, and move them into the other bin. The arm is free to move within and above the bins and sparse rewards are given for grasping a ball and for dropping it in the opposite bin. The environment requires the world model to learn multi-object dynamics in the real world and the sparse reward structure poses a challenge for policy optimization. Dreamer overcomes the challenges of visual localization and sparse rewards on this task, learning a successful strategy within a few hours of autonomous operation. + +$$ +\begin{array} { r l } { r ^ { \mathrm { u p r } } \doteq ( \hat { z } ^ { T } [ 0 , 0 , 1 ] - 1 ) / 2 } & { { } r ^ { \mathrm { h i p } } \doteq 1 - \frac 1 4 \| q ^ { \mathrm { h i p } } + 0 . 2 \| _ { 1 } \quad r ^ { \mathrm { s h o u l d e r } } \doteq 1 - \frac 1 4 \| q ^ { \mathrm { s h o u l d e r } } + 0 . 2 \| _ { 1 } } \end{array} +$$ + +$$ +\begin{array} { r l } { r ^ { \mathrm { k n e e } } \doteq 1 - \frac 1 4 \parallel q ^ { \mathrm { k n e e } } - 1 . 0 \parallel _ { 1 } } & { { } r ^ { \mathrm { v e l o c i t y } } \doteq 5 \big ( \operatorname* { m a x } ( 0 , ^ { \mathcal { B } } v _ { x } ) / \parallel ^ { \mathcal { B } } v \parallel _ { 2 } \cdot \mathrm { c l i p } ( ^ { \mathcal { B } } v _ { x } / 0 . 3 , - 1 , 1 ) + 1 \big ) } \end{array} +$$ + +As shown in Figure 4, after one hour of training, Dreamer learns to consistently flip the robot over from its back, stand up, and walk forward. In the first 5 minutes of training, the robot manages to roll off its back and land on its feet. 20 minutes later, it learns how to stand up on its feet. About 1 hour into training, the robot learns a pronking gait to walk forward at the desired velocity. After succeeding at this task, we tested the robustness of the algorithms by repeatedly knocking the robot off of its feet with a large pole, shown in Figure 8. Within 10 minutes of additional online learning, the robot adapts and withstand pushes or quickly rolls back on its feet. In comparison, SAC quickly learns to roll off its back but fails to stand up or walk given the small data budget. + +# 3.2 UR5 Multi-Object Visual Pick and Place + +Common in warehouse and logistics environments, pick and place tasks require a robot manipulator to transport items from one bin into another. Figure 5 shows a successful pick and place cycle of this task. The task is challenging because of sparse rewards, the need to infer object positions from pixels, and the challenging dynamics of multiple moving objects. The sensory inputs consist of proprioceptive readings (joint angles, gripper position, end effector Cartesian position) and a 3rd person RGB image of the scene. Successfully grasping one of the 3 objects, detected by partial gripper closure, results in a $+ 1$ reward, releasing the object in the same bin gives a $- 1$ reward, and placing in the opposite bin gives a $+ 1 0$ reward. We control the UR5 robot from Universal Robotics at $2 \ \mathrm { H z }$ . Actions are discrete for moving the end effector in increments along X, Y, and $\textsf { Z }$ axes and for toggling the gripper state. Movement in the Z axis is only enabled while holding an object and the gripper automatically opens once above the correct bin. We estimate human teleoperation performance by recording 3 demonstrators for 20 minutes each, controlling the UR5 with a joystick. + +Dreamer reaches an average pick rate of 2.5 objects per minute within 8 hours. The robot initially struggles to learn as the reward signal is very sparse, but begins to gradually improve after 2 hours of training. The robot first learns to localize the objects and toggles the gripper when near an object. Over time, grasping becomes precise and the robot learns to push objects out of corners. Figure 5 shows the learning curves of Dreamer compared to Rainbow DQN, PPO, and the human baseline. Both Rainbow DQN and PPO only learn the short-sighted behavior of grasping and immediately dropping objects in the same bin. In contrast, Dreamer approaches human-level teleoperation performance after 8 hours. We hypothesize that Rainbow DQN and PPO fail because they require larger amounts of experience, which is not feasible for us to collect in the real world. + +# 3.3 XArm Visual Pick and Place + +While the UR5 robot is a high performance industrial robot, the XArm is an accessible low-cost 7 DOF manipulation, which we control at approximately $0 . 5 \ : \mathrm { H z }$ . Similar to Section 3.2, the task requires localizing and grasping a soft object and moving it from one bin to another and back, shown in Figure 6. We connect the object to the gripper with a string, which makes it less likely for the object to get stuck in corners at the cost of more complex dynamics. The sparse reward, discrete action space, and observation space match the UR5 setup except for the addition of depth image observations. + +![](images/dfce202941b6d7b7a3b4e91b152da625264b3b1c43837193ab53e137e11b01f3.jpg) +Figure 6: XArm Visual Pick and Place The XArm is an affordable robot arm that operates slower than the UR5. To demonstrate successful learning on this robot, we use a third-person RealSense camera with RGB and depth modalities, as well as proprioceptive inputs for the robot arm, requiring the world model to learn sensor fusion. The pick and place task uses a soft object. While soft objects would be challenging to model accurately in a simulator, Dreamer avoids this issue by directly learning on the real robot without a simulator. While Rainbow and PPO using R3M visual embeddings converge to the local optimum of grasping and ungrasping the object in the same bin, Dreamer learns a successful pick and place policy from sparse rewards in under 10 hours. + +Dreamer learns a policy that enables the XArm to achieve an average pick rate of 3.1 objects per minute in 10 hours of time, which is comparable to human performance on this task. Figure 6 shows that Dreamer learns to solve the task within 10 hours, whereas the Rainbow algorithm, a top model-free algorithm for discrete control from pixels, fails to learn. We additionally compare Dreamer against a PPO baseline that utilizes R3M (Nair et al., 2022) pretrained visual embeddings for the state, but notice no improvement in performance. Interestingly, we observed that Dreamer learns to sometimes use the string to pull the object out of a corner before grasping it, demonstrating multi-modal behaviors. Moreover, we observed that when lighting conditions change drastically (such as sharp shadows during sunrise), performance initially collapses but Dreamer then adapts to the changing conditions and exceeds its previous performance after a few hours of additional training, reported in Appendix A. + +# 3.4 Sphero Navigation + +We evaluate Dreamer on a visual navigation task that requires maneuvering a wheeled robot to a fixed goal location given only RGB images as input. We use the Sphero Ollie robot, a cylindrical robot with two controllable motors, which we control through continuous torque commands at $2 \ : \mathrm { H z }$ Because the robot is symmetric and the robot only has access to image observations, it has to infer the heading direction from the history of observations. The robot is provided with a dense reward equal to the negative L2 distance, which is computed using a oracle vision pipeline that detects the Sphero’s position (this information is not provided to the agent). As the goal is fixed, after 100 environment steps, we end the episode and randomize the robot’s position through a sequence of high power random motor actions. + +In 2 hours, Dreamer learns to quickly and consistently navigate to the goal and stay near the goal for the remainder of the episode. As shown in Figure 7, Dreamer achieves an average distance to the goal of 0.15, measured in units of the area size and averaged across time steps. We find that DrQv2, a model-free algorithm specifically designed to continuous control from pixels, achieves similar performance. This result matches the simulated experiments of Yarats et al. (2021) that showed the two algorithms to perform similarly for continuous control tasks from images. + +# 4 Related Work + +Existing work on robot learning commonly leverages large amounts of simulated experience before deploying to the real world (Rusu et al., 2016; Peng et al., 2018; OpenAI et al., 2018; Lee et al., 2020; Irpan et al., 2020; Kumar et al., 2021; Siekmann et al., 2021; Escontrela et al., 2022), leverage fleets of robots to collect experience datasets (Kalashnikov et al., 2018; Dasari et al., 2019; Kalashnikov et al., 2021; Ebert et al., 2021), or rely on external information such as human expert demonstrations or task priors to achieve sample-efficient learning (Xie et al., 2019; Schoettler et al., 2019; James et al., 2021; Shah and Levine, 2022; Bohez et al., 2022; Sivakumar et al., 2022). However, designing simulated tasks and collecting expert demonstrations is time-consuming. Moreover, many of these approaches require specialized algorithms for leveraging offline experience, demonstrations, or simulator inaccuracies. In contrast, our experiments show that learning end-to-end from rewards in the physical world is feasible for a diverse range of tasks through world models. + +![](images/372acb53fc3f9ab9f178baac319f8b0cb0c4ee1ca96f374bada7397c4ec23630.jpg) +Figure 7: Sphero Navigation This task requires the Sphero robot to navigate to a goal location given a top-down RGB image as the only input. The task requires the robot to localize itself from raw pixels, to infer its orientation from the sequence of past images because it is ambiguous from a single image, and to control the robot from under-actuated motors that require building up momentum over time. Dreamer learns a successful policy on this task in under 2 hours. + +Relatively few works have demonstrated end-to-end learning from scratch in the physical world. Visual Foresight (Finn et al., 2016; Finn and Levine, 2017; Ebert et al., 2018) learns a video prediction model to solve real world tasks by online planning, but is limited to short-horizon tasks and requires generating images during planning, making it computationally expensive. Yang et al. (2019; 2022) learn quadruped locomotion through a model-based approach by predicting foot placement and leveraging a domain-specific controller to achieve them. Ha et al. (2020) learn a quadruped walking policy by relying on a scripted reset policy, so the robot does not have to learn to stand up. SOLAR (Zhang et al., 2019) learns a latent dynamics model from images and demonstrates reaching and pushing with a robot arm. Nagabandi et al. (2019) learns manipulation policies by planning through a learned dynamics model from state observations. In comparison, our experiments show successful learning across 4 challenging robot tasks that cover a wide range of challenges and sensory modalities, with a single learning algorithm and hyperparameter setting. + +# 5 Discussion + +We applied Dreamer to physical robot learning, finding that modern world models enable sampleefficient robot learning for a range of tasks, from scratch in the real world and without simulators. We also find that the approach is generally applicable in that it can solve robot locomotion, manipulation, and navigation tasks without changing hyperparameters. Dreamer taught a quadruped robot to roll off the back, stand up, and walk in 1 hour from scratch, which previously required extensive training in simulation followed by transfer to the real world or parameterized trajectory generators and given reset policies. We also demonstrate learning to pick and place objects from pixels and sparse rewards on two robot arms in 8–10 hours. + +Limitations While Dreamer shows promising results, learning on hardware over many hours creates wear on robots that may require human intervention or repair. Additionally, more work is required to explore the limits of Dreamer and our baselines by training for a longer time. Finally, we see tackling more challenging tasks, potentially by combining the benefits of fast real world learning with those of simulators, as an impactful future research direction. + +Acknowledgements We thank Stephen James and Justin Kerr for helpful suggestions and help with printing the protective shell of the quadruped robot. We thank Ademi Adeniji for help with setting up the XArm robot and Raven Huang for help with setting up the UR5 robot. This work was supported in part by an NSF Fellowship, NSF NRI #2024675, and the Vanier Canada Graduate Scholarship. + +References +D. Hafner, T. 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In Deep RL Workshop NeurIPS 2021, 2021. +K. Hsu, M. J. Kim, R. Rafailov, J. Wu, and C. Finn. Vision-based manipulators need to also see from their hands, 2022. URL https://arxiv.org/abs/2203.12677. + +# A Adaptation + +Real world robot learning faces practical challenges such as changing environmental conditions and time varying dynamics. We found that Dreamer is able to adapt to the current environmental conditions with no change to the learning algorithm. This shows promise for using Dreamer in continual learning settings (Parisi et al., 2019). Adaptation of the quadruped to external perturbations is reported in Section 3.1 and Figure 8. + +The XArm, situated near large windows, is able to adapt and maintain performance under the presence of changing lighting conditions. The XArm experiments were conducted after sundown to keep the lighting conditions constant throughout training. Figure A.1 shows the learning curve of the XArm. As expected, the performance of the XArm drops during sunrise. However, the XArm is able to adapt to the change in lighting conditions in about 5 hours time and recover the original performance, which is faster than it would be to train from scratch. A careful inspection of the image observations at these times, as shown in Figure A.1, reveals that the robot received observations with strong light rays covering the scene which greatly differs from the original training observations. + +![](images/db6cf431ae9355646aa06c810c30e311d8db38009707b4dea4bad788085ac2bb.jpg) +Figure A.1: The left two images are raw observations consumed by Dreamer. The leftmost image is an image observation as seen by the XArm at night, when it was trained. The next image shows an observation during sunrise. Despite the vast difference in pixel space, the XArm is able to recover, and then surpass, the original performance in approximately 5 hours. Even after 24 hours when the lighting shifts to night time conditions, the XArm is able to maintain performance. + +# B Imagination + +![](images/24fe88bf92baa43778d9defa3450750bc0d2c910fa9c12c5902630d7c2316e1e.jpg) +Figure B.1: To introspect the policy, we can roll out trajectories in the latent space of Dreamer, then decode the images to visualize the intent of the actor network. Each row is an imagined trajectory, showing every 2nd frame. Top: Latent rollouts on the UR5 environment. Multiple objects introduce more visual complexity that the network has to model. Note the second trajectory, which shows a static orange ball becoming a green ball. Bottom: Latent rollouts on the XArm environment. + +# C Detailed Related Work + +RL for locomotion A common approach is to train RL agents from large amounts of simulated data under domain and dynamics randomization (Peng et al., 2018; Lee et al., 2020; Rudin et al., 2021; Siekmann et al., 2021; Escontrela et al., 2022; Miki et al., 2022; Kumar et al., 2021; Rusu et al., 2016; Bohez et al., 2022), then freezing the learned policy and deploying it to the real world. Smith et al. (2021) explored pre-training policies in simulation and fine-tuning them with real world data. Yang et al. (2019) investigate learning a dynamics model using a multi-step loss and using model predictive control to accomplish a specified task. Yang et al. (2022) train locomotion policies in the real world but require a recovery controller trained in simulation to avoid unsafe states. In contrast, we use no simulators or reset policies and directly train on the physical robot. While prior work in locomotion has successfully learned walking behaviors in the real world, these works generally required several domain-specific assumptions or pretraining with simulators. Ha et al. (2020) achieved successful walking on the Minitaur robot in 90 minutes. However, the authors manually programmed a reset policy that was used when the robot fell on its back, while in our work the robot must learn to flip over and stand up. Additionally, the Minitaur robot is simpler than the A1 as it has 8 actuators compared to 12 on the A1. In recent work, Smith et al. (2022) utilize a high update-to-data ratio (UTD) RL algorithm to learn walking from 20 minutes of robot training data. However, their work assumes the availability of a reset policy and therefore comprises of a different learning problem compared to the problem we tackle of learning to flip over and walk from scratch. Additionally, we show our approach generalizes to environments with image observations and sparse rewards. + +RL for manipulation Learning promises to enable robot manipulators to solve contact rich tasks in open real world environments. One class of methods attempts to scale up experience collection through a fleet of robots (Kalashnikov et al., 2018; 2021; Ebert et al., 2021; Dasari et al., 2019; Levine et al., 2018). In contrast, we only leverage one robot, but parallelize an agent’s experience by using the learned world model. Another common approach is to leverage expert demonstrations or other task priors (Pinto and Gupta, 2015; Ha and Song, 2021; Xie et al., 2019; Schoettler et al., 2019; Sivakumar et al., 2022). James and Davison (2021); James et al. (2021) leverages a few demonstrations to increase the sample-efficiency of Q learning by focusing the learner on important aspects of the scene. Other approaches, as in locomotion, first utilize a simulator, then transfer to the real world (Tzeng et al., 2015; Akkaya et al., 2019; OpenAI et al., 2018; Irpan et al., 2020). Our work focuses on single-robot environments where the agent must learn through a small amount of interaction with the world. Meanwhile, the Google Arm Farm line of work by Levine et al. leverages over $5 8 0 \mathrm { k }$ grasp attempts gathered by 7 robots and collected over 4 months. We believe that a method such as Dreamer could benefit greatly from this scale of training data, however it is unlikely that works such as MT-OPT/QT-OPT Kalashnikov et al. (2018; 2021) would work well in the low data regime that Dreamer excels in. + +Model-based RL Due to its higher sample-efficiency over model-free methods, model-based RL is a promising approach to learning on real world robots (Deisenroth et al., 2013). A model based method first learns a dynamics model, which can then be used to plan actions (Nagabandi et al., 2019; Hafner et al., 2018; Chua et al., 2018; Nagabandi et al., 2017; Becker-Ehmck et al., 2020), or be used as a simulator to learn a policy network as in Dreamer (Hafner et al., 2019; 2020). One approach to tackle the high visual complexity of the world is to learn an action conditioned video prediction model (Finn and Levine, 2017; Ebert et al., 2018; Finn et al., 2016). One downside of this approach is the need to directly predict high dimensional observations, which can be computationally inefficient and easily drift. Dreamer learns a dynamics model in a latent space, allowing more efficient rollouts and avoids relying on high quality visual reconstructions for the policy. Another line of work proposes to learn latent dynamics models without having to reconstruct inputs (Deng et al., 2021; Okada and Taniguchi, 2021; Bharadhwaj et al., 2022; Paster et al., 2021), which we see as a promising approach for supporting moving view points in cluttered environments. + +# D Hyperparameters + +
NameSymbolValue
General
Replay capacity (FIFO)Start learningBatch sizeBatch lengthMLP sizeActivationBT10610432324× 512LayerNorm+ELU
World Model
RSSM sizeNumber of latentsClasses per latentKL balancing51232320.8
Actor Critic
Imagination horizonDiscountReturn lambdaTarget update intervalH?150.950.95100
All Optimizers
Gradient clippingLearning rateAdam epsilonE10010-410-6
+ +# E Environment and Hardware Details + +For every robot setup that involved vision (UR5, XArm, Sphero), we used a RealSense D435 camera positioned to offer a fixed 3rd person view of the scene. + +A1 We used the A1 quadrupedal robot by Unitree. The RL policy outputs actions at a frequency that is too high for the PD controller to track, which we overcome by lowpass filtering the action sequence. The joint range allows the legs to self-collide with the body, which can be damaging to the motors and increase battery consumption. We limited the joint range to decrease self-collisions. Finally, the EKF velocity estimator relies on foot-ground contact events to prevent significant drift in the estimates, so we employ a curriculum reward function that does not reward the robot for forward velocity until the robot is upright with extended legs. We also designed a shell which we 3D printed in order to better protect the cables and hardware and provide a smoother rolling over. + +XArm & UR5 We utilized slanted bins to prevent objects from leaving the work area during the long-running pick and place experiments on the UR5, which is common practice Levine et al. (2018); Kalashnikov et al. (2018). We also added a partition behind the setup to keep the background constant. It would be interesting to study how a gripper-mounted camera would impact policy performance Hsu et al. (2022), however we report strong results without this design choice. For the XArm we use the uFactory xArm Gripper. For the UR5, we use the Robotiq 2F-85 parallel jaw gripper. The bin locations are predetermined and provided as part of the environment to prevent the robot from colliding with the bin. In addition, movement in the $\textsf { Z }$ axis is only enabled while holding an object and the gripper automatically opens once above the other bin. + +Sphero We used a rectangular enclosure of $0 . 8 \times 0 . 8 \mathrm { { m ^ { 2 } } }$ to keep the sphero robot within the camera view. We used a simple OpenCV script to estimate the L2 distance between the Sphero and the goal position to provide a dense reward for policy optimization. This positional information was not provided to the agent, which it had to learn from the raw top-down images. \ No newline at end of file diff --git a/parse/dev/6at6rB3IZm/6at6rB3IZm.md b/parse/dev/6at6rB3IZm/6at6rB3IZm.md new file mode 100644 index 0000000000000000000000000000000000000000..b78c26bb15044cb94924c66c1353fac9136c7343 --- /dev/null +++ b/parse/dev/6at6rB3IZm/6at6rB3IZm.md @@ -0,0 +1,302 @@ +# Towards Understanding Grokking: An Effective Theory of Representation Learning + +Ziming Liu, Ouail Kitouni, Niklas Nolte, Eric J. Michaud, Max Tegmark, Mike Williams Department of Physics, Institute for AI and Fundamental Interactions, MIT {zmliu,kitouni,nnolte,ericjm,tegmark,mwill}@mit.edu + +# Abstract + +We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations whose training dynamics and dependence on training set size can be predicted by our effective theory in a toy setting. We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find representation learning to occur only in a “Goldilocks zone” (including comprehension and grokking) between memorization and confusion. We find on transformers the grokking phase stays closer to the memorization phase (compared to the comprehension phase), leading to delayed generalization. The Goldilocks phase is reminiscent of “intelligence from starvation” in Darwinian evolution, where resource limitations drive discovery of more efficient solutions. This study not only provides intuitive explanations of the origin of grokking, but also highlights the usefulness of physics-inspired tools, e.g., effective theories and phase diagrams, for understanding deep learning. + +# 1 Introduction + +Perhaps the central challenge of a scientific understanding of deep learning lies in accounting for neural network generalization. Power et al. [1] recently added a new puzzle to the task of understanding generalization with their discovery of grokking. Grokking refers to the surprising phenomenon of delayed generalization where neural networks, on certain learning problems, generalize long after overfitting their training set. It is a rare albeit striking phenomenon that violates common machine learning intuitions, raising three key puzzles: + +Q1 The origin of generalization: When trained on the algorithmic datasets where grokking occurs, how do models generalize at all? +Q2 The critical training size: Why does the training time needed to “grok” (generalize) diverge as the training set size decreases toward a critical point? +Q3 Delayed generalization: Under what conditions does delayed generalization occur? + +We provide evidence that representation learning is central to answering each of these questions. Our answers can be summarized as follows: + +A1 Generalization can be attributed to learning a good representation of the input embeddings, i.e., a representation that has the appropriate structure for the task and which can be predicted from the theory in Section 3. See Figures 1 and 2. +A2 The critical training set size corresponds to the least amount of training data that can determine such a representation (which, in some cases, is unique up to linear transformations). + +![](images/e3eaca253887fd618f7f1aec0eaf522e3e511cf97173030ad9ee56c1a327e83f.jpg) +Figure 1: Visualization of the first two principal components of the learned input embeddings at different training stages of a transformer learning modular addition. We observe that generalization coincides with the emergence of structure in the embeddings. See Section 4.2 for the training details. + +A3 Grokking is a phase between “comprehension” and “memorization” phases and it can be remedied with proper hyperparmeter tuning, as illustrated by the phase diagrams in Figure 6. + +This paper is organized as follows: In Section 2, we introduce the problem setting and build a simplified toy model. In Section 3, we will use an effective theory approach, a useful tool from theoretical physics, to shed some light on questions Q1 and Q2 and show the relationship between generalization and the learning of structured representations. In Section 4, we explain Q3 by displaying phase diagrams from a grid search of hyperparameters and show how we can “de-delay” generalization by following intuition developed from the phase diagram. We discuss related work in Section 5, followed by conclusions in Section 6.1 + +# 2 Problem Setting + +Power et al. [1] observe grokking on a less common task – learning “algorithmic” binary operations. Given some binary operation ◦, a network is tasked with learning the map $( a , b ) \mapsto c$ where $c = a \circ b$ . They use a decoder-only transformer to predict the second to last token in a tokenized equation of the form “ ”. Each token is represented as a 256-dimensional embedding vector. The embeddings are learnable and initialized randomly. After the transformer, a final linear layer maps the output to class logits for each token. + +Toy Model We primarily study grokking in a simpler toy model, which still retains the key behaviors from the setup of [1]. Although [1] treated this as a classification task, we study both regression (mean-squared error) and classification (cross-entropy). The basic setup is as follows: our model takes as input the symbols $a , b$ and maps them to trainable embedding vectors $\mathbf { E } _ { a } , \mathbf { E } _ { b } \in \mathbb { R } ^ { d _ { \mathrm { i n } } }$ . It then sums $\mathbf { E } _ { a } , \mathbf { E } _ { b }$ and sends the resulting vector through a “decoder” MLP. The target output vector, denoted $\mathbf { Y } _ { c } \in \mathbb { R } ^ { d _ { \mathrm { o u t } } }$ is a fixed random vector (regression task) or a one-hot vector (classification task). Our model architecture can therefore be compactly described as $( a , b ) \mapsto \operatorname { D e c } ( \mathbf { E } _ { a } + \mathbf { E } _ { b } )$ , where the embeddings $\mathbf { E _ { * } }$ and the decoder are trainable. Despite its simplicity, this toy model can generalize to all abelian groups (discussed in Appendix B). In sections 3-4.1, we consider only the binary operation of addition. We consider modular addition in Section 4.2 to generalize some of our results to a transformer architecture and study general non-abelian operations in Appendix H. + +Dataset In our toy setting, we are concerned with learning the addition operation. A data sample corresponding to $i + j$ is denoted as $( i , j )$ for simplicity. If $i , j \in \{ 0 , \ldots , p - 1 \}$ , there are in total $p ( p + 1 ) / 2$ different samples since we consider $i + j$ and $j + i$ to be the same sample. A dataset $D$ is a set of non-repeating data samples. We denote the full dataset as $D _ { 0 }$ and split it into a training dataset $D$ and a validation dataset $D ^ { \prime }$ , i.e., $D \bigcup D ^ { \prime } = D _ { 0 } , D \bigcap D ^ { \prime } = \emptyset$ . We define training data fraction $= | D | / | D _ { 0 } |$ where $| \cdot |$ denotes the cardinality of the set. + +![](images/b448f7b12d80bc5a5de6a38e337185955ee926dc22a5e7bfb1f8a7006e5d2a6f.jpg) + +Figure 2: Visualization of the learned set of embeddings $( p = 1 1 $ ) and the decoder function associated with it for the case of 2D embeddings. Axes refer to each dimension of the learned embeddings. The decoder is evaluated on a grid of points in embedding-space and the color at each point represents the highest probability class. For visualization purposes, the decoder is trained on inputs of the form $( \mathbf { E } _ { i } + \mathbf { \bar { E } } _ { j } ) / 2$ . One can read off the output of the decoder when fed the operation $i \circ j$ from this figure simply by taking the midpoint between the respective embeddings of $i$ and $j$ . + +![](images/e72e69e51ef60695b4b64c66126a4e2a24d598c9eb27d1a2a44995c7866ba4af.jpg) +(d) Generalization in toy modular addition + +# 3 Why Generalization Occurs: Representations and Dynamics + +We can see that generalization appears to be linked to the emergence of highly-structured embeddings in Figure 2. In particular, Figure 2 (a, b) shows parallelograms in toy addition, and (c, d) shows a circle in toy modular addition. We now restrict ourselves to the toy addition setup and formalize a notion of representation quality and show that it predicts the model’s performance. We then develop a physics-inspired effective theory of learning which can accurately predict the critical training set size and training trajectories of representations. The concept of an effective theory in physics is similar to model reduction in computational methods in that it aims to describe complex phenomena with simple yet intuitive pictures. In our effective theory, we will model the dynamics of representation learning not as gradient descent of the true task loss but rather a simpler effective loss function $\ell _ { \mathrm { e f f } }$ which depends only on the representations in embedding space and not on the decoder. + +# 3.1 Representation quality predicts generalization for the toy model + +A rigorous definition for structure in the learned representation is necessary. We propose the following definition, + +Definition 1. $( i , j , m , n )$ is a $\delta$ -parallelogram in the representation $\mathbf { R } \equiv \left[ \mathbf { E } _ { 0 } , \cdots , \mathbf { E } _ { p - 1 } \right] i f$ + +$$ +| ( \mathbf { E } _ { i } + \mathbf { E } _ { j } ) - ( \mathbf { E } _ { m } + \mathbf { E } _ { n } ) | \leq \delta . +$$ + +In the following derivations, we can take $\delta$ , which is a small threshold to tolerate numerical errors, to be zero. + +Proposition 1. When the training loss is zero, any parallelogram $( i , j , m , n )$ in representation R satisfies $i + j = m + n$ . + +Proof. Suppose that this is not the case, i.e., suppose $\mathbf { E } _ { i } + \mathbf { E } _ { j } = \mathbf { E } _ { m } + \mathbf { E } _ { n }$ but $i + j \neq m + n$ , then $\mathbf { Y } _ { i + j } = \bar { \mathrm { D e c } } ( \mathbf { E } _ { i } + \mathbf { E } _ { j } ) = \mathrm { D e c } ( \mathbf { E } _ { m } + \mathbf { E } _ { n } ) = \bar { \mathbf { Y } } _ { m + n }$ where the first and last equalities come from the zero training loss assumption. However, since $i + j \neq m + n$ , we have $\mathbf { Y } _ { i + j } \neq \mathbf { Y } _ { n + m }$ (almost surely in the regression task), a contradiction. □ + +![](images/d8a505118aa04e4fe7eee8c0bb3a01ebac68a692b2db9a5ad5553b7c00dbb680.jpg) +Figure 3: We compute accuracy (of the full dataset) either measured empirically Acc, or predicted from the representation of the embeddings $\widehat { \mathrm { A c c } }$ . These two accuracies as a function of training data fraction are plotted in (a)(b), and their agreement is shown in (c). + +It is convenient to define the permissible parallelogram set associated with a training dataset $D$ (“permissible” means consistent with $100 \%$ training accuracy) as + +$$ +P _ { 0 } ( D ) = \{ ( i , j , m , n ) | ( i , j ) \in D , ( m , n ) \in D , i + j = m + n \} . +$$ + +For simplicity, we denote $P _ { 0 } \equiv P _ { 0 } ( D _ { 0 } )$ . Given a representation $\mathbf { R }$ , we can check how many permissible parallelograms actually exist in $\mathbf { R }$ within error $\delta$ , so we define the parallelogram set corresponding to $\mathbf { R }$ as + +$$ +\begin{array} { r } { P ( \mathbf { R } , \delta ) = \{ ( i , j , m , n ) | ( i , j , m , n ) \in P _ { 0 } , | ( \mathbf { E } _ { i } + \mathbf { E } _ { j } ) - ( \mathbf { E } _ { m } + \mathbf { E } _ { n } ) | \leq \delta \} . } \end{array} +$$ + +For brevity we will write $P ( \mathbf { R } )$ , suppressing the dependence on $\delta$ . We define the representation quality index (RQI) as + +$$ +\mathrm { R Q I } ( \mathbf { R } ) = \frac { | P ( \mathbf { R } ) | } { | P _ { 0 } | } \in [ 0 , 1 ] . +$$ + +We will use the term linear representation or linear structure to refer to a representation whose embeddings are of the form $\mathbf { E } _ { k } = \mathbf { a } + k \mathbf { b } \left( k = 0 , \cdots , p - 1 ; \mathbf { a } , \mathbf { b } \in \mathbb { R } ^ { d _ { \mathrm { i n } } } \right)$ . A linear representation has $\operatorname { R Q I } = 1$ , while a random representation (sampled from, say, a normal dstribution) has $\mathrm { R Q I } = 0$ with high probability. + +Quantitatively, we denote the “predicted accuracy” $\widehat { \mathrm { A c c } }$ as the accuracy achievable on the whole dataset given the representation $\mathbf { R }$ (see Appendix D for the full details). In Figure 3, we see that the predicted $\widehat { \mathrm { A c c } }$ aligns well with the true accuracy Acc, establishing good evidence that structured representation of input embeddings leads to generalization. We use an example to illustrate the origin of generalization here. In the setup of Figure 2 (b), suppose the decoder can achieve zero training loss and $\mathbf { E } _ { 6 } + \mathbf { E } _ { 8 }$ is a training sample hence $\mathrm { D e c } ( { \bf E } _ { 6 } + { \bf E } _ { 8 } ) = { \bf Y } _ { 1 4 }$ . At validation time, the decoder is tasked with predicting a validation sample $\mathbf { E } _ { 5 } + \mathbf { E } _ { 9 }$ . Since $( 5 , 9 , 6 , 8 )$ forms a parallelogram such that $\mathbf { E } _ { 5 } + \mathbf { E } _ { 9 } = \mathbf { E } _ { 6 } + \mathbf { E } _ { 8 }$ , the decoder can predict the validation sample correctly because $\mathrm { D e c } ( \mathbf { E } _ { 5 } + \mathbf { E } _ { 9 } ) = \mathrm { D e c } ( \mathbf { E } _ { 6 } + \mathbf { E } _ { 8 } ) = \mathbf { Y } _ { 1 4 } ,$ . + +# 3.2 The dynamics of embedding vectors + +Suppose that we have an ideal model $\mathcal { M } ^ { * } = ( \mathrm { D e c } ^ { * } , { \bf R } ^ { * } )$ such that:2 + +• (1) ${ \mathfrak { M } } ^ { * }$ can achieve zero training loss; +• (2) ${ \mathfrak { M } } ^ { * }$ has an injective decoder, i.e., $\mathrm { D e c } ^ { * } ( \mathbf { x } _ { 1 } ) \neq \mathrm { D e c } ^ { * } ( \mathbf { x } _ { 2 } )$ for any $\mathbf { x } _ { 1 } \neq \mathbf { x } _ { 2 }$ . + +Then Proposition 2 provides a mechanism for the formation of parallelograms. + +![](images/8682de6492ababd71a56cbc9713e1dbc840e7595ade48ed9156adc9f347f337d.jpg) +Figure 4: (a) The effective theory predicts a phase transition in the probability of obtaining a linear representation around $r _ { c } = 0 . 4$ . (b) Empirical results display a phase transition of RQI around $r _ { c } = 0 . 4$ , in agreement with the theory (the blue line shows the median of multiple random seeds). The evolution of 1D representations predicted by the effective theory or obtained from neural network training (shown in (c) and (d) respectively) agree creditably well. + +Proposition 2. If a training set $D$ contains two samples $( i , j )$ and $( m , n )$ with $i + j = m + n $ then ${ \mathfrak { M } } ^ { * }$ learns a representation $\mathbf { R } ^ { * }$ such that $\mathbf { E } _ { i } + \mathbf { E } _ { j } = \mathbf { E } _ { m } + \mathbf { E } _ { n }$ , i.e., $( i , j , m , n )$ forms $a$ parallelogram. + +Proof. Due to the zero training loss assumption, we have $\mathrm { D e c } ^ { * } ( \mathbf { E } _ { i } + \mathbf { E } _ { j } ) = \mathbf { Y } _ { i + j } = \mathbf { Y } _ { m + n } =$ $\mathrm { D e c } ^ { * } ( \mathbf { E } _ { m } + \mathbf { E } _ { n } )$ . Then the injectivity of ${ \mathrm { D e c } } ^ { * }$ implies $\mathbf { E } _ { i } + \mathbf { E } _ { j } = \mathbf { E } _ { m } + \mathbf { \bar { E } } _ { n }$ . + +The dynamics of the trained embedding vectors are determined by various factors interacting in complex ways, for instance: the details of the decoder architecture, the optimizer hyperparameters, and the various kinds of implicit regularization induced by the training procedure. We will see that the dynamics of normalized quantities, namely, the normalized embeddings at time $t$ , defined as $\begin{array} { r } { \tilde { \mathbf { E } } _ { k } ^ { ( t ) } = \frac { \mathbf { E } _ { k } ^ { ( t ) } - \mu _ { t } } { \sigma _ { t } } } \end{array}$ , where $\begin{array} { r } { \mu _ { t } = \frac { 1 } { p } \sum _ { k } \mathbf { E } _ { k } ^ { ( t ) } } \end{array}$ and $\begin{array} { r } { \sigma _ { t } = \frac { 1 } { p } \sum _ { k } | \mathbf { E } _ { k } ^ { ( t ) } - \mu _ { t } | ^ { 2 } } \end{array}$ , can be qualitatively described by a simple effective loss (in the physics effective theory sense). We will assume that the normalized embedding vectors obey a gradient flow for an effective loss function of the form + +$$ +\frac { d \tilde { \mathbf { E } } _ { i } } { d t } = - \frac { \partial \ell _ { \mathrm { e f f } } } { \partial \tilde { \mathbf { E } } _ { i } } , +$$ + +$$ +\ell _ { \mathrm { e f f } } = \frac { \ell _ { 0 } } { Z _ { 0 } } , \quad \ell _ { 0 } \equiv \sum _ { ( i , j , m , n ) \in P _ { 0 } ( D ) } | \tilde { \mathbf { E } } _ { i } + \tilde { \mathbf { E } } _ { j } - \tilde { \mathbf { E } } _ { m } - \tilde { \mathbf { E } } _ { n } | ^ { 2 } / | P _ { 0 } ( D ) | , \quad Z _ { 0 } \equiv \sum _ { k } | \tilde { \mathbf { E } } _ { k } | ^ { 2 } , +$$ + +where $| \cdot |$ denotes Euclidean vector norm. Note that the embeddings do not collapse to the trivial solution $\mathbf { E } _ { 0 } = \cdot \cdot \cdot = \mathbf { E } _ { p - 1 } = 0$ unless initialized as such, because two conserved quantities exist, as proven in Appendix F: + +$$ +\mathbf { C } = \sum _ { k } \mathbf { E } _ { k } , \quad Z _ { 0 } = \sum _ { k } | \mathbf { E } _ { k } | ^ { 2 } . +$$ + +We shall now use the effective dynamics to explain empirical observations such as the existence of a critical training set size for generalization. + +Degeneracy of ground states (loss optima) We define ground states as those representations satisfying $\ell _ { \mathrm { e f f } } = 0$ , which requires the following linear equations to hold: + +$$ +A ( P ) = \{ \mathbf { E } _ { i } + \mathbf { E } _ { j } = \mathbf { E } _ { m } + \mathbf { E } _ { n } | ( i , j , m , n ) \in P \} . +$$ + +Since each embedding dimension obeys the same set of linear equations, we will assume, without loss of generality, that $d _ { \mathrm { i n } } = 1$ . The dimension of the null space of $A ( P )$ , denoted as $n _ { 0 }$ , is the number of degrees of freedom of the ground states. Given a set of parallelograms implied by a training dataset $D$ , the nullity of $A ( P ( D ) )$ could be obtained by computing the singular values $0 \leq \sigma _ { 1 } \leq \cdot \cdot \cdot \leq \sigma _ { p }$ We always have $n _ { 0 } \geq 2$ , i.e., $\sigma _ { 1 } = \sigma _ { 2 } = 0$ because the nullity of $A ( P _ { 0 } )$ , the set of linear equations given by all possible parallelograms, is $\mathrm { N u l l i t y } ( A ( P _ { 0 } ) ) = 2$ which can be attributed to two degrees of freedom (translation and scaling). If $n _ { 0 } = 2$ , the representation is unique up to translations and scaling factors, and the embeddings have the form $\mathbf { E } _ { k } = \mathbf { a } + k \mathbf { b }$ . Otherwise, when $n _ { 0 } > 2$ , the representation is not constrained enough such that all the embeddings lie on a line. + +We present theoretical predictions alongside empirical results for addition $\boldsymbol { p } = 1 0 ^ { \circ } ,$ ) in Figure 4. As shown in Figure 4 (a), our effective theory predicts that the probability that the training set implies a unique linear structure (which would result in perfect generalization) depends on the training data fraction and has a phase transition around $r _ { c } = 0 . 4$ . Empirical results from training different models are shown in Figure 4 (b). The number of steps to reach $\mathrm { R Q I } > 0 . 9 5$ is seen to have a phase transition at $r _ { c } = 0 . 4$ , agreeing with the proposed effective theory and with the empirical findings in [1]. + +Time towards the linear structure We define the Hessian matrix of $\ell _ { 0 }$ as + +$$ +\mathbf { H } _ { i j } = \frac { 1 } { Z _ { 0 } } \frac { \partial ^ { 2 } \ell _ { 0 } } { \partial \mathbf { E } _ { i } \partial \mathbf { E } _ { j } } , +$$ + +Note that $\begin{array} { r } { \ell _ { \mathrm { e f f } } = \frac { 1 } { 2 } \mathbf { R } ^ { T } \mathbf { H } \mathbf { R } } \end{array}$ , $\mathbf { R } = [ \mathbf { E } _ { 0 } , \mathbf { E } _ { 1 } , \cdots , \mathbf { E } _ { p - 1 } ]$ , so the gradient descent is linear, i.e., + +$$ +\frac { d \mathbf { R } } { d t } = - \mathbf { H } \mathbf { R } . +$$ + +If $\mathbf { H }$ has eigenvalues $\lambda _ { i } = \sigma _ { i } ^ { 2 }$ (sorted in increasing order) and eigenvectors $\bar { \bf v } _ { i }$ , and we have the initial condition $\begin{array} { r } { { \bf R } ( t = 0 ) = \sum _ { i } a _ { i } { \bar { \bf v } } _ { i } } \end{array}$ , then we have $\begin{array} { r } { { \bf R } ( t ) = \sum _ { i } a _ { i } \bar { \bf v } _ { i } e ^ { - \lambda _ { i } t } } \end{array}$ . The first two eigenvalues vanish and $t _ { h } = 1 / \lambda _ { 3 }$ determines the timescale for the slowest component to decrease by a factor of $e$ . We call $\lambda _ { 3 }$ the grokking rate. When the step size is $\eta$ , the corresponding number of steps is $n _ { h } = t _ { h } / \eta = 1 / ( \lambda _ { 3 } \bar { \eta _ { } } )$ . + +We verify the above analysis with empirical results. Figure 4 (c)(d) show the trajectories obtained from the effective theory and from neural network training, respectively. The 1D neural representation in Figure 4 (d) are manually normalized to zero mean and unit variance. The two trajectories agree qualitatively, and it takes about $3 n _ { h }$ steps for two trajectories to converge to the linear structure. The quantitative differences might be due to the absence of the decoder in the effective theory, which assumes the decoder to take infinitesimal step sizes. + +Dependence of grokking on data size Note that $\ell _ { \mathrm { e f f } }$ involves averaging over parallelograms in the training set, it is dependent on training data size, so is $\lambda _ { 3 }$ . In Figure 5 (a), we plot the dependence of $\lambda _ { 3 }$ on training data fraction. There are many datasets with the same data size, so $\lambda _ { 3 }$ is a probabilistic function of data size. + +Two insights on grokking can be extracted from this plot: (i) When the data fraction is below some threshold (around 0.4), $\lambda _ { 3 }$ is zero with high probability, corresponding to no generalization. This again verifies our critical point in Figure 4. (ii) When data size is above the threshold, $\lambda _ { 3 }$ (on average) is an increasing function of data size. This implies that grokking time $t \sim 1 / \lambda _ { 3 }$ decreases as training data size becomes larger, an important observation from [1]. + +To verify our effective theory, we compare the grokking steps obtained from real neural network training (defined as steps to $\mathrm { R Q I } > 0 . 9 5 $ , and those predicted by our theory $\begin{array} { r } { t _ { \mathrm { t h } } \sim \frac { 1 } { \lambda _ { 3 } \eta } } \end{array}$ $\dot { \eta }$ is the embedding learning rate), shown in Figure 5 (b). The theory agrees qualitatively with neural networks, showing the trend of decreasing grokking steps as increasing data size. The quantitative differences might be explained as the gap between our effective loss and actual loss. + +Limitations of the effective theory While our theory defines an effective loss based on the Euclidean distance between embeddings $\mathbf { E } _ { i } + \mathbf { E } _ { j }$ and $ { \mathbf { E } } _ { n } + { \mathbf { E } } _ { m }$ , one could imagine generalizing the theory to define a broader notion of parallogram given by some other metric on the representation space. For instance, if we have a decoder like in Figure 2 (d) then the distance between distinct representations within the same “pizza slice” is low, meaning that representations arranged not in parallelograms w.r.t. the Euclidean metric may be parallelograms with respect to the metric defined by the decoder. + +# 4 Delayed Generalization: A Phase Diagram + +So far, we have (1) observed empirically that generalization on algorithmic datasets corresponds with the emergence of well-structured representations, (2) defined a notion of representation quality in a toy setting and shown that it predicts generalization, and (3) developed an effective theory to describe the learning dynamics of the representations in the same toy setting. We now study how optimizer hyperparameters affect high-level learning performance. In particular, we develop phase diagrams for how learning performance depends on the representation learning rate, decoder learning rate and the decoder weight decay. These parameters are of interest since they most explicitly regulate a kind of competition between the encoder and decoder, as we elaborate below. + +![](images/790df465da9461408a8b7bb795949bbbd6da09d372793ab426437bf1da28416a.jpg) +Figure 5: Effective theory explains the dependence of grokking time on data size, for the addition task. (a) Dependence of $\lambda _ { 3 }$ on training data fraction. Above the critical data fraction (around 0.4), as data size becomes larger, $\lambda _ { 3 }$ increases hence grokking time $t \sim 1 / \lambda _ { 3 }$ (predicted by our effective theory) decreases. (b) Comparing grokking steps (defined as $\mathrm { R Q I } > 0 . 9 5 ) ,$ predicted by the effective theory with real neural network results. $\eta = 1 \bar { 0 } ^ { - 3 }$ is the learning rate of the embeddings. + +# 4.1 Phase diagram of a toy model + +Training details We update the representation and the decoder with different optimizers. For the 1D embeddings, we use the Adam optimizer with learning rate $[ 1 0 ^ { - 5 } , 1 0 ^ { - 2 } ]$ and zero weight decay. For the decoder, we use an AdamW optimizer with the learning rate in $[ 1 0 ^ { - 5 } , 1 0 ^ { - 2 } ]$ and the weight decay in [0, 10] (regression) or $[ 0 , 2 0 ]$ (classification). For training/validation spliting, we choose 45/10 for non-modular addition $\begin{array} { r } { p = 1 0 , } \end{array}$ ) and 24/12 for the permutation group $S _ { 3 }$ . We hard-code addition or matrix multiplication (details in Appendix H) in the decoder for the addition group and the permutation group, respectively. + +For each choice of learning rate and weight decay, we compute the number of steps to reach high $( 9 0 \% )$ training/validation accuracy. The 2D plane is split into four phases: comprehension, grokking, memorization and confusion, defined in Table 1 in Appendix A. Both comprehension and grokking are able to generalize (in the “Goldilocks zone”), although the grokking phase has delayed generalization. Memorization is also called overfitting, and confusion means failure to even memorize training data. Figure 6 shows the phase diagrams for the addition group and the permutation group. They display quite rich phenomena. + +Competition between representation learning and decoder overfitting In the regression setup of the addition dataset, we show how the competition between representation learning and decoder learning (which depend on both learning rate and weight decay, among other things) lead to different learning phases in Figure 6 (a). As expected, a fast decoder coupled with slow representation learning (bottom right) lead to memorization. In the opposite extreme, although an extremely slow decoder coupled with fast representation learning (top left) will generalize in the end, the generalization time is long due to the inefficient decoder training. The ideal phase (comprehension) requires representation learning to be faster, but not too much, than the decoder. + +Drawing from an analogy to physical systems, one can think of embedding vectors as a group of particles. In our effective theory from Section 3.2, the dynamics of the particles are described only by their relative positions, in that sense, structure forms mainly due to inter-particle interactions (in reality, these interactions are mediated by the decoder and the loss). The decoder plays the role of an environment exerting external forces on the embeddings. If the magnitude of the external forces are small/large one can expect better/worse representations. + +![](images/9369694209dd16f219ef635b4e040381732aee4942d26c6e5b15ac545eac2aa0.jpg) +Figure 6: Phase diagrams of learning for the addition group and the permutation group. (a) shows the competition between representation and decoder. (b)(c)(d): each phase diagram contains four phases: comprehension, grokking, memorization and confusion, defined in Table 1. In (b)(c)(d), grokking is sandwiched between comprehension and memorization. + +Universality of phase diagrams We fix the embedding learning rate to be $1 0 ^ { - 3 }$ and sweep instead decoder weight decay in Figure 6 (b)(c)(d). The phase diagrams correspond to addition regression (b), addition classification (c) and permutation regression (d), respectively. Common phenomena emerge from these different tasks: (i) they all include four phases; (ii) The top right corner (a fast and capable decoder) is the memorization phase; (iii) the bottom right corner (a fast and simple decoder) is the confusion phase; (iv) grokking is sandwiched between comprehension and memorization, which seems to imply that it is an undesirable phase that stems from improperly tuned hyperparameters. + +# 4.2 Beyond the toy model + +We conjecture that many of the principles which we saw dictate the training dynamics in the toy model also apply more generally. Below, we will see how our framework generalizes to transformer architectures for the task of addition modulo $p$ , a minimal reproducible example of the original grokking paper [1]. + +We first encode $p = 5 3$ integers into 256D learnable embeddings, then pass two integers to a decoderonly transformer architecture. For simplicity, we do not encode the operation symbols here. The outputs from the last layer are concatenated and passed to a linear layer for classification. Training both the encoder and the decoder with the same optimizer (i.e., with the same hyperparameters) leads to the grokking phenomenon. Generalization appears much earlier once we lower the effective decoder capacity with weight decay (full phase diagram in Figure 7). + +Early on, the model is able to perfectly fit the training set while having no generalization. We study the embeddings at different training times and find that neither PCA (shown in Figure 1) nor t-SNE (not shown here) reveal any structure. Eventually, validation accuracy starts to increase, and perfect generalization coincides with the PCA projecting the embeddings into a circle in 2D. Of course, no choice of dimensionality reduction is guaranteed to find any structure, and thus, it is challenging to show explicitly that generalization only occurs when a structure exists. Nevertheless, the fact that, when coupled with the implicit regularization of the optimizer for sparse solutions, such a clear structure appears in a simple PCA so quickly at generalization time suggests that our analysis in the toy setting is applicable here as well. This is also seen in the evolution of the entropy of the explained variance ratio in the PCA of the embeddings (defined as $\begin{array} { r } { S = - \sum _ { i } \sigma _ { i } \log \sigma _ { i } } \end{array}$ where $\sigma _ { i }$ is the fractional variance explained by the ith principal component). As seen in Figure 7, the entropy increases up to generalization time then decreases drastically afterwards which would be consistent with the conjecture that generalization occurs when a low-dimensional structure is discovered. The decoder then primarily relies on the information in this low-dimensional manifold and essentially “prunes” the rest of the high-dimensional embedding space. Another interesting insight appears when we project the embeddings at initialization onto the principal axes at the end of training. Some of the structure required for generalization exists before training hinting at a connection with the Lottery Ticket Hypothesis. See Appendix K for more details. + +![](images/f97ba3b50d927fef5d6356bddf07766187a7aa148dc09b2096966b4a90acf17f.jpg) +Figure 7: Left: Evolution of the effective dimension of the embeddings (defined as the exponential of the entropy) during training and evaluated over 100 seeds. Center: Effect of dropout on speeding up generalization. Right: Phase diagram of the transformer architecture. A scan is performed over the weight decay and learning rate of the decoder while the learning rate of the embeddings is kept fixed at $1 \mathrm { { 0 } ^ { - 3 } }$ (with zero weight decay). + +In Figure 7 (right), we show a comparable phase diagram to Figure 6 evaluated now in the transformer setting. Note that, as opposed to the setting in [1], weight decay has only been applied to the decoder and not to the embedding layer. Contrary to the toy model, a certain amount of weight decay proves beneficial to generalization and speeds it up significantly. We conjecture that this difference comes from the different embedding dimensions. With a highly over-parameterized setting, a non-zero weight decay gives a crucial incentive to reduce complexity in the decoder and help generalize in fewer steps. This is subject to further investigation. We also explore the effect of dropout layers in the decoder blocks of the transformer. With a significant dropout rate, the generalization time can be brought down to under $1 0 ^ { 3 }$ steps and the grokking phenomenon vanishes completely. The overall trend suggests that constraining the decoder with the same tools used to avoid overfitting reduces generalization time and can avoid the grokking phenomenon. This is also observed in an image classification task where we were able to induce grokking. See Appendix J for more details. + +# 4.3 Grokking Experiment on MNIST + +We now demonstrate, for the first time, that grokking (significantly delayed generalization) is a more general phenomenon in machine learning that can occur not only on algorithmic datasets, but also on mainstream benchmark datasets. In particular, we exhibit grokking on MNIST in Figure 8 and demonstrate that we can control grokking by varying optimization hyperparameters. More details on the experimental setup are in Appendix J. + +# 5 Related work + +Relatively few works have analyzed the phenomenon of grokking. [2] describe the circuit that transformers use to perform modular addition, track its formation over training, and broadly suggest that grokking is related to the phenomenon of “phase changes” in neural network training. [3, 4] provided earlier speculative, informal conjectures on grokking [3, 4]. Our work is related to the following broad research directions: + +![](images/ce9758fc290cfb65a5d604010e550b1f2db4f6660cf61449cd96630f3e657035.jpg) +Figure 8: Left: Training curves for a run on MNIST, in the setting where we observe grokking. Right: Phase diagram with the four phases of learning dynamics on MNIST. + +Learning mathematical structures [5] trains a neural network to learn arithmetic operation from pictures of digits, but they do not observe grokking due to their abundant training data. Beyond arithmetic relations, machine learning has been applied to learn other mathematical structures, including geometry [6], knot theory [7] and group theory [8]. + +Double descent Grokking is somewhat reminiscent of the phenomena of “epoch-wise” double descent [9], where generalization can improve after a period of overfitting. [10] find that regularization can mitigate double descent, similar perhaps to how weight decay influences grokking. + +Representation learning Representation learning lies at the core of machine learning [11–14]. Representation quality is usually measured by (perhaps vague) semantic meanings or performance on downstream tasks. In our study, the simplicity of arithmetic datasets allows us to define representation quality and study evolution of representations in a quantitative way. + +Physics of learning Physics-inspired tools have proved to be useful in understanding deep learning from a theoretical perspective. These tools include effective theories [15, 16], conservation laws [17] and free energy principle [18]. In addition, statistical physics has been identified as a powerful tool in studying generalization in neural networks [19–22]. Our work connects a low-level understanding of models with their high-level performance. In a recent work, researchers at Anthropic [23], connect a sudden decrease in loss during training with the emergence of induction heads within their models. They analogize their work to statistical physics, since it bridges a “microscopic”, mechanistic understanding of networks with “macroscopic” facts about overall model performance. + +# 6 Conclusion + +We have shown how, in both toy models and general settings, that representation enables generalization when it reflects structure in the data. We developed an effective theory of representation learning dynamics (in a toy setting) which predicts the critical dependence of learning on the training data fraction. We then presented four learning phases (comprehension, grokking, memorization and confusion) which depend on the decoder capacity and learning speed (given by, among other things, learning rate and weight decay) in decoder-only architectures. While we have mostly focused on a toy model, we find preliminary evidence that our results generalize to the setting of [1]. + +Our work can be viewed as a step towards a statistical physics of deep learning, connecting the “microphysics” of low-level network dynamics with the “thermodynamics” of high-level model behavior. We view the application of theoretical tools from physics, such as effective theories [24], to be a rich area for further work. The broader impact of such work, if successful, could be to make models more transparent and predictable [23, 25, 26], crucial to the task of ensuring the safety of advanced AI systems. + +# References + +[1] Alethea Power, Yuri Burda, Harri Edwards, Igor Babuschkin, and Vedant Misra. Grokking: Generalization beyond overfitting on small algorithmic datasets. arXiv preprint arXiv:2201.02177, 2022. +[2] Neel Nanda and Tom Lieberum. A mechanistic interpretability analysis of grokking, 2022. 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Lee, Edward Moroshko, Pedro Savarese, Itay Golan, Daniel Soudry, and Nathan Srebro. Kernel and rich regimes in overparametrized models. In Jacob Abernethy and Shivani Agarwal, editors, Proceedings of Thirty Third Conference on Learning Theory, volume 125 of Proceedings of Machine Learning Research, pages 3635–3673. PMLR, 09–12 Jul 2020. URL https://proceedings.mlr.press/v125/woodworth20a. html. + +# Checklist + +1. For all authors... + +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] +(b) Did you describe the limitations of your work? [Yes] +(c) Did you discuss any potential negative societal impacts of your work? [N/A] +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? 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[N/A] \ No newline at end of file diff --git a/parse/dev/6at6rB3IZm/6at6rB3IZm_content_list.json b/parse/dev/6at6rB3IZm/6at6rB3IZm_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d3abd2479ca4563b763900faf734415399ce7f94 --- /dev/null +++ b/parse/dev/6at6rB3IZm/6at6rB3IZm_content_list.json @@ -0,0 +1,1438 @@ +[ + { + "type": "text", + "text": "Towards Understanding Grokking: An Effective Theory of Representation Learning ", + "text_level": 1, + "bbox": [ + 207, + 122, + 790, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ziming Liu, Ouail Kitouni, Niklas Nolte, Eric J. Michaud, Max Tegmark, Mike Williams Department of Physics, Institute for AI and Fundamental Interactions, MIT {zmliu,kitouni,nnolte,ericjm,tegmark,mwill}@mit.edu ", + "bbox": [ + 191, + 224, + 807, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 303, + 535, + 320 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations whose training dynamics and dependence on training set size can be predicted by our effective theory in a toy setting. We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find representation learning to occur only in a “Goldilocks zone” (including comprehension and grokking) between memorization and confusion. We find on transformers the grokking phase stays closer to the memorization phase (compared to the comprehension phase), leading to delayed generalization. The Goldilocks phase is reminiscent of “intelligence from starvation” in Darwinian evolution, where resource limitations drive discovery of more efficient solutions. This study not only provides intuitive explanations of the origin of grokking, but also highlights the usefulness of physics-inspired tools, e.g., effective theories and phase diagrams, for understanding deep learning. ", + "bbox": [ + 232, + 334, + 766, + 554 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 577, + 310, + 594 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Perhaps the central challenge of a scientific understanding of deep learning lies in accounting for neural network generalization. Power et al. [1] recently added a new puzzle to the task of understanding generalization with their discovery of grokking. Grokking refers to the surprising phenomenon of delayed generalization where neural networks, on certain learning problems, generalize long after overfitting their training set. It is a rare albeit striking phenomenon that violates common machine learning intuitions, raising three key puzzles: ", + "bbox": [ + 176, + 608, + 825, + 691 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Q1 The origin of generalization: When trained on the algorithmic datasets where grokking occurs, how do models generalize at all? \nQ2 The critical training size: Why does the training time needed to “grok” (generalize) diverge as the training set size decreases toward a critical point? \nQ3 Delayed generalization: Under what conditions does delayed generalization occur? ", + "bbox": [ + 191, + 702, + 825, + 781 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We provide evidence that representation learning is central to answering each of these questions. Our answers can be summarized as follows: ", + "bbox": [ + 173, + 791, + 823, + 819 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A1 Generalization can be attributed to learning a good representation of the input embeddings, i.e., a representation that has the appropriate structure for the task and which can be predicted from the theory in Section 3. See Figures 1 and 2. \nA2 The critical training set size corresponds to the least amount of training data that can determine such a representation (which, in some cases, is unique up to linear transformations). ", + "bbox": [ + 192, + 830, + 825, + 905 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/e3eaca253887fd618f7f1aec0eaf522e3e511cf97173030ad9ee56c1a327e83f.jpg", + "image_caption": [ + "Figure 1: Visualization of the first two principal components of the learned input embeddings at different training stages of a transformer learning modular addition. We observe that generalization coincides with the emergence of structure in the embeddings. See Section 4.2 for the training details. " + ], + "image_footnote": [], + "bbox": [ + 183, + 90, + 816, + 250 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A3 Grokking is a phase between “comprehension” and “memorization” phases and it can be remedied with proper hyperparmeter tuning, as illustrated by the phase diagrams in Figure 6. ", + "bbox": [ + 192, + 338, + 823, + 366 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This paper is organized as follows: In Section 2, we introduce the problem setting and build a simplified toy model. In Section 3, we will use an effective theory approach, a useful tool from theoretical physics, to shed some light on questions Q1 and Q2 and show the relationship between generalization and the learning of structured representations. In Section 4, we explain Q3 by displaying phase diagrams from a grid search of hyperparameters and show how we can “de-delay” generalization by following intuition developed from the phase diagram. We discuss related work in Section 5, followed by conclusions in Section 6.1 ", + "bbox": [ + 174, + 383, + 825, + 481 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Problem Setting ", + "text_level": 1, + "bbox": [ + 176, + 506, + 339, + 523 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Power et al. [1] observe grokking on a less common task – learning “algorithmic” binary operations. Given some binary operation ◦, a network is tasked with learning the map $( a , b ) \\mapsto c$ where $c = a \\circ b$ . They use a decoder-only transformer to predict the second to last token in a tokenized equation of the form “ ”. Each token is represented as a 256-dimensional embedding vector. The embeddings are learnable and initialized randomly. After the transformer, a final linear layer maps the output to class logits for each token. ", + "bbox": [ + 174, + 542, + 826, + 626 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Toy Model We primarily study grokking in a simpler toy model, which still retains the key behaviors from the setup of [1]. Although [1] treated this as a classification task, we study both regression (mean-squared error) and classification (cross-entropy). The basic setup is as follows: our model takes as input the symbols $a , b$ and maps them to trainable embedding vectors $\\mathbf { E } _ { a } , \\mathbf { E } _ { b } \\in \\mathbb { R } ^ { d _ { \\mathrm { i n } } }$ . It then sums $\\mathbf { E } _ { a } , \\mathbf { E } _ { b }$ and sends the resulting vector through a “decoder” MLP. The target output vector, denoted $\\mathbf { Y } _ { c } \\in \\mathbb { R } ^ { d _ { \\mathrm { o u t } } }$ is a fixed random vector (regression task) or a one-hot vector (classification task). Our model architecture can therefore be compactly described as $( a , b ) \\mapsto \\operatorname { D e c } ( \\mathbf { E } _ { a } + \\mathbf { E } _ { b } )$ , where the embeddings $\\mathbf { E _ { * } }$ and the decoder are trainable. Despite its simplicity, this toy model can generalize to all abelian groups (discussed in Appendix B). In sections 3-4.1, we consider only the binary operation of addition. We consider modular addition in Section 4.2 to generalize some of our results to a transformer architecture and study general non-abelian operations in Appendix H. ", + "bbox": [ + 174, + 632, + 825, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Dataset In our toy setting, we are concerned with learning the addition operation. A data sample corresponding to $i + j$ is denoted as $( i , j )$ for simplicity. If $i , j \\in \\{ 0 , \\ldots , p - 1 \\}$ , there are in total $p ( p + 1 ) / 2$ different samples since we consider $i + j$ and $j + i$ to be the same sample. A dataset $D$ is a set of non-repeating data samples. We denote the full dataset as $D _ { 0 }$ and split it into a training dataset $D$ and a validation dataset $D ^ { \\prime }$ , i.e., $D \\bigcup D ^ { \\prime } = D _ { 0 } , D \\bigcap D ^ { \\prime } = \\emptyset$ . We define training data fraction $= | D | / | D _ { 0 } |$ where $| \\cdot |$ denotes the cardinality of the set. ", + "bbox": [ + 173, + 791, + 825, + 875 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/b448f7b12d80bc5a5de6a38e337185955ee926dc22a5e7bfb1f8a7006e5d2a6f.jpg", + "image_caption": [ + "", + "Figure 2: Visualization of the learned set of embeddings $( p = 1 1 $ ) and the decoder function associated with it for the case of 2D embeddings. Axes refer to each dimension of the learned embeddings. The decoder is evaluated on a grid of points in embedding-space and the color at each point represents the highest probability class. For visualization purposes, the decoder is trained on inputs of the form $( \\mathbf { E } _ { i } + \\mathbf { \\bar { E } } _ { j } ) / 2$ . One can read off the output of the decoder when fed the operation $i \\circ j$ from this figure simply by taking the midpoint between the respective embeddings of $i$ and $j$ . " + ], + "image_footnote": [], + "bbox": [ + 210, + 103, + 464, + 364 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/e72e69e51ef60695b4b64c66126a4e2a24d598c9eb27d1a2a44995c7866ba4af.jpg", + "image_caption": [ + "(d) Generalization in toy modular addition " + ], + "image_footnote": [], + "bbox": [ + 537, + 101, + 789, + 354 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Why Generalization Occurs: Representations and Dynamics ", + "text_level": 1, + "bbox": [ + 173, + 483, + 709, + 501 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We can see that generalization appears to be linked to the emergence of highly-structured embeddings in Figure 2. In particular, Figure 2 (a, b) shows parallelograms in toy addition, and (c, d) shows a circle in toy modular addition. We now restrict ourselves to the toy addition setup and formalize a notion of representation quality and show that it predicts the model’s performance. We then develop a physics-inspired effective theory of learning which can accurately predict the critical training set size and training trajectories of representations. The concept of an effective theory in physics is similar to model reduction in computational methods in that it aims to describe complex phenomena with simple yet intuitive pictures. In our effective theory, we will model the dynamics of representation learning not as gradient descent of the true task loss but rather a simpler effective loss function $\\ell _ { \\mathrm { e f f } }$ which depends only on the representations in embedding space and not on the decoder. ", + "bbox": [ + 173, + 515, + 826, + 654 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Representation quality predicts generalization for the toy model ", + "text_level": 1, + "bbox": [ + 174, + 670, + 656, + 685 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A rigorous definition for structure in the learned representation is necessary. We propose the following definition, ", + "bbox": [ + 173, + 695, + 823, + 724 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 1. $( i , j , m , n )$ is a $\\delta$ -parallelogram in the representation $\\mathbf { R } \\equiv \\left[ \\mathbf { E } _ { 0 } , \\cdots , \\mathbf { E } _ { p - 1 } \\right] i f$ ", + "bbox": [ + 174, + 727, + 779, + 743 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b64c0de131873fc3371a9c196107311b16acc9f198345f19a6ffaf7617556fc7.jpg", + "text": "$$\n| ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) - ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) | \\leq \\delta .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 750, + 606, + 767 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the following derivations, we can take $\\delta$ , which is a small threshold to tolerate numerical errors, to be zero. ", + "bbox": [ + 173, + 779, + 825, + 809 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Proposition 1. When the training loss is zero, any parallelogram $( i , j , m , n )$ in representation R satisfies $i + j = m + n$ . ", + "bbox": [ + 173, + 811, + 823, + 840 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Proof. Suppose that this is not the case, i.e., suppose $\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n }$ but $i + j \\neq m + n$ , then $\\mathbf { Y } _ { i + j } = \\bar { \\mathrm { D e c } } ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) = \\mathrm { D e c } ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) = \\bar { \\mathbf { Y } } _ { m + n }$ where the first and last equalities come from the zero training loss assumption. However, since $i + j \\neq m + n$ , we have $\\mathbf { Y } _ { i + j } \\neq \\mathbf { Y } _ { n + m }$ (almost surely in the regression task), a contradiction. □ ", + "bbox": [ + 173, + 854, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/d8a505118aa04e4fe7eee8c0bb3a01ebac68a692b2db9a5ad5553b7c00dbb680.jpg", + "image_caption": [ + "Figure 3: We compute accuracy (of the full dataset) either measured empirically Acc, or predicted from the representation of the embeddings $\\widehat { \\mathrm { A c c } }$ . These two accuracies as a function of training data fraction are plotted in (a)(b), and their agreement is shown in (c). " + ], + "image_footnote": [], + "bbox": [ + 178, + 89, + 823, + 258 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is convenient to define the permissible parallelogram set associated with a training dataset $D$ (“permissible” means consistent with $100 \\%$ training accuracy) as ", + "bbox": [ + 171, + 342, + 823, + 371 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7fa867f1cdc1bd0019ba795c4ddb07c75262c2620d55f40046c7dd8d3930f5a7.jpg", + "text": "$$\nP _ { 0 } ( D ) = \\{ ( i , j , m , n ) | ( i , j ) \\in D , ( m , n ) \\in D , i + j = m + n \\} .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 378, + 714, + 397 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For simplicity, we denote $P _ { 0 } \\equiv P _ { 0 } ( D _ { 0 } )$ . Given a representation $\\mathbf { R }$ , we can check how many permissible parallelograms actually exist in $\\mathbf { R }$ within error $\\delta$ , so we define the parallelogram set corresponding to $\\mathbf { R }$ as ", + "bbox": [ + 173, + 404, + 825, + 446 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b3f96f1564f4a6f78d567da6df91d6512407e51bc2a4b8bc543e48c51f00d510.jpg", + "text": "$$\n\\begin{array} { r } { P ( \\mathbf { R } , \\delta ) = \\{ ( i , j , m , n ) | ( i , j , m , n ) \\in P _ { 0 } , | ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) - ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) | \\leq \\delta \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 454, + 748, + 472 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For brevity we will write $P ( \\mathbf { R } )$ , suppressing the dependence on $\\delta$ . We define the representation quality index (RQI) as ", + "bbox": [ + 174, + 478, + 823, + 507 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/612893b67a60eba7b2dbb60aa0644975b0ac84a4f0603eb946186a8a77c12fb0.jpg", + "text": "$$\n\\mathrm { R Q I } ( \\mathbf { R } ) = \\frac { | P ( \\mathbf { R } ) | } { | P _ { 0 } | } \\in [ 0 , 1 ] .\n$$", + "text_format": "latex", + "bbox": [ + 401, + 507, + 596, + 541 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We will use the term linear representation or linear structure to refer to a representation whose embeddings are of the form $\\mathbf { E } _ { k } = \\mathbf { a } + k \\mathbf { b } \\left( k = 0 , \\cdots , p - 1 ; \\mathbf { a } , \\mathbf { b } \\in \\mathbb { R } ^ { d _ { \\mathrm { i n } } } \\right)$ . A linear representation has $\\operatorname { R Q I } = 1$ , while a random representation (sampled from, say, a normal dstribution) has $\\mathrm { R Q I } = 0$ with high probability. ", + "bbox": [ + 173, + 545, + 825, + 602 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Quantitatively, we denote the “predicted accuracy” $\\widehat { \\mathrm { A c c } }$ as the accuracy achievable on the whole dataset given the representation $\\mathbf { R }$ (see Appendix D for the full details). In Figure 3, we see that the predicted $\\widehat { \\mathrm { A c c } }$ aligns well with the true accuracy Acc, establishing good evidence that structured representation of input embeddings leads to generalization. We use an example to illustrate the origin of generalization here. In the setup of Figure 2 (b), suppose the decoder can achieve zero training loss and $\\mathbf { E } _ { 6 } + \\mathbf { E } _ { 8 }$ is a training sample hence $\\mathrm { D e c } ( { \\bf E } _ { 6 } + { \\bf E } _ { 8 } ) = { \\bf Y } _ { 1 4 }$ . At validation time, the decoder is tasked with predicting a validation sample $\\mathbf { E } _ { 5 } + \\mathbf { E } _ { 9 }$ . Since $( 5 , 9 , 6 , 8 )$ forms a parallelogram such that $\\mathbf { E } _ { 5 } + \\mathbf { E } _ { 9 } = \\mathbf { E } _ { 6 } + \\mathbf { E } _ { 8 }$ , the decoder can predict the validation sample correctly because $\\mathrm { D e c } ( \\mathbf { E } _ { 5 } + \\mathbf { E } _ { 9 } ) = \\mathrm { D e c } ( \\mathbf { E } _ { 6 } + \\mathbf { E } _ { 8 } ) = \\mathbf { Y } _ { 1 4 } ,$ . ", + "bbox": [ + 173, + 608, + 826, + 739 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 The dynamics of embedding vectors ", + "text_level": 1, + "bbox": [ + 176, + 756, + 462, + 771 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Suppose that we have an ideal model $\\mathcal { M } ^ { * } = ( \\mathrm { D e c } ^ { * } , { \\bf R } ^ { * } )$ such that:2 ", + "bbox": [ + 173, + 781, + 614, + 797 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• (1) ${ \\mathfrak { M } } ^ { * }$ can achieve zero training loss; \n• (2) ${ \\mathfrak { M } } ^ { * }$ has an injective decoder, i.e., $\\mathrm { D e c } ^ { * } ( \\mathbf { x } _ { 1 } ) \\neq \\mathrm { D e c } ^ { * } ( \\mathbf { x } _ { 2 } )$ for any $\\mathbf { x } _ { 1 } \\neq \\mathbf { x } _ { 2 }$ . ", + "bbox": [ + 204, + 809, + 730, + 845 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Then Proposition 2 provides a mechanism for the formation of parallelograms. ", + "bbox": [ + 173, + 857, + 689, + 873 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/8682de6492ababd71a56cbc9713e1dbc840e7595ade48ed9156adc9f347f337d.jpg", + "image_caption": [ + "Figure 4: (a) The effective theory predicts a phase transition in the probability of obtaining a linear representation around $r _ { c } = 0 . 4$ . (b) Empirical results display a phase transition of RQI around $r _ { c } = 0 . 4$ , in agreement with the theory (the blue line shows the median of multiple random seeds). The evolution of 1D representations predicted by the effective theory or obtained from neural network training (shown in (c) and (d) respectively) agree creditably well. " + ], + "image_footnote": [], + "bbox": [ + 176, + 90, + 820, + 234 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 2. If a training set $D$ contains two samples $( i , j )$ and $( m , n )$ with $i + j = m + n $ then ${ \\mathfrak { M } } ^ { * }$ learns a representation $\\mathbf { R } ^ { * }$ such that $\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n }$ , i.e., $( i , j , m , n )$ forms $a$ parallelogram. ", + "bbox": [ + 174, + 339, + 823, + 383 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. Due to the zero training loss assumption, we have $\\mathrm { D e c } ^ { * } ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) = \\mathbf { Y } _ { i + j } = \\mathbf { Y } _ { m + n } =$ $\\mathrm { D e c } ^ { * } ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } )$ . Then the injectivity of ${ \\mathrm { D e c } } ^ { * }$ implies $\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { \\bar { E } } _ { n }$ . ", + "bbox": [ + 174, + 397, + 823, + 426 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The dynamics of the trained embedding vectors are determined by various factors interacting in complex ways, for instance: the details of the decoder architecture, the optimizer hyperparameters, and the various kinds of implicit regularization induced by the training procedure. We will see that the dynamics of normalized quantities, namely, the normalized embeddings at time $t$ , defined as $\\begin{array} { r } { \\tilde { \\mathbf { E } } _ { k } ^ { ( t ) } = \\frac { \\mathbf { E } _ { k } ^ { ( t ) } - \\mu _ { t } } { \\sigma _ { t } } } \\end{array}$ , where $\\begin{array} { r } { \\mu _ { t } = \\frac { 1 } { p } \\sum _ { k } \\mathbf { E } _ { k } ^ { ( t ) } } \\end{array}$ and $\\begin{array} { r } { \\sigma _ { t } = \\frac { 1 } { p } \\sum _ { k } | \\mathbf { E } _ { k } ^ { ( t ) } - \\mu _ { t } | ^ { 2 } } \\end{array}$ , can be qualitatively described by a simple effective loss (in the physics effective theory sense). We will assume that the normalized embedding vectors obey a gradient flow for an effective loss function of the form ", + "bbox": [ + 173, + 440, + 826, + 546 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a7c857a9a60847b64ea2e1ecfcdca4edddd8d88ef5504150afa30d4eab1f2d44.jpg", + "text": "$$\n\\frac { d \\tilde { \\mathbf { E } } _ { i } } { d t } = - \\frac { \\partial \\ell _ { \\mathrm { e f f } } } { \\partial \\tilde { \\mathbf { E } } _ { i } } ,\n$$", + "text_format": "latex", + "bbox": [ + 446, + 553, + 550, + 590 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f4512923e2fc038d22faf621acb71b7638e8519a046774680cf01b6f1c8338b7.jpg", + "text": "$$\n\\ell _ { \\mathrm { e f f } } = \\frac { \\ell _ { 0 } } { Z _ { 0 } } , \\quad \\ell _ { 0 } \\equiv \\sum _ { ( i , j , m , n ) \\in P _ { 0 } ( D ) } | \\tilde { \\mathbf { E } } _ { i } + \\tilde { \\mathbf { E } } _ { j } - \\tilde { \\mathbf { E } } _ { m } - \\tilde { \\mathbf { E } } _ { n } | ^ { 2 } / | P _ { 0 } ( D ) | , \\quad Z _ { 0 } \\equiv \\sum _ { k } | \\tilde { \\mathbf { E } } _ { k } | ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 199, + 607, + 777, + 645 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $| \\cdot |$ denotes Euclidean vector norm. Note that the embeddings do not collapse to the trivial solution $\\mathbf { E } _ { 0 } = \\cdot \\cdot \\cdot = \\mathbf { E } _ { p - 1 } = 0$ unless initialized as such, because two conserved quantities exist, as proven in Appendix F: ", + "bbox": [ + 176, + 651, + 826, + 693 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a4ac744d22ff186459f66ccaddae4eddd12bd1b6f97816018102518a5ea1bc9b.jpg", + "text": "$$\n\\mathbf { C } = \\sum _ { k } \\mathbf { E } _ { k } , \\quad Z _ { 0 } = \\sum _ { k } | \\mathbf { E } _ { k } | ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 691, + 607, + 726 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We shall now use the effective dynamics to explain empirical observations such as the existence of a critical training set size for generalization. ", + "bbox": [ + 174, + 734, + 823, + 763 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Degeneracy of ground states (loss optima) We define ground states as those representations satisfying $\\ell _ { \\mathrm { e f f } } = 0$ , which requires the following linear equations to hold: ", + "bbox": [ + 171, + 768, + 821, + 797 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7e79e47f682c75a686eb0934e0007bdec7347837ca9f328cdd40f2a6f945258c.jpg", + "text": "$$\nA ( P ) = \\{ \\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n } | ( i , j , m , n ) \\in P \\} .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 804, + 666, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since each embedding dimension obeys the same set of linear equations, we will assume, without loss of generality, that $d _ { \\mathrm { i n } } = 1$ . The dimension of the null space of $A ( P )$ , denoted as $n _ { 0 }$ , is the number of degrees of freedom of the ground states. Given a set of parallelograms implied by a training dataset $D$ , the nullity of $A ( P ( D ) )$ could be obtained by computing the singular values $0 \\leq \\sigma _ { 1 } \\leq \\cdot \\cdot \\cdot \\leq \\sigma _ { p }$ We always have $n _ { 0 } \\geq 2$ , i.e., $\\sigma _ { 1 } = \\sigma _ { 2 } = 0$ because the nullity of $A ( P _ { 0 } )$ , the set of linear equations given by all possible parallelograms, is $\\mathrm { N u l l i t y } ( A ( P _ { 0 } ) ) = 2$ which can be attributed to two degrees of freedom (translation and scaling). If $n _ { 0 } = 2$ , the representation is unique up to translations and scaling factors, and the embeddings have the form $\\mathbf { E } _ { k } = \\mathbf { a } + k \\mathbf { b }$ . Otherwise, when $n _ { 0 } > 2$ , the representation is not constrained enough such that all the embeddings lie on a line. ", + "bbox": [ + 173, + 827, + 825, + 912 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 823, + 133 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We present theoretical predictions alongside empirical results for addition $\\boldsymbol { p } = 1 0 ^ { \\circ } ,$ ) in Figure 4. As shown in Figure 4 (a), our effective theory predicts that the probability that the training set implies a unique linear structure (which would result in perfect generalization) depends on the training data fraction and has a phase transition around $r _ { c } = 0 . 4$ . Empirical results from training different models are shown in Figure 4 (b). The number of steps to reach $\\mathrm { R Q I } > 0 . 9 5$ is seen to have a phase transition at $r _ { c } = 0 . 4$ , agreeing with the proposed effective theory and with the empirical findings in [1]. ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Time towards the linear structure We define the Hessian matrix of $\\ell _ { 0 }$ as ", + "bbox": [ + 174, + 228, + 661, + 243 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/710f2852fcdda452d0414166012100586158db1f24bc385f00b7e401dbdc8ad4.jpg", + "text": "$$\n\\mathbf { H } _ { i j } = \\frac { 1 } { Z _ { 0 } } \\frac { \\partial ^ { 2 } \\ell _ { 0 } } { \\partial \\mathbf { E } _ { i } \\partial \\mathbf { E } _ { j } } ,\n$$", + "text_format": "latex", + "bbox": [ + 428, + 247, + 566, + 284 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that $\\begin{array} { r } { \\ell _ { \\mathrm { e f f } } = \\frac { 1 } { 2 } \\mathbf { R } ^ { T } \\mathbf { H } \\mathbf { R } } \\end{array}$ , $\\mathbf { R } = [ \\mathbf { E } _ { 0 } , \\mathbf { E } _ { 1 } , \\cdots , \\mathbf { E } _ { p - 1 } ]$ , so the gradient descent is linear, i.e., ", + "bbox": [ + 173, + 287, + 767, + 305 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c3aa6593093d402a3cba032466f3984819bede5c32c03fac13b3bb0c0bb6475f.jpg", + "text": "$$\n\\frac { d \\mathbf { R } } { d t } = - \\mathbf { H } \\mathbf { R } .\n$$", + "text_format": "latex", + "bbox": [ + 449, + 310, + 547, + 340 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "If $\\mathbf { H }$ has eigenvalues $\\lambda _ { i } = \\sigma _ { i } ^ { 2 }$ (sorted in increasing order) and eigenvectors $\\bar { \\bf v } _ { i }$ , and we have the initial condition $\\begin{array} { r } { { \\bf R } ( t = 0 ) = \\sum _ { i } a _ { i } { \\bar { \\bf v } } _ { i } } \\end{array}$ , then we have $\\begin{array} { r } { { \\bf R } ( t ) = \\sum _ { i } a _ { i } \\bar { \\bf v } _ { i } e ^ { - \\lambda _ { i } t } } \\end{array}$ . The first two eigenvalues vanish and $t _ { h } = 1 / \\lambda _ { 3 }$ determines the timescale for the slowest component to decrease by a factor of $e$ . We call $\\lambda _ { 3 }$ the grokking rate. When the step size is $\\eta$ , the corresponding number of steps is $n _ { h } = t _ { h } / \\eta = 1 / ( \\lambda _ { 3 } \\bar { \\eta _ { } } )$ . ", + "bbox": [ + 173, + 353, + 825, + 426 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We verify the above analysis with empirical results. Figure 4 (c)(d) show the trajectories obtained from the effective theory and from neural network training, respectively. The 1D neural representation in Figure 4 (d) are manually normalized to zero mean and unit variance. The two trajectories agree qualitatively, and it takes about $3 n _ { h }$ steps for two trajectories to converge to the linear structure. The quantitative differences might be due to the absence of the decoder in the effective theory, which assumes the decoder to take infinitesimal step sizes. ", + "bbox": [ + 173, + 430, + 825, + 513 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Dependence of grokking on data size Note that $\\ell _ { \\mathrm { e f f } }$ involves averaging over parallelograms in the training set, it is dependent on training data size, so is $\\lambda _ { 3 }$ . In Figure 5 (a), we plot the dependence of $\\lambda _ { 3 }$ on training data fraction. There are many datasets with the same data size, so $\\lambda _ { 3 }$ is a probabilistic function of data size. ", + "bbox": [ + 174, + 520, + 825, + 575 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Two insights on grokking can be extracted from this plot: (i) When the data fraction is below some threshold (around 0.4), $\\lambda _ { 3 }$ is zero with high probability, corresponding to no generalization. This again verifies our critical point in Figure 4. (ii) When data size is above the threshold, $\\lambda _ { 3 }$ (on average) is an increasing function of data size. This implies that grokking time $t \\sim 1 / \\lambda _ { 3 }$ decreases as training data size becomes larger, an important observation from [1]. ", + "bbox": [ + 174, + 582, + 825, + 652 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To verify our effective theory, we compare the grokking steps obtained from real neural network training (defined as steps to $\\mathrm { R Q I } > 0 . 9 5 $ , and those predicted by our theory $\\begin{array} { r } { t _ { \\mathrm { t h } } \\sim \\frac { 1 } { \\lambda _ { 3 } \\eta } } \\end{array}$ $\\dot { \\eta }$ is the embedding learning rate), shown in Figure 5 (b). The theory agrees qualitatively with neural networks, showing the trend of decreasing grokking steps as increasing data size. The quantitative differences might be explained as the gap between our effective loss and actual loss. ", + "bbox": [ + 174, + 657, + 825, + 731 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Limitations of the effective theory While our theory defines an effective loss based on the Euclidean distance between embeddings $\\mathbf { E } _ { i } + \\mathbf { E } _ { j }$ and $ { \\mathbf { E } } _ { n } + { \\mathbf { E } } _ { m }$ , one could imagine generalizing the theory to define a broader notion of parallogram given by some other metric on the representation space. For instance, if we have a decoder like in Figure 2 (d) then the distance between distinct representations within the same “pizza slice” is low, meaning that representations arranged not in parallelograms w.r.t. the Euclidean metric may be parallelograms with respect to the metric defined by the decoder. ", + "bbox": [ + 174, + 734, + 825, + 820 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Delayed Generalization: A Phase Diagram ", + "text_level": 1, + "bbox": [ + 173, + 838, + 560, + 856 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "So far, we have (1) observed empirically that generalization on algorithmic datasets corresponds with the emergence of well-structured representations, (2) defined a notion of representation quality in a toy setting and shown that it predicts generalization, and (3) developed an effective theory to describe the learning dynamics of the representations in the same toy setting. We now study how optimizer hyperparameters affect high-level learning performance. In particular, we develop phase diagrams for how learning performance depends on the representation learning rate, decoder learning rate and the decoder weight decay. These parameters are of interest since they most explicitly regulate a kind of competition between the encoder and decoder, as we elaborate below. ", + "bbox": [ + 174, + 869, + 823, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/790df465da9461408a8b7bb795949bbbd6da09d372793ab426437bf1da28416a.jpg", + "image_caption": [ + "Figure 5: Effective theory explains the dependence of grokking time on data size, for the addition task. (a) Dependence of $\\lambda _ { 3 }$ on training data fraction. Above the critical data fraction (around 0.4), as data size becomes larger, $\\lambda _ { 3 }$ increases hence grokking time $t \\sim 1 / \\lambda _ { 3 }$ (predicted by our effective theory) decreases. (b) Comparing grokking steps (defined as $\\mathrm { R Q I } > 0 . 9 5 ) ,$ predicted by the effective theory with real neural network results. $\\eta = 1 \\bar { 0 } ^ { - 3 }$ is the learning rate of the embeddings. " + ], + "image_footnote": [], + "bbox": [ + 207, + 92, + 790, + 262 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 377, + 825, + 446 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 Phase diagram of a toy model ", + "text_level": 1, + "bbox": [ + 176, + 474, + 418, + 489 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Training details We update the representation and the decoder with different optimizers. For the 1D embeddings, we use the Adam optimizer with learning rate $[ 1 0 ^ { - 5 } , 1 0 ^ { - 2 } ]$ and zero weight decay. For the decoder, we use an AdamW optimizer with the learning rate in $[ 1 0 ^ { - 5 } , 1 0 ^ { - 2 } ]$ and the weight decay in [0, 10] (regression) or $[ 0 , 2 0 ]$ (classification). For training/validation spliting, we choose 45/10 for non-modular addition $\\begin{array} { r } { p = 1 0 , } \\end{array}$ ) and 24/12 for the permutation group $S _ { 3 }$ . We hard-code addition or matrix multiplication (details in Appendix H) in the decoder for the addition group and the permutation group, respectively. ", + "bbox": [ + 173, + 503, + 825, + 602 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For each choice of learning rate and weight decay, we compute the number of steps to reach high $( 9 0 \\% )$ training/validation accuracy. The 2D plane is split into four phases: comprehension, grokking, memorization and confusion, defined in Table 1 in Appendix A. Both comprehension and grokking are able to generalize (in the “Goldilocks zone”), although the grokking phase has delayed generalization. Memorization is also called overfitting, and confusion means failure to even memorize training data. Figure 6 shows the phase diagrams for the addition group and the permutation group. They display quite rich phenomena. ", + "bbox": [ + 174, + 607, + 825, + 704 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Competition between representation learning and decoder overfitting In the regression setup of the addition dataset, we show how the competition between representation learning and decoder learning (which depend on both learning rate and weight decay, among other things) lead to different learning phases in Figure 6 (a). As expected, a fast decoder coupled with slow representation learning (bottom right) lead to memorization. In the opposite extreme, although an extremely slow decoder coupled with fast representation learning (top left) will generalize in the end, the generalization time is long due to the inefficient decoder training. The ideal phase (comprehension) requires representation learning to be faster, but not too much, than the decoder. ", + "bbox": [ + 173, + 710, + 825, + 821 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Drawing from an analogy to physical systems, one can think of embedding vectors as a group of particles. In our effective theory from Section 3.2, the dynamics of the particles are described only by their relative positions, in that sense, structure forms mainly due to inter-particle interactions (in reality, these interactions are mediated by the decoder and the loss). The decoder plays the role of an environment exerting external forces on the embeddings. If the magnitude of the external forces are small/large one can expect better/worse representations. ", + "bbox": [ + 174, + 828, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/9369694209dd16f219ef635b4e040381732aee4942d26c6e5b15ac545eac2aa0.jpg", + "image_caption": [ + "Figure 6: Phase diagrams of learning for the addition group and the permutation group. (a) shows the competition between representation and decoder. (b)(c)(d): each phase diagram contains four phases: comprehension, grokking, memorization and confusion, defined in Table 1. In (b)(c)(d), grokking is sandwiched between comprehension and memorization. " + ], + "image_footnote": [], + "bbox": [ + 269, + 92, + 725, + 473 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Universality of phase diagrams We fix the embedding learning rate to be $1 0 ^ { - 3 }$ and sweep instead decoder weight decay in Figure 6 (b)(c)(d). The phase diagrams correspond to addition regression (b), addition classification (c) and permutation regression (d), respectively. Common phenomena emerge from these different tasks: (i) they all include four phases; (ii) The top right corner (a fast and capable decoder) is the memorization phase; (iii) the bottom right corner (a fast and simple decoder) is the confusion phase; (iv) grokking is sandwiched between comprehension and memorization, which seems to imply that it is an undesirable phase that stems from improperly tuned hyperparameters. ", + "bbox": [ + 174, + 565, + 825, + 662 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 Beyond the toy model ", + "text_level": 1, + "bbox": [ + 176, + 678, + 362, + 693 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We conjecture that many of the principles which we saw dictate the training dynamics in the toy model also apply more generally. Below, we will see how our framework generalizes to transformer architectures for the task of addition modulo $p$ , a minimal reproducible example of the original grokking paper [1]. ", + "bbox": [ + 174, + 704, + 825, + 761 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We first encode $p = 5 3$ integers into 256D learnable embeddings, then pass two integers to a decoderonly transformer architecture. For simplicity, we do not encode the operation symbols here. The outputs from the last layer are concatenated and passed to a linear layer for classification. Training both the encoder and the decoder with the same optimizer (i.e., with the same hyperparameters) leads to the grokking phenomenon. Generalization appears much earlier once we lower the effective decoder capacity with weight decay (full phase diagram in Figure 7). ", + "bbox": [ + 174, + 765, + 825, + 849 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Early on, the model is able to perfectly fit the training set while having no generalization. We study the embeddings at different training times and find that neither PCA (shown in Figure 1) nor t-SNE (not shown here) reveal any structure. Eventually, validation accuracy starts to increase, and perfect generalization coincides with the PCA projecting the embeddings into a circle in 2D. Of course, no choice of dimensionality reduction is guaranteed to find any structure, and thus, it is challenging to show explicitly that generalization only occurs when a structure exists. Nevertheless, the fact that, when coupled with the implicit regularization of the optimizer for sparse solutions, such a clear structure appears in a simple PCA so quickly at generalization time suggests that our analysis in the toy setting is applicable here as well. This is also seen in the evolution of the entropy of the explained variance ratio in the PCA of the embeddings (defined as $\\begin{array} { r } { S = - \\sum _ { i } \\sigma _ { i } \\log \\sigma _ { i } } \\end{array}$ where $\\sigma _ { i }$ is the fractional variance explained by the ith principal component). As seen in Figure 7, the entropy increases up to generalization time then decreases drastically afterwards which would be consistent with the conjecture that generalization occurs when a low-dimensional structure is discovered. The decoder then primarily relies on the information in this low-dimensional manifold and essentially “prunes” the rest of the high-dimensional embedding space. Another interesting insight appears when we project the embeddings at initialization onto the principal axes at the end of training. Some of the structure required for generalization exists before training hinting at a connection with the Lottery Ticket Hypothesis. See Appendix K for more details. ", + "bbox": [ + 174, + 856, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/f97ba3b50d927fef5d6356bddf07766187a7aa148dc09b2096966b4a90acf17f.jpg", + "image_caption": [ + "Figure 7: Left: Evolution of the effective dimension of the embeddings (defined as the exponential of the entropy) during training and evaluated over 100 seeds. Center: Effect of dropout on speeding up generalization. Right: Phase diagram of the transformer architecture. A scan is performed over the weight decay and learning rate of the decoder while the learning rate of the embeddings is kept fixed at $1 \\mathrm { { 0 } ^ { - 3 } }$ (with zero weight decay). " + ], + "image_footnote": [], + "bbox": [ + 205, + 94, + 792, + 233 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 338, + 825, + 532 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Figure 7 (right), we show a comparable phase diagram to Figure 6 evaluated now in the transformer setting. Note that, as opposed to the setting in [1], weight decay has only been applied to the decoder and not to the embedding layer. Contrary to the toy model, a certain amount of weight decay proves beneficial to generalization and speeds it up significantly. We conjecture that this difference comes from the different embedding dimensions. With a highly over-parameterized setting, a non-zero weight decay gives a crucial incentive to reduce complexity in the decoder and help generalize in fewer steps. This is subject to further investigation. We also explore the effect of dropout layers in the decoder blocks of the transformer. With a significant dropout rate, the generalization time can be brought down to under $1 0 ^ { 3 }$ steps and the grokking phenomenon vanishes completely. The overall trend suggests that constraining the decoder with the same tools used to avoid overfitting reduces generalization time and can avoid the grokking phenomenon. This is also observed in an image classification task where we were able to induce grokking. See Appendix J for more details. ", + "bbox": [ + 174, + 539, + 825, + 704 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 Grokking Experiment on MNIST ", + "text_level": 1, + "bbox": [ + 176, + 722, + 444, + 737 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We now demonstrate, for the first time, that grokking (significantly delayed generalization) is a more general phenomenon in machine learning that can occur not only on algorithmic datasets, but also on mainstream benchmark datasets. In particular, we exhibit grokking on MNIST in Figure 8 and demonstrate that we can control grokking by varying optimization hyperparameters. More details on the experimental setup are in Appendix J. ", + "bbox": [ + 174, + 747, + 825, + 818 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Related work ", + "text_level": 1, + "bbox": [ + 174, + 837, + 316, + 854 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Relatively few works have analyzed the phenomenon of grokking. [2] describe the circuit that transformers use to perform modular addition, track its formation over training, and broadly suggest that grokking is related to the phenomenon of “phase changes” in neural network training. [3, 4] provided earlier speculative, informal conjectures on grokking [3, 4]. Our work is related to the following broad research directions: ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/ce9758fc290cfb65a5d604010e550b1f2db4f6660cf61449cd96630f3e657035.jpg", + "image_caption": [ + "Figure 8: Left: Training curves for a run on MNIST, in the setting where we observe grokking. Right: Phase diagram with the four phases of learning dynamics on MNIST. " + ], + "image_footnote": [], + "bbox": [ + 205, + 99, + 795, + 290 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 354, + 823, + 383 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Learning mathematical structures [5] trains a neural network to learn arithmetic operation from pictures of digits, but they do not observe grokking due to their abundant training data. Beyond arithmetic relations, machine learning has been applied to learn other mathematical structures, including geometry [6], knot theory [7] and group theory [8]. ", + "bbox": [ + 174, + 390, + 825, + 445 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Double descent Grokking is somewhat reminiscent of the phenomena of “epoch-wise” double descent [9], where generalization can improve after a period of overfitting. [10] find that regularization can mitigate double descent, similar perhaps to how weight decay influences grokking. ", + "bbox": [ + 174, + 452, + 821, + 494 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Representation learning Representation learning lies at the core of machine learning [11–14]. Representation quality is usually measured by (perhaps vague) semantic meanings or performance on downstream tasks. In our study, the simplicity of arithmetic datasets allows us to define representation quality and study evolution of representations in a quantitative way. ", + "bbox": [ + 174, + 500, + 825, + 555 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Physics of learning Physics-inspired tools have proved to be useful in understanding deep learning from a theoretical perspective. These tools include effective theories [15, 16], conservation laws [17] and free energy principle [18]. In addition, statistical physics has been identified as a powerful tool in studying generalization in neural networks [19–22]. Our work connects a low-level understanding of models with their high-level performance. In a recent work, researchers at Anthropic [23], connect a sudden decrease in loss during training with the emergence of induction heads within their models. They analogize their work to statistical physics, since it bridges a “microscopic”, mechanistic understanding of networks with “macroscopic” facts about overall model performance. ", + "bbox": [ + 174, + 561, + 825, + 672 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 693, + 299, + 709 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We have shown how, in both toy models and general settings, that representation enables generalization when it reflects structure in the data. We developed an effective theory of representation learning dynamics (in a toy setting) which predicts the critical dependence of learning on the training data fraction. We then presented four learning phases (comprehension, grokking, memorization and confusion) which depend on the decoder capacity and learning speed (given by, among other things, learning rate and weight decay) in decoder-only architectures. While we have mostly focused on a toy model, we find preliminary evidence that our results generalize to the setting of [1]. ", + "bbox": [ + 174, + 724, + 825, + 821 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Our work can be viewed as a step towards a statistical physics of deep learning, connecting the “microphysics” of low-level network dynamics with the “thermodynamics” of high-level model behavior. We view the application of theoretical tools from physics, such as effective theories [24], to be a rich area for further work. The broader impact of such work, if successful, could be to make models more transparent and predictable [23, 25, 26], crucial to the task of ensuring the safety of advanced AI systems. ", + "bbox": [ + 174, + 828, + 823, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 90, + 267, + 106 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "[1] Alethea Power, Yuri Burda, Harri Edwards, Igor Babuschkin, and Vedant Misra. Grokking: Generalization beyond overfitting on small algorithmic datasets. arXiv preprint arXiv:2201.02177, 2022. \n[2] Neel Nanda and Tom Lieberum. A mechanistic interpretability analysis of grokking, 2022. URL https://www.alignmentforum.org/posts/N6WM6hs7RQMKDhYjB/ a-mechanistic-interpretability-analysis-of-grokking. \n[3] Beren Millidge. Grokking ’grokking’. https://beren.io/ 2022-01-11-Grokking-Grokking/, 2022. \n[4] Rohin Shah. Alignment Newsletter #159. https: //www.alignmentforum.org/posts/zvWqPmQasssaAWkrj/ an-159-building-agents-that-know-how-to-experiment-by#DEEP_LEARNING_, 2021. \n[5] Yedid Hoshen and Shmuel Peleg. Visual learning of arithmetic operation. 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For all authors... ", + "bbox": [ + 200, + 227, + 328, + 241 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 240, + 246, + 825, + 338 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 199, + 342, + 482, + 356 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] ", + "bbox": [ + 240, + 359, + 736, + 391 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 200, + 395, + 382, + 410 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] All experiments were run on a workstation with two NVIDIA A6000 GPUs within a few days. ", + "bbox": [ + 240, + 414, + 825, + 545 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... 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See Figures 1 and 2.", + "type": "text" + } + ], + "index": 38, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 693, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 117, + 693, + 505, + 707 + ], + "score": 1.0, + "content": "A2 The critical training set size corresponds to the least amount of training data that can determine", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 135, + 705, + 471, + 717 + ], + "spans": [ + { + "bbox": [ + 135, + 705, + 471, + 717 + ], + "score": 1.0, + "content": "such a representation (which, in some cases, is unique up to linear transformations).", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + } + ], + "index": 38, + "bbox_fs": [ + 117, + 656, + 506, + 717 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 72, + 500, + 198 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 72, + 500, + 198 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 72, + 500, + 198 + ], + "spans": [ + { + "bbox": [ + 112, + 72, + 500, + 198 + ], + "score": 0.945, + "type": "image", + "image_path": "e3eaca253887fd618f7f1aec0eaf522e3e511cf97173030ad9ee56c1a327e83f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 72, + 500, + 114.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 114.0, + 500, + 156.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 156.0, + 500, + 198.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 208, + 505, + 241 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "Figure 1: Visualization of the first two principal components of the learned input embeddings at", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "different training stages of a transformer learning modular addition. We observe that generalization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "coincides with the emergence of structure in the embeddings. See Section 4.2 for the training details.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 118, + 268, + 504, + 290 + ], + "lines": [ + { + "bbox": [ + 117, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 117, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "A3 Grokking is a phase between “comprehension” and “memorization” phases and it can be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 135, + 280, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 135, + 280, + 506, + 291 + ], + "score": 1.0, + "content": "remedied with proper hyperparmeter tuning, as illustrated by the phase diagrams in Figure 6.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "This paper is organized as follows: In Section 2, we introduce the problem setting and build a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 315, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 328 + ], + "score": 1.0, + "content": "simplified toy model. In Section 3, we will use an effective theory approach, a useful tool from", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "theoretical physics, to shed some light on questions Q1 and Q2 and show the relationship between", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "generalization and the learning of structured representations. 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We discuss related work in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 370, + 303, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 303, + 381 + ], + "score": 1.0, + "content": "Section 5, followed by conclusions in Section 6.1", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 401, + 208, + 415 + ], + "lines": [ + { + "bbox": [ + 104, + 399, + 211, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 211, + 419 + ], + "score": 1.0, + "content": "2 Problem Setting", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 506, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 507, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 507, + 443 + ], + "score": 1.0, + "content": "Power et al. [1] observe grokking on a less common task – learning “algorithmic” binary operations.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 396, + 453 + ], + "score": 1.0, + "content": "Given some binary operation ◦, a network is tasked with learning the map", + "type": "text" + }, + { + "bbox": [ + 397, + 441, + 439, + 453 + ], + "score": 0.93, + "content": "( a , b ) \\mapsto c", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 442, + 466, + 453 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 466, + 442, + 503, + 451 + ], + "score": 0.9, + "content": "c = a \\circ b", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 442, + 506, + 453 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "They use a decoder-only transformer to predict the second to last token in a tokenized equation of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "form “ ”. Each token is represented as a 256-dimensional", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "embedding vector. The embeddings are learnable and initialized randomly. After the transformer, a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 484, + 359, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 359, + 498 + ], + "score": 1.0, + "content": "final linear layer maps the output to class logits for each token.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "Toy Model We primarily study grokking in a simpler toy model, which still retains the key behaviors", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "from the setup of [1]. 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Despite its simplicity, this toy model can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "generalize to all abelian groups (discussed in Appendix B). In sections 3-4.1, we consider only the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "binary operation of addition. 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A data sample", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 177, + 650 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 177, + 638, + 199, + 649 + ], + "score": 0.9, + "content": "i + j", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 637, + 255, + 650 + ], + "score": 1.0, + "content": "is denoted as", + "type": "text" + }, + { + "bbox": [ + 256, + 638, + 276, + 649 + ], + "score": 0.92, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 637, + 347, + 650 + ], + "score": 1.0, + "content": "for simplicity. If", + "type": "text" + }, + { + "bbox": [ + 348, + 638, + 432, + 649 + ], + "score": 0.92, + "content": "i , j \\in \\{ 0 , \\ldots , p - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 637, + 506, + 650 + ], + "score": 1.0, + "content": ", there are in total", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 648, + 504, + 661 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 153, + 660 + ], + "score": 0.91, + "content": "p ( p + 1 ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 648, + 298, + 661 + ], + "score": 1.0, + "content": "different samples since we consider", + "type": "text" + }, + { + "bbox": [ + 298, + 649, + 320, + 660 + ], + "score": 0.91, + "content": "i + j", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 648, + 338, + 661 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 338, + 649, + 360, + 660 + ], + "score": 0.91, + "content": "j + i", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 648, + 494, + 661 + ], + "score": 1.0, + "content": "to be the same sample. 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[1] observe grokking on a less common task – learning “algorithmic” binary operations.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 396, + 453 + ], + "score": 1.0, + "content": "Given some binary operation ◦, a network is tasked with learning the map", + "type": "text" + }, + { + "bbox": [ + 397, + 441, + 439, + 453 + ], + "score": 0.93, + "content": "( a , b ) \\mapsto c", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 442, + 466, + 453 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 466, + 442, + 503, + 451 + ], + "score": 0.9, + "content": "c = a \\circ b", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 442, + 506, + 453 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "They use a decoder-only transformer to predict the second to last token in a tokenized equation of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "form “ ”. 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Axes refer to each dimension of the learned embeddings. The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "decoder is evaluated on a grid of points in embedding-space and the color at each point represents", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "the highest probability class. 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In particular, Figure 2 (a, b) shows parallelograms in toy addition, and (c, d) shows a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "circle in toy modular addition. We now restrict ourselves to the toy addition setup and formalize a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "notion of representation quality and show that it predicts the model’s performance. We then develop a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "physics-inspired effective theory of learning which can accurately predict the critical training set size", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "and training trajectories of representations. The concept of an effective theory in physics is similar", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "to model reduction in computational methods in that it aims to describe complex phenomena with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "simple yet intuitive pictures. In our effective theory, we will model the dynamics of representation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 495, + 504, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 490, + 509 + ], + "score": 1.0, + "content": "learning not as gradient descent of the true task loss but rather a simpler effective loss function", + "type": "text" + }, + { + "bbox": [ + 490, + 496, + 504, + 507 + ], + "score": 0.86, + "content": "\\ell _ { \\mathrm { e f f } }", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 506, + 456, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 456, + 519 + ], + "score": 1.0, + "content": "which depends only on the representations in embedding space and not on the decoder.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5 + }, + { + "type": "title", + "bbox": [ + 107, + 531, + 402, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 403, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 403, + 545 + ], + "score": 1.0, + "content": "3.1 Representation quality predicts generalization for the toy model", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 504, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 565 + ], + "score": 1.0, + "content": "A rigorous definition for structure in the learned representation is necessary. We propose the following", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 560, + 151, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 151, + 575 + ], + "score": 1.0, + "content": "definition,", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5 + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 477, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 474, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 164, + 591 + ], + "score": 1.0, + "content": "Definition 1.", + "type": "text" + }, + { + "bbox": [ + 164, + 577, + 208, + 589 + ], + "score": 0.92, + "content": "( i , j , m , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 575, + 226, + 591 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 226, + 577, + 232, + 587 + ], + "score": 0.73, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 575, + 379, + 591 + ], + "score": 1.0, + "content": "-parallelogram in the representation", + "type": "text" + }, + { + "bbox": [ + 380, + 576, + 474, + 589 + ], + "score": 0.91, + "content": "\\mathbf { R } \\equiv \\left[ \\mathbf { E } _ { 0 } , \\cdots , \\mathbf { E } _ { p - 1 } \\right] i f", + "type": "inline_equation" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 594, + 371, + 608 + ], + "lines": [ + { + "bbox": [ + 240, + 594, + 371, + 608 + ], + "spans": [ + { + "bbox": [ + 240, + 594, + 371, + 608 + ], + "score": 0.9, + "content": "| ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) - ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) | \\leq \\delta .", + "type": "interline_equation", + "image_path": "b64c0de131873fc3371a9c196107311b16acc9f198345f19a6ffaf7617556fc7.jpg" + } + ] + } + ], + "index": 53, + "virtual_lines": [ + { + "bbox": [ + 240, + 594, + 371, + 608 + ], + "spans": [], + "index": 53 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 104, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 616, + 268, + 631 + ], + "score": 1.0, + "content": "In the following derivations, we can take", + "type": "text" + }, + { + "bbox": [ + 269, + 618, + 275, + 628 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 616, + 506, + 631 + ], + "score": 1.0, + "content": ", which is a small threshold to tolerate numerical errors, to", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 628, + 141, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 141, + 641 + ], + "score": 1.0, + "content": "be zero.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 504, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 377, + 657 + ], + "score": 1.0, + "content": "Proposition 1. When the training loss is zero, any parallelogram", + "type": "text" + }, + { + "bbox": [ + 377, + 644, + 421, + 656 + ], + "score": 0.92, + "content": "( i , j , m , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "in representation R", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 655, + 206, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 140, + 668 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 141, + 655, + 202, + 666 + ], + "score": 0.92, + "content": "i + j = m + n", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 655, + 206, + 668 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 56.5 + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 316, + 690 + ], + "score": 1.0, + "content": "Proof. Suppose that this is not the case, i.e., suppose", + "type": "text" + }, + { + "bbox": [ + 316, + 678, + 405, + 690 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 677, + 421, + 690 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 421, + 678, + 482, + 689 + ], + "score": 0.91, + "content": "i + j \\neq m + n", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 677, + 505, + 690 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 320, + 702 + ], + "score": 0.9, + "content": "\\mathbf { Y } _ { i + j } = \\bar { \\mathrm { D e c } } ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) = \\mathrm { D e c } ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) = \\bar { \\mathbf { Y } } _ { m + n }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "where the first and last equalities come from", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 307, + 712 + ], + "score": 1.0, + "content": "the zero training loss assumption. 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In particular, Figure 2 (a, b) shows parallelograms in toy addition, and (c, d) shows a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "circle in toy modular addition. We now restrict ourselves to the toy addition setup and formalize a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "notion of representation quality and show that it predicts the model’s performance. We then develop a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "physics-inspired effective theory of learning which can accurately predict the critical training set size", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "and training trajectories of representations. The concept of an effective theory in physics is similar", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "to model reduction in computational methods in that it aims to describe complex phenomena with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "simple yet intuitive pictures. In our effective theory, we will model the dynamics of representation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 495, + 504, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 490, + 509 + ], + "score": 1.0, + "content": "learning not as gradient descent of the true task loss but rather a simpler effective loss function", + "type": "text" + }, + { + "bbox": [ + 490, + 496, + 504, + 507 + ], + "score": 0.86, + "content": "\\ell _ { \\mathrm { e f f } }", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 506, + 456, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 456, + 519 + ], + "score": 1.0, + "content": "which depends only on the representations in embedding space and not on the decoder.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 407, + 506, + 519 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 531, + 402, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 403, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 403, + 545 + ], + "score": 1.0, + "content": "3.1 Representation quality predicts generalization for the toy model", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 504, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 565 + ], + "score": 1.0, + "content": "A rigorous definition for structure in the learned representation is necessary. We propose the following", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 560, + 151, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 151, + 575 + ], + "score": 1.0, + "content": "definition,", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 549, + 506, + 575 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 477, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 474, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 164, + 591 + ], + "score": 1.0, + "content": "Definition 1.", + "type": "text" + }, + { + "bbox": [ + 164, + 577, + 208, + 589 + ], + "score": 0.92, + "content": "( i , j , m , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 575, + 226, + 591 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 226, + 577, + 232, + 587 + ], + "score": 0.73, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 575, + 379, + 591 + ], + "score": 1.0, + "content": "-parallelogram in the representation", + "type": "text" + }, + { + "bbox": [ + 380, + 576, + 474, + 589 + ], + "score": 0.91, + "content": "\\mathbf { R } \\equiv \\left[ \\mathbf { E } _ { 0 } , \\cdots , \\mathbf { E } _ { p - 1 } \\right] i f", + "type": "inline_equation" + } + ], + "index": 52 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 575, + 474, + 591 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 594, + 371, + 608 + ], + "lines": [ + { + "bbox": [ + 240, + 594, + 371, + 608 + ], + "spans": [ + { + "bbox": [ + 240, + 594, + 371, + 608 + ], + "score": 0.9, + "content": "| ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) - ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) | \\leq \\delta .", + "type": "interline_equation", + "image_path": "b64c0de131873fc3371a9c196107311b16acc9f198345f19a6ffaf7617556fc7.jpg" + } + ] + } + ], + "index": 53, + "virtual_lines": [ + { + "bbox": [ + 240, + 594, + 371, + 608 + ], + "spans": [], + "index": 53 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 104, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 616, + 268, + 631 + ], + "score": 1.0, + "content": "In the following derivations, we can take", + "type": "text" + }, + { + "bbox": [ + 269, + 618, + 275, + 628 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 616, + 506, + 631 + ], + "score": 1.0, + "content": ", which is a small threshold to tolerate numerical errors, to", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 628, + 141, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 141, + 641 + ], + "score": 1.0, + "content": "be zero.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54.5, + "bbox_fs": [ + 104, + 616, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 504, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 377, + 657 + ], + "score": 1.0, + "content": "Proposition 1. When the training loss is zero, any parallelogram", + "type": "text" + }, + { + "bbox": [ + 377, + 644, + 421, + 656 + ], + "score": 0.92, + "content": "( i , j , m , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "in representation R", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 655, + 206, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 140, + 668 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 141, + 655, + 202, + 666 + ], + "score": 0.92, + "content": "i + j = m + n", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 655, + 206, + 668 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 56.5, + "bbox_fs": [ + 105, + 642, + 505, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 316, + 690 + ], + "score": 1.0, + "content": "Proof. Suppose that this is not the case, i.e., suppose", + "type": "text" + }, + { + "bbox": [ + 316, + 678, + 405, + 690 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 677, + 421, + 690 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 421, + 678, + 482, + 689 + ], + "score": 0.91, + "content": "i + j \\neq m + n", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 677, + 505, + 690 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 320, + 702 + ], + "score": 0.9, + "content": "\\mathbf { Y } _ { i + j } = \\bar { \\mathrm { D e c } } ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) = \\mathrm { D e c } ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) = \\bar { \\mathbf { Y } } _ { m + n }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "where the first and last equalities come from", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 307, + 712 + ], + "score": 1.0, + "content": "the zero training loss assumption. However, since", + "type": "text" + }, + { + "bbox": [ + 307, + 700, + 369, + 712 + ], + "score": 0.92, + "content": "i + j \\neq m + n", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 699, + 408, + 712 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 408, + 700, + 471, + 712 + ], + "score": 0.93, + "content": "\\mathbf { Y } _ { i + j } \\neq \\mathbf { Y } _ { n + m }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "(almost", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 290, + 723 + ], + "score": 1.0, + "content": "surely in the regression task), a contradiction.", + "type": "text" + }, + { + "bbox": [ + 494, + 711, + 505, + 721 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 61 + } + ], + "index": 59.5, + "bbox_fs": [ + 105, + 677, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 71, + 504, + 205 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 71, + 504, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 71, + 504, + 205 + ], + "spans": [ + { + "bbox": [ + 109, + 71, + 504, + 205 + ], + "score": 0.974, + "type": "image", + "image_path": "d8a505118aa04e4fe7eee8c0bb3a01ebac68a692b2db9a5ad5553b7c00dbb680.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 71, + 504, + 115.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 115.66666666666666, + 504, + 160.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 160.33333333333331, + 504, + 204.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 213, + 506, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "Figure 3: We compute accuracy (of the full dataset) either measured empirically Acc, or predicted", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 278, + 239 + ], + "score": 1.0, + "content": "from the representation of the embeddings", + "type": "text" + }, + { + "bbox": [ + 279, + 224, + 296, + 237 + ], + "score": 0.8, + "content": "\\widehat { \\mathrm { A c c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 226, + 505, + 239 + ], + "score": 1.0, + "content": ". These two accuracies as a function of training data", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 367, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 367, + 251 + ], + "score": 1.0, + "content": "fraction are plotted in (a)(b), and their agreement is shown in (c).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 105, + 271, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 494, + 284 + ], + "score": 1.0, + "content": "It is convenient to define the permissible parallelogram set associated with a training dataset", + "type": "text" + }, + { + "bbox": [ + 494, + 272, + 504, + 282 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 282, + 369, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 258, + 296 + ], + "score": 1.0, + "content": "(“permissible” means consistent with", + "type": "text" + }, + { + "bbox": [ + 258, + 283, + 282, + 293 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 282, + 369, + 296 + ], + "score": 1.0, + "content": "training accuracy) as", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 300, + 437, + 315 + ], + "lines": [ + { + "bbox": [ + 173, + 300, + 437, + 315 + ], + "spans": [ + { + "bbox": [ + 173, + 300, + 437, + 315 + ], + "score": 0.85, + "content": "P _ { 0 } ( D ) = \\{ ( i , j , m , n ) | ( i , j ) \\in D , ( m , n ) \\in D , i + j = m + n \\} .", + "type": "interline_equation", + "image_path": "7fa867f1cdc1bd0019ba795c4ddb07c75262c2620d55f40046c7dd8d3930f5a7.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 173, + 300, + 437, + 315 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 217, + 334 + ], + "score": 1.0, + "content": "For simplicity, we denote", + "type": "text" + }, + { + "bbox": [ + 217, + 320, + 278, + 332 + ], + "score": 0.94, + "content": "P _ { 0 } \\equiv P _ { 0 } ( D _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 319, + 383, + 334 + ], + "score": 1.0, + "content": ". Given a representation", + "type": "text" + }, + { + "bbox": [ + 384, + 321, + 393, + 330 + ], + "score": 0.67, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 319, + 505, + 334 + ], + "score": 1.0, + "content": ", we can check how many", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 288, + 344 + ], + "score": 1.0, + "content": "permissible parallelograms actually exist in", + "type": "text" + }, + { + "bbox": [ + 288, + 332, + 298, + 342 + ], + "score": 0.6, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 331, + 351, + 344 + ], + "score": 1.0, + "content": "within error", + "type": "text" + }, + { + "bbox": [ + 351, + 332, + 357, + 342 + ], + "score": 0.79, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 331, + 506, + 344 + ], + "score": 1.0, + "content": ", so we define the parallelogram set", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 342, + 199, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 176, + 355 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 177, + 343, + 186, + 352 + ], + "score": 0.73, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 342, + 199, + 355 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 360, + 458, + 374 + ], + "lines": [ + { + "bbox": [ + 153, + 360, + 458, + 374 + ], + "spans": [ + { + "bbox": [ + 153, + 360, + 458, + 374 + ], + "score": 0.88, + "content": "\\begin{array} { r } { P ( \\mathbf { R } , \\delta ) = \\{ ( i , j , m , n ) | ( i , j , m , n ) \\in P _ { 0 } , | ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) - ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) | \\leq \\delta \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "b3f96f1564f4a6f78d567da6df91d6512407e51bc2a4b8bc543e48c51f00d510.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 153, + 360, + 458, + 374 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 504, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 213, + 393 + ], + "score": 1.0, + "content": "For brevity we will write", + "type": "text" + }, + { + "bbox": [ + 213, + 380, + 238, + 392 + ], + "score": 0.92, + "content": "P ( \\mathbf { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 379, + 373, + 393 + ], + "score": 1.0, + "content": ", suppressing the dependence on", + "type": "text" + }, + { + "bbox": [ + 373, + 381, + 379, + 390 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 379, + 505, + 393 + ], + "score": 1.0, + "content": ". We define the representation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 391, + 200, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 200, + 405 + ], + "score": 1.0, + "content": "quality index (RQI) as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 402, + 365, + 429 + ], + "lines": [ + { + "bbox": [ + 246, + 402, + 365, + 429 + ], + "spans": [ + { + "bbox": [ + 246, + 402, + 365, + 429 + ], + "score": 0.94, + "content": "\\mathrm { R Q I } ( \\mathbf { R } ) = \\frac { | P ( \\mathbf { R } ) | } { | P _ { 0 } | } \\in [ 0 , 1 ] .", + "type": "interline_equation", + "image_path": "612893b67a60eba7b2dbb60aa0644975b0ac84a4f0603eb946186a8a77c12fb0.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 246, + 402, + 365, + 429 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "We will use the term linear representation or linear structure to refer to a representation whose", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 219, + 456 + ], + "score": 1.0, + "content": "embeddings are of the form", + "type": "text" + }, + { + "bbox": [ + 219, + 443, + 405, + 455 + ], + "score": 0.76, + "content": "\\mathbf { E } _ { k } = \\mathbf { a } + k \\mathbf { b } \\left( k = 0 , \\cdots , p - 1 ; \\mathbf { a } , \\mathbf { b } \\in \\mathbb { R } ^ { d _ { \\mathrm { i n } } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 443, + 505, + 456 + ], + "score": 1.0, + "content": ". A linear representation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 454, + 504, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 122, + 467 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 122, + 455, + 160, + 466 + ], + "score": 0.88, + "content": "\\operatorname { R Q I } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 454, + 466, + 467 + ], + "score": 1.0, + "content": ", while a random representation (sampled from, say, a normal dstribution) has", + "type": "text" + }, + { + "bbox": [ + 466, + 454, + 504, + 466 + ], + "score": 0.88, + "content": "\\mathrm { R Q I } = 0", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 464, + 195, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 195, + 478 + ], + "score": 1.0, + "content": "with high probability.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 506, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 318, + 496 + ], + "score": 1.0, + "content": "Quantitatively, we denote the “predicted accuracy”", + "type": "text" + }, + { + "bbox": [ + 318, + 481, + 336, + 494 + ], + "score": 0.75, + "content": "\\widehat { \\mathrm { A c c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "as the accuracy achievable on the whole", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 238, + 508 + ], + "score": 1.0, + "content": "dataset given the representation", + "type": "text" + }, + { + "bbox": [ + 239, + 495, + 248, + 505 + ], + "score": 0.54, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "(see Appendix D for the full details). In Figure 3, we see that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 160, + 520 + ], + "score": 1.0, + "content": "the predicted", + "type": "text" + }, + { + "bbox": [ + 160, + 505, + 178, + 518 + ], + "score": 0.82, + "content": "\\widehat { \\mathrm { A c c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "aligns well with the true accuracy Acc, establishing good evidence that structured", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "representation of input embeddings leads to generalization. We use an example to illustrate the origin", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "of generalization here. In the setup of Figure 2 (b), suppose the decoder can achieve zero training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 541, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 141, + 553 + ], + "score": 1.0, + "content": "loss and", + "type": "text" + }, + { + "bbox": [ + 141, + 541, + 178, + 552 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { 6 } + \\mathbf { E } _ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 541, + 284, + 553 + ], + "score": 1.0, + "content": "is a training sample hence", + "type": "text" + }, + { + "bbox": [ + 285, + 541, + 376, + 552 + ], + "score": 0.91, + "content": "\\mathrm { D e c } ( { \\bf E } _ { 6 } + { \\bf E } _ { 8 } ) = { \\bf Y } _ { 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 541, + 506, + 553 + ], + "score": 1.0, + "content": ". At validation time, the decoder", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 295, + 565 + ], + "score": 1.0, + "content": "is tasked with predicting a validation sample", + "type": "text" + }, + { + "bbox": [ + 295, + 552, + 334, + 563 + ], + "score": 0.89, + "content": "\\mathbf { E } _ { 5 } + \\mathbf { E } _ { 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 551, + 367, + 565 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 367, + 552, + 408, + 563 + ], + "score": 0.86, + "content": "( 5 , 9 , 6 , 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "forms a parallelogram", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 146, + 575 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 147, + 563, + 237, + 574 + ], + "score": 0.89, + "content": "\\mathbf { E } _ { 5 } + \\mathbf { E } _ { 9 } = \\mathbf { E } _ { 6 } + \\mathbf { E } _ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 563, + 505, + 575 + ], + "score": 1.0, + "content": ", the decoder can predict the validation sample correctly because", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 572, + 276, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 273, + 586 + ], + "score": 0.88, + "content": "\\mathrm { D e c } ( \\mathbf { E } _ { 5 } + \\mathbf { E } _ { 9 } ) = \\mathrm { D e c } ( \\mathbf { E } _ { 6 } + \\mathbf { E } _ { 8 } ) = \\mathbf { Y } _ { 1 4 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 572, + 276, + 587 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 599, + 283, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 285, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 285, + 613 + ], + "score": 1.0, + "content": "3.2 The dynamics of embedding vectors", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 619, + 376, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 378, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 256, + 633 + ], + "score": 1.0, + "content": "Suppose that we have an ideal model", + "type": "text" + }, + { + "bbox": [ + 256, + 619, + 332, + 631 + ], + "score": 0.92, + "content": "\\mathcal { M } ^ { * } = ( \\mathrm { D e c } ^ { * } , { \\bf R } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 618, + 378, + 633 + ], + "score": 1.0, + "content": "such that:2", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 125, + 641, + 447, + 670 + ], + "lines": [ + { + "bbox": [ + 126, + 640, + 289, + 655 + ], + "spans": [ + { + "bbox": [ + 126, + 640, + 149, + 655 + ], + "score": 1.0, + "content": "• (1)", + "type": "text" + }, + { + "bbox": [ + 149, + 642, + 165, + 652 + ], + "score": 0.82, + "content": "{ \\mathfrak { M } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 640, + 289, + 655 + ], + "score": 1.0, + "content": "can achieve zero training loss;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 125, + 654, + 448, + 672 + ], + "spans": [ + { + "bbox": [ + 125, + 654, + 148, + 672 + ], + "score": 1.0, + "content": "• (2)", + "type": "text" + }, + { + "bbox": [ + 149, + 657, + 164, + 668 + ], + "score": 0.8, + "content": "{ \\mathfrak { M } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 654, + 283, + 672 + ], + "score": 1.0, + "content": "has an injective decoder, i.e.,", + "type": "text" + }, + { + "bbox": [ + 283, + 657, + 376, + 669 + ], + "score": 0.92, + "content": "\\mathrm { D e c } ^ { * } ( \\mathbf { x } _ { 1 } ) \\neq \\mathrm { D e c } ^ { * } ( \\mathbf { x } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 654, + 408, + 672 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 408, + 658, + 444, + 669 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 1 } \\neq \\mathbf { x } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 654, + 448, + 672 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 679, + 422, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 422, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 422, + 695 + ], + "score": 1.0, + "content": "Then Proposition 2 provides a mechanism for the formation of parallelograms.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 117, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 117, + 699, + 428, + 714 + ], + "score": 1.0, + "content": "2One can verify a posteriori if a trained model M is close to being an ideal model", + "type": "text" + }, + { + "bbox": [ + 429, + 702, + 443, + 711 + ], + "score": 0.86, + "content": "{ \\mathcal { M } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 699, + 506, + 714 + ], + "score": 1.0, + "content": ". 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These two accuracies as a function of training data", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 367, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 367, + 251 + ], + "score": 1.0, + "content": "fraction are plotted in (a)(b), and their agreement is shown in (c).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 105, + 271, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 494, + 284 + ], + "score": 1.0, + "content": "It is convenient to define the permissible parallelogram set associated with a training dataset", + "type": "text" + }, + { + "bbox": [ + 494, + 272, + 504, + 282 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 282, + 369, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 258, + 296 + ], + "score": 1.0, + "content": "(“permissible” means consistent with", + "type": "text" + }, + { + "bbox": [ + 258, + 283, + 282, + 293 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 282, + 369, + 296 + ], + "score": 1.0, + "content": "training accuracy) as", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 271, + 504, + 296 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 300, + 437, + 315 + ], + "lines": [ + { + "bbox": [ + 173, + 300, + 437, + 315 + ], + "spans": [ + { + "bbox": [ + 173, + 300, + 437, + 315 + ], + "score": 0.85, + "content": "P _ { 0 } ( D ) = \\{ ( i , j , m , n ) | ( i , j ) \\in D , ( m , n ) \\in D , i + j = m + n \\} .", + "type": "interline_equation", + "image_path": "7fa867f1cdc1bd0019ba795c4ddb07c75262c2620d55f40046c7dd8d3930f5a7.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 173, + 300, + 437, + 315 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 217, + 334 + ], + "score": 1.0, + "content": "For simplicity, we denote", + "type": "text" + }, + { + "bbox": [ + 217, + 320, + 278, + 332 + ], + "score": 0.94, + "content": "P _ { 0 } \\equiv P _ { 0 } ( D _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 319, + 383, + 334 + ], + "score": 1.0, + "content": ". Given a representation", + "type": "text" + }, + { + "bbox": [ + 384, + 321, + 393, + 330 + ], + "score": 0.67, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 319, + 505, + 334 + ], + "score": 1.0, + "content": ", we can check how many", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 288, + 344 + ], + "score": 1.0, + "content": "permissible parallelograms actually exist in", + "type": "text" + }, + { + "bbox": [ + 288, + 332, + 298, + 342 + ], + "score": 0.6, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 331, + 351, + 344 + ], + "score": 1.0, + "content": "within error", + "type": "text" + }, + { + "bbox": [ + 351, + 332, + 357, + 342 + ], + "score": 0.79, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 331, + 506, + 344 + ], + "score": 1.0, + "content": ", so we define the parallelogram set", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 342, + 199, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 176, + 355 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 177, + 343, + 186, + 352 + ], + "score": 0.73, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 342, + 199, + 355 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 319, + 506, + 355 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 360, + 458, + 374 + ], + "lines": [ + { + "bbox": [ + 153, + 360, + 458, + 374 + ], + "spans": [ + { + "bbox": [ + 153, + 360, + 458, + 374 + ], + "score": 0.88, + "content": "\\begin{array} { r } { P ( \\mathbf { R } , \\delta ) = \\{ ( i , j , m , n ) | ( i , j , m , n ) \\in P _ { 0 } , | ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) - ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } ) | \\leq \\delta \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "b3f96f1564f4a6f78d567da6df91d6512407e51bc2a4b8bc543e48c51f00d510.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 153, + 360, + 458, + 374 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 504, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 213, + 393 + ], + "score": 1.0, + "content": "For brevity we will write", + "type": "text" + }, + { + "bbox": [ + 213, + 380, + 238, + 392 + ], + "score": 0.92, + "content": "P ( \\mathbf { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 379, + 373, + 393 + ], + "score": 1.0, + "content": ", suppressing the dependence on", + "type": "text" + }, + { + "bbox": [ + 373, + 381, + 379, + 390 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 379, + 505, + 393 + ], + "score": 1.0, + "content": ". We define the representation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 391, + 200, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 200, + 405 + ], + "score": 1.0, + "content": "quality index (RQI) as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 379, + 505, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 402, + 365, + 429 + ], + "lines": [ + { + "bbox": [ + 246, + 402, + 365, + 429 + ], + "spans": [ + { + "bbox": [ + 246, + 402, + 365, + 429 + ], + "score": 0.94, + "content": "\\mathrm { R Q I } ( \\mathbf { R } ) = \\frac { | P ( \\mathbf { R } ) | } { | P _ { 0 } | } \\in [ 0 , 1 ] .", + "type": "interline_equation", + "image_path": "612893b67a60eba7b2dbb60aa0644975b0ac84a4f0603eb946186a8a77c12fb0.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 246, + 402, + 365, + 429 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "We will use the term linear representation or linear structure to refer to a representation whose", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 219, + 456 + ], + "score": 1.0, + "content": "embeddings are of the form", + "type": "text" + }, + { + "bbox": [ + 219, + 443, + 405, + 455 + ], + "score": 0.76, + "content": "\\mathbf { E } _ { k } = \\mathbf { a } + k \\mathbf { b } \\left( k = 0 , \\cdots , p - 1 ; \\mathbf { a } , \\mathbf { b } \\in \\mathbb { R } ^ { d _ { \\mathrm { i n } } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 443, + 505, + 456 + ], + "score": 1.0, + "content": ". A linear representation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 454, + 504, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 122, + 467 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 122, + 455, + 160, + 466 + ], + "score": 0.88, + "content": "\\operatorname { R Q I } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 454, + 466, + 467 + ], + "score": 1.0, + "content": ", while a random representation (sampled from, say, a normal dstribution) has", + "type": "text" + }, + { + "bbox": [ + 466, + 454, + 504, + 466 + ], + "score": 0.88, + "content": "\\mathrm { R Q I } = 0", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 464, + 195, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 195, + 478 + ], + "score": 1.0, + "content": "with high probability.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 432, + 505, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 506, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 318, + 496 + ], + "score": 1.0, + "content": "Quantitatively, we denote the “predicted accuracy”", + "type": "text" + }, + { + "bbox": [ + 318, + 481, + 336, + 494 + ], + "score": 0.75, + "content": "\\widehat { \\mathrm { A c c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "as the accuracy achievable on the whole", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 238, + 508 + ], + "score": 1.0, + "content": "dataset given the representation", + "type": "text" + }, + { + "bbox": [ + 239, + 495, + 248, + 505 + ], + "score": 0.54, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "(see Appendix D for the full details). In Figure 3, we see that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 160, + 520 + ], + "score": 1.0, + "content": "the predicted", + "type": "text" + }, + { + "bbox": [ + 160, + 505, + 178, + 518 + ], + "score": 0.82, + "content": "\\widehat { \\mathrm { A c c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "aligns well with the true accuracy Acc, establishing good evidence that structured", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "representation of input embeddings leads to generalization. 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In the setup of Figure 2 (b), suppose the decoder can achieve zero training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 541, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 141, + 553 + ], + "score": 1.0, + "content": "loss and", + "type": "text" + }, + { + "bbox": [ + 141, + 541, + 178, + 552 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { 6 } + \\mathbf { E } _ { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 541, + 284, + 553 + ], + "score": 1.0, + "content": "is a training sample hence", + "type": "text" + }, + { + "bbox": [ + 285, + 541, + 376, + 552 + ], + "score": 0.91, + "content": "\\mathrm { D e c } ( { \\bf E } _ { 6 } + { \\bf E } _ { 8 } ) = { \\bf Y } _ { 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 541, + 506, + 553 + ], + "score": 1.0, + "content": ". At validation time, the decoder", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 295, + 565 + ], + "score": 1.0, + "content": "is tasked with predicting a validation sample", + "type": "text" + }, + { + "bbox": [ + 295, + 552, + 334, + 563 + ], + "score": 0.89, + "content": "\\mathbf { E } _ { 5 } + \\mathbf { E } _ { 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 551, + 367, + 565 + ], + "score": 1.0, + "content": ". 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(b) Empirical results display a phase transition of RQI around", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 107, + 216, + 142, + 226 + ], + "score": 0.9, + "content": "r _ { c } = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 215, + 506, + 228 + ], + "score": 1.0, + "content": ", in agreement with the theory (the blue line shows the median of multiple random seeds).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "The evolution of 1D representations predicted by the effective theory or obtained from neural network", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 237, + 369, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 369, + 250 + ], + "score": 1.0, + "content": "training (shown in (c) and (d) respectively) agree creditably well.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 504, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 503, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 241, + 283 + ], + "score": 1.0, + "content": "Proposition 2. If a training set", + "type": "text" + }, + { + "bbox": [ + 241, + 271, + 250, + 280 + ], + "score": 0.71, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 268, + 344, + 283 + ], + "score": 1.0, + "content": "contains two samples", + "type": "text" + }, + { + "bbox": [ + 345, + 270, + 365, + 281 + ], + "score": 0.9, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 268, + 386, + 283 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 270, + 414, + 282 + ], + "score": 0.91, + "content": "( m , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 268, + 436, + 283 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 437, + 271, + 503, + 281 + ], + "score": 0.89, + "content": "i + j = m + n ", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 280, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 127, + 294 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 127, + 281, + 143, + 291 + ], + "score": 0.84, + "content": "{ \\mathfrak { M } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 280, + 245, + 294 + ], + "score": 1.0, + "content": "learns a representation", + "type": "text" + }, + { + "bbox": [ + 245, + 281, + 259, + 291 + ], + "score": 0.84, + "content": "\\mathbf { R } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 280, + 303, + 294 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 304, + 281, + 400, + 294 + ], + "score": 0.91, + "content": "\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 280, + 424, + 294 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 424, + 281, + 468, + 293 + ], + "score": 0.91, + "content": "( i , j , m , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 280, + 497, + 294 + ], + "score": 1.0, + "content": "forms", + "type": "text" + }, + { + "bbox": [ + 497, + 283, + 504, + 291 + ], + "score": 0.34, + "content": "a", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 292, + 169, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 169, + 306 + ], + "score": 1.0, + "content": "parallelogram.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 104, + 311, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 346, + 331 + ], + "score": 1.0, + "content": "Proof. Due to the zero training loss assumption, we have", + "type": "text" + }, + { + "bbox": [ + 346, + 315, + 505, + 327 + ], + "score": 0.88, + "content": "\\mathrm { D e c } ^ { * } ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) = \\mathbf { Y } _ { i + j } = \\mathbf { Y } _ { m + n } =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 324, + 422, + 340 + ], + "spans": [ + { + "bbox": [ + 107, + 326, + 177, + 338 + ], + "score": 0.91, + "content": "\\mathrm { D e c } ^ { * } ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 324, + 272, + 340 + ], + "score": 1.0, + "content": ". Then the injectivity of", + "type": "text" + }, + { + "bbox": [ + 272, + 327, + 294, + 337 + ], + "score": 0.74, + "content": "{ \\mathrm { D e c } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 324, + 327, + 340 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 328, + 327, + 417, + 339 + ], + "score": 0.9, + "content": "\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { \\bar { E } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 324, + 422, + 340 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "The dynamics of the trained embedding vectors are determined by various factors interacting in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "complex ways, for instance: the details of the decoder architecture, the optimizer hyperparameters,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "and the various kinds of implicit regularization induced by the training procedure. We will see that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 451, + 395 + ], + "score": 1.0, + "content": "the dynamics of normalized quantities, namely, the normalized embeddings at time", + "type": "text" + }, + { + "bbox": [ + 452, + 384, + 457, + 392 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 382, + 505, + 395 + ], + "score": 1.0, + "content": ", defined as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 391, + 508, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 169, + 412 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\tilde { \\mathbf { E } } _ { k } ^ { ( t ) } = \\frac { \\mathbf { E } _ { k } ^ { ( t ) } - \\mu _ { t } } { \\sigma _ { t } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 391, + 200, + 418 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 200, + 395, + 266, + 413 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mu _ { t } = \\frac { 1 } { p } \\sum _ { k } \\mathbf { E } _ { k } ^ { ( t ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 391, + 284, + 418 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 395, + 382, + 413 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\sigma _ { t } = \\frac { 1 } { p } \\sum _ { k } | \\mathbf { E } _ { k } ^ { ( t ) } - \\mu _ { t } | ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 391, + 508, + 418 + ], + "score": 1.0, + "content": ", can be qualitatively described", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "by a simple effective loss (in the physics effective theory sense). We will assume that the normalized", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 421, + 432, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 432, + 434 + ], + "score": 1.0, + "content": "embedding vectors obey a gradient flow for an effective loss function of the form", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 273, + 438, + 337, + 468 + ], + "lines": [ + { + "bbox": [ + 273, + 438, + 337, + 468 + ], + "spans": [ + { + "bbox": [ + 273, + 438, + 337, + 468 + ], + "score": 0.95, + "content": "\\frac { d \\tilde { \\mathbf { E } } _ { i } } { d t } = - \\frac { \\partial \\ell _ { \\mathrm { e f f } } } { \\partial \\tilde { \\mathbf { E } } _ { i } } ,", + "type": "interline_equation", + "image_path": "a7c857a9a60847b64ea2e1ecfcdca4edddd8d88ef5504150afa30d4eab1f2d44.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 273, + 438, + 337, + 453.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 273, + 453.0, + 337, + 468.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 481, + 476, + 511 + ], + "lines": [ + { + "bbox": [ + 122, + 481, + 476, + 511 + ], + "spans": [ + { + "bbox": [ + 122, + 481, + 476, + 511 + ], + "score": 0.9, + "content": "\\ell _ { \\mathrm { e f f } } = \\frac { \\ell _ { 0 } } { Z _ { 0 } } , \\quad \\ell _ { 0 } \\equiv \\sum _ { ( i , j , m , n ) \\in P _ { 0 } ( D ) } | \\tilde { \\mathbf { E } } _ { i } + \\tilde { \\mathbf { E } } _ { j } - \\tilde { \\mathbf { E } } _ { m } - \\tilde { \\mathbf { E } } _ { n } | ^ { 2 } / | P _ { 0 } ( D ) | , \\quad Z _ { 0 } \\equiv \\sum _ { k } | \\tilde { \\mathbf { E } } _ { k } | ^ { 2 } ,", + "type": "interline_equation", + "image_path": "f4512923e2fc038d22faf621acb71b7638e8519a046774680cf01b6f1c8338b7.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 122, + 481, + 476, + 491.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 122, + 491.0, + 476, + 501.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 122, + 501.0, + 476, + 511.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 516, + 506, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 133, + 529 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 517, + 148, + 529 + ], + "score": 0.84, + "content": "| \\cdot |", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "denotes Euclidean vector norm. Note that the embeddings do not collapse to the trivial", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 141, + 541 + ], + "score": 1.0, + "content": "solution", + "type": "text" + }, + { + "bbox": [ + 141, + 528, + 233, + 540 + ], + "score": 0.91, + "content": "\\mathbf { E } _ { 0 } = \\cdot \\cdot \\cdot = \\mathbf { E } _ { p - 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "unless initialized as such, because two conserved quantities exist, as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 539, + 200, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 200, + 551 + ], + "score": 1.0, + "content": "proven in Appendix F:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 548, + 372, + 575 + ], + "lines": [ + { + "bbox": [ + 239, + 548, + 372, + 575 + ], + "spans": [ + { + "bbox": [ + 239, + 548, + 372, + 575 + ], + "score": 0.94, + "content": "\\mathbf { C } = \\sum _ { k } \\mathbf { E } _ { k } , \\quad Z _ { 0 } = \\sum _ { k } | \\mathbf { E } _ { k } | ^ { 2 } .", + "type": "interline_equation", + "image_path": "a4ac744d22ff186459f66ccaddae4eddd12bd1b6f97816018102518a5ea1bc9b.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 239, + 548, + 372, + 561.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 239, + 561.5, + 372, + 575.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "We shall now use the effective dynamics to explain empirical observations such as the existence of a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 277, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 277, + 606 + ], + "score": 1.0, + "content": "critical training set size for generalization.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 105, + 609, + 503, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "Degeneracy of ground states (loss optima) We define ground states as those representations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 400, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 147, + 633 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 147, + 621, + 179, + 632 + ], + "score": 0.91, + "content": "\\ell _ { \\mathrm { e f f } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 620, + 400, + 633 + ], + "score": 1.0, + "content": ", which requires the following linear equations to hold:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 637, + 408, + 651 + ], + "lines": [ + { + "bbox": [ + 202, + 637, + 408, + 651 + ], + "spans": [ + { + "bbox": [ + 202, + 637, + 408, + 651 + ], + "score": 0.91, + "content": "A ( P ) = \\{ \\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n } | ( i , j , m , n ) \\in P \\} .", + "type": "interline_equation", + "image_path": "7e79e47f682c75a686eb0934e0007bdec7347837ca9f328cdd40f2a6f945258c.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 202, + 637, + 408, + 651 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "Since each embedding dimension obeys the same set of linear equations, we will assume, without loss", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 177, + 680 + ], + "score": 1.0, + "content": "of generality, that", + "type": "text" + }, + { + "bbox": [ + 178, + 667, + 209, + 678 + ], + "score": 0.92, + "content": "d _ { \\mathrm { i n } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 667, + 352, + 680 + ], + "score": 1.0, + "content": ". The dimension of the null space of", + "type": "text" + }, + { + "bbox": [ + 352, + 667, + 376, + 679 + ], + "score": 0.92, + "content": "A ( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 667, + 423, + 680 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 423, + 669, + 435, + 678 + ], + "score": 0.85, + "content": "n _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 667, + 506, + 680 + ], + "score": 1.0, + "content": ", is the number of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "degrees of freedom of the ground states. 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(b) Empirical results display a phase transition of RQI around", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 107, + 216, + 142, + 226 + ], + "score": 0.9, + "content": "r _ { c } = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 215, + 506, + 228 + ], + "score": 1.0, + "content": ", in agreement with the theory (the blue line shows the median of multiple random seeds).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "The evolution of 1D representations predicted by the effective theory or obtained from neural network", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 237, + 369, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 369, + 250 + ], + "score": 1.0, + "content": "training (shown in (c) and (d) respectively) agree creditably well.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 504, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 503, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 241, + 283 + ], + "score": 1.0, + "content": "Proposition 2. If a training set", + "type": "text" + }, + { + "bbox": [ + 241, + 271, + 250, + 280 + ], + "score": 0.71, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 268, + 344, + 283 + ], + "score": 1.0, + "content": "contains two samples", + "type": "text" + }, + { + "bbox": [ + 345, + 270, + 365, + 281 + ], + "score": 0.9, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 268, + 386, + 283 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 387, + 270, + 414, + 282 + ], + "score": 0.91, + "content": "( m , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 268, + 436, + 283 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 437, + 271, + 503, + 281 + ], + "score": 0.89, + "content": "i + j = m + n ", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 280, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 127, + 294 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 127, + 281, + 143, + 291 + ], + "score": 0.84, + "content": "{ \\mathfrak { M } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 280, + 245, + 294 + ], + "score": 1.0, + "content": "learns a representation", + "type": "text" + }, + { + "bbox": [ + 245, + 281, + 259, + 291 + ], + "score": 0.84, + "content": "\\mathbf { R } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 280, + 303, + 294 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 304, + 281, + 400, + 294 + ], + "score": 0.91, + "content": "\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 280, + 424, + 294 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 424, + 281, + 468, + 293 + ], + "score": 0.91, + "content": "( i , j , m , n )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 280, + 497, + 294 + ], + "score": 1.0, + "content": "forms", + "type": "text" + }, + { + "bbox": [ + 497, + 283, + 504, + 291 + ], + "score": 0.34, + "content": "a", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 292, + 169, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 169, + 306 + ], + "score": 1.0, + "content": "parallelogram.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 268, + 504, + 306 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 104, + 311, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 346, + 331 + ], + "score": 1.0, + "content": "Proof. Due to the zero training loss assumption, we have", + "type": "text" + }, + { + "bbox": [ + 346, + 315, + 505, + 327 + ], + "score": 0.88, + "content": "\\mathrm { D e c } ^ { * } ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) = \\mathbf { Y } _ { i + j } = \\mathbf { Y } _ { m + n } =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 324, + 422, + 340 + ], + "spans": [ + { + "bbox": [ + 107, + 326, + 177, + 338 + ], + "score": 0.91, + "content": "\\mathrm { D e c } ^ { * } ( \\mathbf { E } _ { m } + \\mathbf { E } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 324, + 272, + 340 + ], + "score": 1.0, + "content": ". Then the injectivity of", + "type": "text" + }, + { + "bbox": [ + 272, + 327, + 294, + 337 + ], + "score": 0.74, + "content": "{ \\mathrm { D e c } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 324, + 327, + 340 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 328, + 327, + 417, + 339 + ], + "score": 0.9, + "content": "\\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { \\bar { E } } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 324, + 422, + 340 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 311, + 505, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "The dynamics of the trained embedding vectors are determined by various factors interacting in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "complex ways, for instance: the details of the decoder architecture, the optimizer hyperparameters,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "and the various kinds of implicit regularization induced by the training procedure. We will see that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 451, + 395 + ], + "score": 1.0, + "content": "the dynamics of normalized quantities, namely, the normalized embeddings at time", + "type": "text" + }, + { + "bbox": [ + 452, + 384, + 457, + 392 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 382, + 505, + 395 + ], + "score": 1.0, + "content": ", defined as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 391, + 508, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 169, + 412 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\tilde { \\mathbf { E } } _ { k } ^ { ( t ) } = \\frac { \\mathbf { E } _ { k } ^ { ( t ) } - \\mu _ { t } } { \\sigma _ { t } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 391, + 200, + 418 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 200, + 395, + 266, + 413 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mu _ { t } = \\frac { 1 } { p } \\sum _ { k } \\mathbf { E } _ { k } ^ { ( t ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 391, + 284, + 418 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 395, + 382, + 413 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\sigma _ { t } = \\frac { 1 } { p } \\sum _ { k } | \\mathbf { E } _ { k } ^ { ( t ) } - \\mu _ { t } | ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 391, + 508, + 418 + ], + "score": 1.0, + "content": ", can be qualitatively described", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "by a simple effective loss (in the physics effective theory sense). We will assume that the normalized", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 421, + 432, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 432, + 434 + ], + "score": 1.0, + "content": "embedding vectors obey a gradient flow for an effective loss function of the form", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 349, + 508, + 434 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 273, + 438, + 337, + 468 + ], + "lines": [ + { + "bbox": [ + 273, + 438, + 337, + 468 + ], + "spans": [ + { + "bbox": [ + 273, + 438, + 337, + 468 + ], + "score": 0.95, + "content": "\\frac { d \\tilde { \\mathbf { E } } _ { i } } { d t } = - \\frac { \\partial \\ell _ { \\mathrm { e f f } } } { \\partial \\tilde { \\mathbf { E } } _ { i } } ,", + "type": "interline_equation", + "image_path": "a7c857a9a60847b64ea2e1ecfcdca4edddd8d88ef5504150afa30d4eab1f2d44.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 273, + 438, + 337, + 453.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 273, + 453.0, + 337, + 468.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 481, + 476, + 511 + ], + "lines": [ + { + "bbox": [ + 122, + 481, + 476, + 511 + ], + "spans": [ + { + "bbox": [ + 122, + 481, + 476, + 511 + ], + "score": 0.9, + "content": "\\ell _ { \\mathrm { e f f } } = \\frac { \\ell _ { 0 } } { Z _ { 0 } } , \\quad \\ell _ { 0 } \\equiv \\sum _ { ( i , j , m , n ) \\in P _ { 0 } ( D ) } | \\tilde { \\mathbf { E } } _ { i } + \\tilde { \\mathbf { E } } _ { j } - \\tilde { \\mathbf { E } } _ { m } - \\tilde { \\mathbf { E } } _ { n } | ^ { 2 } / | P _ { 0 } ( D ) | , \\quad Z _ { 0 } \\equiv \\sum _ { k } | \\tilde { \\mathbf { E } } _ { k } | ^ { 2 } ,", + "type": "interline_equation", + "image_path": "f4512923e2fc038d22faf621acb71b7638e8519a046774680cf01b6f1c8338b7.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 122, + 481, + 476, + 491.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 122, + 491.0, + 476, + 501.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 122, + 501.0, + 476, + 511.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 516, + 506, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 133, + 529 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 517, + 148, + 529 + ], + "score": 0.84, + "content": "| \\cdot |", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "denotes Euclidean vector norm. Note that the embeddings do not collapse to the trivial", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 141, + 541 + ], + "score": 1.0, + "content": "solution", + "type": "text" + }, + { + "bbox": [ + 141, + 528, + 233, + 540 + ], + "score": 0.91, + "content": "\\mathbf { E } _ { 0 } = \\cdot \\cdot \\cdot = \\mathbf { E } _ { p - 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "unless initialized as such, because two conserved quantities exist, as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 539, + 200, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 200, + 551 + ], + "score": 1.0, + "content": "proven in Appendix F:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 516, + 506, + 551 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 548, + 372, + 575 + ], + "lines": [ + { + "bbox": [ + 239, + 548, + 372, + 575 + ], + "spans": [ + { + "bbox": [ + 239, + 548, + 372, + 575 + ], + "score": 0.94, + "content": "\\mathbf { C } = \\sum _ { k } \\mathbf { E } _ { k } , \\quad Z _ { 0 } = \\sum _ { k } | \\mathbf { E } _ { k } | ^ { 2 } .", + "type": "interline_equation", + "image_path": "a4ac744d22ff186459f66ccaddae4eddd12bd1b6f97816018102518a5ea1bc9b.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 239, + 548, + 372, + 561.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 239, + 561.5, + 372, + 575.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "We shall now use the effective dynamics to explain empirical observations such as the existence of a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 277, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 277, + 606 + ], + "score": 1.0, + "content": "critical training set size for generalization.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 582, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 609, + 503, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "Degeneracy of ground states (loss optima) We define ground states as those representations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 400, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 147, + 633 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 147, + 621, + 179, + 632 + ], + "score": 0.91, + "content": "\\ell _ { \\mathrm { e f f } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 620, + 400, + 633 + ], + "score": 1.0, + "content": ", which requires the following linear equations to hold:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 609, + 505, + 633 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 637, + 408, + 651 + ], + "lines": [ + { + "bbox": [ + 202, + 637, + 408, + 651 + ], + "spans": [ + { + "bbox": [ + 202, + 637, + 408, + 651 + ], + "score": 0.91, + "content": "A ( P ) = \\{ \\mathbf { E } _ { i } + \\mathbf { E } _ { j } = \\mathbf { E } _ { m } + \\mathbf { E } _ { n } | ( i , j , m , n ) \\in P \\} .", + "type": "interline_equation", + "image_path": "7e79e47f682c75a686eb0934e0007bdec7347837ca9f328cdd40f2a6f945258c.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 202, + 637, + 408, + 651 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "Since each embedding dimension obeys the same set of linear equations, we will assume, without loss", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 177, + 680 + ], + "score": 1.0, + "content": "of generality, that", + "type": "text" + }, + { + "bbox": [ + 178, + 667, + 209, + 678 + ], + "score": 0.92, + "content": "d _ { \\mathrm { i n } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 667, + 352, + 680 + ], + "score": 1.0, + "content": ". The dimension of the null space of", + "type": "text" + }, + { + "bbox": [ + 352, + 667, + 376, + 679 + ], + "score": 0.92, + "content": "A ( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 667, + 423, + 680 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 423, + 669, + 435, + 678 + ], + "score": 0.85, + "content": "n _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 667, + 506, + 680 + ], + "score": 1.0, + "content": ", is the number of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "degrees of freedom of the ground states. Given a set of parallelograms implied by a training dataset", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 686, + 503, + 704 + ], + "spans": [ + { + "bbox": [ + 107, + 689, + 116, + 699 + ], + "score": 0.79, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 686, + 174, + 704 + ], + "score": 1.0, + "content": ", the nullity of", + "type": "text" + }, + { + "bbox": [ + 174, + 689, + 214, + 701 + ], + "score": 0.91, + "content": "A ( P ( D ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 686, + 425, + 704 + ], + "score": 1.0, + "content": "could be obtained by computing the singular values", + "type": "text" + }, + { + "bbox": [ + 425, + 689, + 503, + 701 + ], + "score": 0.9, + "content": "0 \\leq \\sigma _ { 1 } \\leq \\cdot \\cdot \\cdot \\leq \\sigma _ { p }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 173, + 713 + ], + "score": 1.0, + "content": "We always have", + "type": "text" + }, + { + "bbox": [ + 173, + 701, + 203, + 711 + ], + "score": 0.9, + "content": "n _ { 0 } \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 699, + 225, + 713 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 225, + 700, + 277, + 711 + ], + "score": 0.9, + "content": "\\sigma _ { 1 } = \\sigma _ { 2 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 699, + 368, + 713 + ], + "score": 1.0, + "content": "because the nullity of", + "type": "text" + }, + { + "bbox": [ + 368, + 700, + 395, + 712 + ], + "score": 0.92, + "content": "A ( P _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 699, + 506, + 713 + ], + "score": 1.0, + "content": ", the set of linear equations", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 264, + 723 + ], + "score": 1.0, + "content": "given by all possible parallelograms, is", + "type": "text" + }, + { + "bbox": [ + 264, + 711, + 348, + 723 + ], + "score": 0.74, + "content": "\\mathrm { N u l l i t y } ( A ( P _ { 0 } ) ) = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "which can be attributed to two degrees", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 267, + 85 + ], + "score": 1.0, + "content": "of freedom (translation and scaling). If", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 267, + 73, + 297, + 84 + ], + "score": 0.91, + "content": "n _ { 0 } = 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 297, + 73, + 505, + 85 + ], + "score": 1.0, + "content": ", the representation is unique up to translations and", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 316, + 96 + ], + "score": 1.0, + "content": "scaling factors, and the embeddings have the form", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 317, + 84, + 375, + 95 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { k } = \\mathbf { a } + k \\mathbf { b }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 376, + 83, + 453, + 96 + ], + "score": 1.0, + "content": ". 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If", + "type": "text" + }, + { + "bbox": [ + 267, + 73, + 297, + 84 + ], + "score": 0.91, + "content": "n _ { 0 } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 73, + 505, + 85 + ], + "score": 1.0, + "content": ", the representation is unique up to translations and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 316, + 96 + ], + "score": 1.0, + "content": "scaling factors, and the embeddings have the form", + "type": "text" + }, + { + "bbox": [ + 317, + 84, + 375, + 95 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { k } = \\mathbf { a } + k \\mathbf { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 83, + 453, + 96 + ], + "score": 1.0, + "content": ". Otherwise, when", + "type": "text" + }, + { + "bbox": [ + 454, + 84, + 486, + 95 + ], + "score": 0.9, + "content": "n _ { 0 } > 2", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 83, + 506, + 96 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 437, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 437, + 106 + ], + "score": 1.0, + "content": "representation is not constrained enough such that all the embeddings lie on a line.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 407, + 123 + ], + "score": 1.0, + "content": "We present theoretical predictions alongside empirical results for addition", + "type": "text" + }, + { + "bbox": [ + 407, + 111, + 439, + 122 + ], + "score": 0.86, + "content": "\\boldsymbol { p } = 1 0 ^ { \\circ } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 111, + 505, + 123 + ], + "score": 1.0, + "content": ") in Figure 4. As", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 135 + ], + "score": 1.0, + "content": "shown in Figure 4 (a), our effective theory predicts that the probability that the training set implies a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "unique linear structure (which would result in perfect generalization) depends on the training data", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 273, + 156 + ], + "score": 1.0, + "content": "fraction and has a phase transition around", + "type": "text" + }, + { + "bbox": [ + 273, + 144, + 309, + 155 + ], + "score": 0.9, + "content": "r _ { c } = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 144, + 505, + 156 + ], + "score": 1.0, + "content": ". Empirical results from training different models", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 325, + 167 + ], + "score": 1.0, + "content": "are shown in Figure 4 (b). The number of steps to reach", + "type": "text" + }, + { + "bbox": [ + 325, + 155, + 375, + 166 + ], + "score": 0.89, + "content": "\\mathrm { R Q I } > 0 . 9 5", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "is seen to have a phase transition", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 487, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 116, + 178 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 116, + 166, + 152, + 177 + ], + "score": 0.9, + "content": "r _ { c } = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 165, + 487, + 178 + ], + "score": 1.0, + "content": ", agreeing with the proposed effective theory and with the empirical findings in [1].", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 181, + 405, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 405, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 382, + 195 + ], + "score": 1.0, + "content": "Time towards the linear structure We define the Hessian matrix of", + "type": "text" + }, + { + "bbox": [ + 383, + 182, + 392, + 193 + ], + "score": 0.89, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 180, + 405, + 195 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 196, + 347, + 225 + ], + "lines": [ + { + "bbox": [ + 262, + 196, + 347, + 225 + ], + "spans": [ + { + "bbox": [ + 262, + 196, + 347, + 225 + ], + "score": 0.95, + "content": "\\mathbf { H } _ { i j } = \\frac { 1 } { Z _ { 0 } } \\frac { \\partial ^ { 2 } \\ell _ { 0 } } { \\partial \\mathbf { E } _ { i } \\partial \\mathbf { E } _ { j } } ,", + "type": "interline_equation", + "image_path": "710f2852fcdda452d0414166012100586158db1f24bc385f00b7e401dbdc8ad4.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 262, + 196, + 347, + 210.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 262, + 210.5, + 347, + 225.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 228, + 470, + 242 + ], + "lines": [ + { + "bbox": [ + 104, + 227, + 472, + 244 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 145, + 244 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 228, + 212, + 243 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\ell _ { \\mathrm { e f f } } = \\frac { 1 } { 2 } \\mathbf { R } ^ { T } \\mathbf { H } \\mathbf { R } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 227, + 216, + 244 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 216, + 229, + 320, + 243 + ], + "score": 0.81, + "content": "\\mathbf { R } = [ \\mathbf { E } _ { 0 } , \\mathbf { E } _ { 1 } , \\cdots , \\mathbf { E } _ { p - 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 227, + 472, + 244 + ], + "score": 1.0, + "content": ", so the gradient descent is linear, i.e.,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 275, + 246, + 335, + 270 + ], + "lines": [ + { + "bbox": [ + 275, + 246, + 335, + 270 + ], + "spans": [ + { + "bbox": [ + 275, + 246, + 335, + 270 + ], + "score": 0.92, + "content": "\\frac { d \\mathbf { R } } { d t } = - \\mathbf { H } \\mathbf { R } .", + "type": "interline_equation", + "image_path": "c3aa6593093d402a3cba032466f3984819bede5c32c03fac13b3bb0c0bb6475f.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 275, + 246, + 335, + 270 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 104, + 278, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 115, + 294 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 115, + 281, + 126, + 291 + ], + "score": 0.41, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 278, + 190, + 294 + ], + "score": 1.0, + "content": "has eigenvalues", + "type": "text" + }, + { + "bbox": [ + 190, + 280, + 224, + 292 + ], + "score": 0.93, + "content": "\\lambda _ { i } = \\sigma _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 278, + 400, + 294 + ], + "score": 1.0, + "content": "(sorted in increasing order) and eigenvectors", + "type": "text" + }, + { + "bbox": [ + 401, + 281, + 411, + 291 + ], + "score": 0.88, + "content": "\\bar { \\bf v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 278, + 506, + 294 + ], + "score": 1.0, + "content": ", and we have the initial", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 289, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 148, + 306 + ], + "score": 1.0, + "content": "condition", + "type": "text" + }, + { + "bbox": [ + 148, + 292, + 238, + 305 + ], + "score": 0.9, + "content": "\\begin{array} { r } { { \\bf R } ( t = 0 ) = \\sum _ { i } a _ { i } { \\bar { \\bf v } } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 289, + 300, + 306 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + }, + { + "bbox": [ + 301, + 291, + 393, + 305 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { \\bf R } ( t ) = \\sum _ { i } a _ { i } \\bar { \\bf v } _ { i } e ^ { - \\lambda _ { i } t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 289, + 506, + 306 + ], + "score": 1.0, + "content": ". The first two eigenvalues", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 152, + 316 + ], + "score": 1.0, + "content": "vanish and", + "type": "text" + }, + { + "bbox": [ + 152, + 304, + 196, + 315 + ], + "score": 0.9, + "content": "t _ { h } = 1 / \\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "determines the timescale for the slowest component to decrease by a factor", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 118, + 328 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 316, + 124, + 324 + ], + "score": 0.58, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 313, + 162, + 328 + ], + "score": 1.0, + "content": ". We call", + "type": "text" + }, + { + "bbox": [ + 162, + 315, + 174, + 325 + ], + "score": 0.85, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 313, + 340, + 328 + ], + "score": 1.0, + "content": "the grokking rate. When the step size is", + "type": "text" + }, + { + "bbox": [ + 341, + 316, + 348, + 326 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 313, + 506, + 328 + ], + "score": 1.0, + "content": ", the corresponding number of steps is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 324, + 201, + 338 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 198, + 338 + ], + "score": 0.92, + "content": "n _ { h } = t _ { h } / \\eta = 1 / ( \\lambda _ { 3 } \\bar { \\eta _ { } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 324, + 201, + 338 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "We verify the above analysis with empirical results. Figure 4 (c)(d) show the trajectories obtained", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "from the effective theory and from neural network training, respectively. The 1D neural representation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "in Figure 4 (d) are manually normalized to zero mean and unit variance. The two trajectories agree", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 231, + 387 + ], + "score": 1.0, + "content": "qualitatively, and it takes about", + "type": "text" + }, + { + "bbox": [ + 231, + 375, + 249, + 385 + ], + "score": 0.89, + "content": "3 n _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "steps for two trajectories to converge to the linear structure. The", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "quantitative differences might be due to the absence of the decoder in the effective theory, which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 316, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 316, + 409 + ], + "score": 1.0, + "content": "assumes the decoder to take infinitesimal step sizes.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 304, + 425 + ], + "score": 1.0, + "content": "Dependence of grokking on data size Note that", + "type": "text" + }, + { + "bbox": [ + 305, + 413, + 320, + 424 + ], + "score": 0.9, + "content": "\\ell _ { \\mathrm { e f f } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "involves averaging over parallelograms in the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 320, + 435 + ], + "score": 1.0, + "content": "training set, it is dependent on training data size, so is", + "type": "text" + }, + { + "bbox": [ + 321, + 424, + 332, + 434 + ], + "score": 0.88, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 424, + 505, + 435 + ], + "score": 1.0, + "content": ". In Figure 5 (a), we plot the dependence of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 118, + 445 + ], + "score": 0.86, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 435, + 424, + 446 + ], + "score": 1.0, + "content": "on training data fraction. There are many datasets with the same data size, so", + "type": "text" + }, + { + "bbox": [ + 425, + 435, + 436, + 445 + ], + "score": 0.87, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 435, + 505, + 446 + ], + "score": 1.0, + "content": "is a probabilistic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 445, + 193, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 193, + 457 + ], + "score": 1.0, + "content": "function of data size.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "Two insights on grokking can be extracted from this plot: (i) When the data fraction is below some", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 202, + 485 + ], + "score": 1.0, + "content": "threshold (around 0.4),", + "type": "text" + }, + { + "bbox": [ + 203, + 473, + 214, + 484 + ], + "score": 0.87, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "is zero with high probability, corresponding to no generalization. This", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 442, + 497 + ], + "score": 1.0, + "content": "again verifies our critical point in Figure 4. (ii) When data size is above the threshold,", + "type": "text" + }, + { + "bbox": [ + 442, + 484, + 453, + 495 + ], + "score": 0.86, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "(on average)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 493, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 381, + 509 + ], + "score": 1.0, + "content": "is an increasing function of data size. This implies that grokking time", + "type": "text" + }, + { + "bbox": [ + 382, + 494, + 420, + 506 + ], + "score": 0.93, + "content": "t \\sim 1 / \\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 493, + 506, + 509 + ], + "score": 1.0, + "content": "decreases as training", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 505, + 350, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 350, + 517 + ], + "score": 1.0, + "content": "data size becomes larger, an important observation from [1].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "To verify our effective theory, we compare the grokking steps obtained from real neural network", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 527, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 223, + 551 + ], + "score": 1.0, + "content": "training (defined as steps to", + "type": "text" + }, + { + "bbox": [ + 223, + 533, + 276, + 544 + ], + "score": 0.87, + "content": "\\mathrm { R Q I } > 0 . 9 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 527, + 423, + 551 + ], + "score": 1.0, + "content": ", and those predicted by our theory", + "type": "text" + }, + { + "bbox": [ + 424, + 532, + 467, + 547 + ], + "score": 0.92, + "content": "\\begin{array} { r } { t _ { \\mathrm { t h } } \\sim \\frac { 1 } { \\lambda _ { 3 } \\eta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 534, + 479, + 544 + ], + "score": 0.57, + "content": "\\dot { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 527, + 505, + 551 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 544, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 507, + 559 + ], + "score": 1.0, + "content": "embedding learning rate), shown in Figure 5 (b). The theory agrees qualitatively with neural networks,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "showing the trend of decreasing grokking steps as increasing data size. The quantitative differences", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 567, + 397, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 397, + 579 + ], + "score": 1.0, + "content": "might be explained as the gap between our effective loss and actual loss.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "Limitations of the effective theory While our theory defines an effective loss based on the Euclidean", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 226, + 608 + ], + "score": 1.0, + "content": "distance between embeddings", + "type": "text" + }, + { + "bbox": [ + 227, + 595, + 263, + 606 + ], + "score": 0.94, + "content": "\\mathbf { E } _ { i } + \\mathbf { E } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 592, + 281, + 608 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 594, + 322, + 606 + ], + "score": 0.92, + "content": " { \\mathbf { E } } _ { n } + { \\mathbf { E } } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 592, + 506, + 608 + ], + "score": 1.0, + "content": ", one could imagine generalizing the theory to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "define a broader notion of parallogram given by some other metric on the representation space. For", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "instance, if we have a decoder like in Figure 2 (d) then the distance between distinct representations", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "within the same “pizza slice” is low, meaning that representations arranged not in parallelograms", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 639, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 506, + 650 + ], + "score": 1.0, + "content": "w.r.t. the Euclidean metric may be parallelograms with respect to the metric defined by the decoder.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 106, + 664, + 343, + 678 + ], + "lines": [ + { + "bbox": [ + 104, + 663, + 345, + 682 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 345, + 682 + ], + "score": 1.0, + "content": "4 Delayed Generalization: A Phase Diagram", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "So far, we have (1) observed empirically that generalization on algorithmic datasets corresponds with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "the emergence of well-structured representations, (2) defined a notion of representation quality in a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "toy setting and shown that it predicts generalization, and (3) developed an effective theory to describe", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 504, + 106 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 73, + 506, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 407, + 123 + ], + "score": 1.0, + "content": "We present theoretical predictions alongside empirical results for addition", + "type": "text" + }, + { + "bbox": [ + 407, + 111, + 439, + 122 + ], + "score": 0.86, + "content": "\\boldsymbol { p } = 1 0 ^ { \\circ } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 111, + 505, + 123 + ], + "score": 1.0, + "content": ") in Figure 4. As", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 135 + ], + "score": 1.0, + "content": "shown in Figure 4 (a), our effective theory predicts that the probability that the training set implies a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "unique linear structure (which would result in perfect generalization) depends on the training data", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 273, + 156 + ], + "score": 1.0, + "content": "fraction and has a phase transition around", + "type": "text" + }, + { + "bbox": [ + 273, + 144, + 309, + 155 + ], + "score": 0.9, + "content": "r _ { c } = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 144, + 505, + 156 + ], + "score": 1.0, + "content": ". Empirical results from training different models", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 325, + 167 + ], + "score": 1.0, + "content": "are shown in Figure 4 (b). The number of steps to reach", + "type": "text" + }, + { + "bbox": [ + 325, + 155, + 375, + 166 + ], + "score": 0.89, + "content": "\\mathrm { R Q I } > 0 . 9 5", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "is seen to have a phase transition", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 487, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 116, + 178 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 116, + 166, + 152, + 177 + ], + "score": 0.9, + "content": "r _ { c } = 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 165, + 487, + 178 + ], + "score": 1.0, + "content": ", agreeing with the proposed effective theory and with the empirical findings in [1].", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 111, + 506, + 178 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 181, + 405, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 405, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 382, + 195 + ], + "score": 1.0, + "content": "Time towards the linear structure We define the Hessian matrix of", + "type": "text" + }, + { + "bbox": [ + 383, + 182, + 392, + 193 + ], + "score": 0.89, + "content": "\\ell _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 180, + 405, + 195 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 180, + 405, + 195 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 196, + 347, + 225 + ], + "lines": [ + { + "bbox": [ + 262, + 196, + 347, + 225 + ], + "spans": [ + { + "bbox": [ + 262, + 196, + 347, + 225 + ], + "score": 0.95, + "content": "\\mathbf { H } _ { i j } = \\frac { 1 } { Z _ { 0 } } \\frac { \\partial ^ { 2 } \\ell _ { 0 } } { \\partial \\mathbf { E } _ { i } \\partial \\mathbf { E } _ { j } } ,", + "type": "interline_equation", + "image_path": "710f2852fcdda452d0414166012100586158db1f24bc385f00b7e401dbdc8ad4.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 262, + 196, + 347, + 210.5 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 262, + 210.5, + 347, + 225.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 228, + 470, + 242 + ], + "lines": [ + { + "bbox": [ + 104, + 227, + 472, + 244 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 145, + 244 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 228, + 212, + 243 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\ell _ { \\mathrm { e f f } } = \\frac { 1 } { 2 } \\mathbf { R } ^ { T } \\mathbf { H } \\mathbf { R } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 227, + 216, + 244 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 216, + 229, + 320, + 243 + ], + "score": 0.81, + "content": "\\mathbf { R } = [ \\mathbf { E } _ { 0 } , \\mathbf { E } _ { 1 } , \\cdots , \\mathbf { E } _ { p - 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 227, + 472, + 244 + ], + "score": 1.0, + "content": ", so the gradient descent is linear, i.e.,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 227, + 472, + 244 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 275, + 246, + 335, + 270 + ], + "lines": [ + { + "bbox": [ + 275, + 246, + 335, + 270 + ], + "spans": [ + { + "bbox": [ + 275, + 246, + 335, + 270 + ], + "score": 0.92, + "content": "\\frac { d \\mathbf { R } } { d t } = - \\mathbf { H } \\mathbf { R } .", + "type": "interline_equation", + "image_path": "c3aa6593093d402a3cba032466f3984819bede5c32c03fac13b3bb0c0bb6475f.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 275, + 246, + 335, + 270 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 104, + 278, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 115, + 294 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 115, + 281, + 126, + 291 + ], + "score": 0.41, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 278, + 190, + 294 + ], + "score": 1.0, + "content": "has eigenvalues", + "type": "text" + }, + { + "bbox": [ + 190, + 280, + 224, + 292 + ], + "score": 0.93, + "content": "\\lambda _ { i } = \\sigma _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 278, + 400, + 294 + ], + "score": 1.0, + "content": "(sorted in increasing order) and eigenvectors", + "type": "text" + }, + { + "bbox": [ + 401, + 281, + 411, + 291 + ], + "score": 0.88, + "content": "\\bar { \\bf v } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 278, + 506, + 294 + ], + "score": 1.0, + "content": ", and we have the initial", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 289, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 148, + 306 + ], + "score": 1.0, + "content": "condition", + "type": "text" + }, + { + "bbox": [ + 148, + 292, + 238, + 305 + ], + "score": 0.9, + "content": "\\begin{array} { r } { { \\bf R } ( t = 0 ) = \\sum _ { i } a _ { i } { \\bar { \\bf v } } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 289, + 300, + 306 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + }, + { + "bbox": [ + 301, + 291, + 393, + 305 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { \\bf R } ( t ) = \\sum _ { i } a _ { i } \\bar { \\bf v } _ { i } e ^ { - \\lambda _ { i } t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 289, + 506, + 306 + ], + "score": 1.0, + "content": ". The first two eigenvalues", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 152, + 316 + ], + "score": 1.0, + "content": "vanish and", + "type": "text" + }, + { + "bbox": [ + 152, + 304, + 196, + 315 + ], + "score": 0.9, + "content": "t _ { h } = 1 / \\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "determines the timescale for the slowest component to decrease by a factor", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 118, + 328 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 316, + 124, + 324 + ], + "score": 0.58, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 313, + 162, + 328 + ], + "score": 1.0, + "content": ". We call", + "type": "text" + }, + { + "bbox": [ + 162, + 315, + 174, + 325 + ], + "score": 0.85, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 313, + 340, + 328 + ], + "score": 1.0, + "content": "the grokking rate. When the step size is", + "type": "text" + }, + { + "bbox": [ + 341, + 316, + 348, + 326 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 313, + 506, + 328 + ], + "score": 1.0, + "content": ", the corresponding number of steps is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 324, + 201, + 338 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 198, + 338 + ], + "score": 0.92, + "content": "n _ { h } = t _ { h } / \\eta = 1 / ( \\lambda _ { 3 } \\bar { \\eta _ { } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 324, + 201, + 338 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 278, + 506, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "We verify the above analysis with empirical results. Figure 4 (c)(d) show the trajectories obtained", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "from the effective theory and from neural network training, respectively. The 1D neural representation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "in Figure 4 (d) are manually normalized to zero mean and unit variance. The two trajectories agree", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 231, + 387 + ], + "score": 1.0, + "content": "qualitatively, and it takes about", + "type": "text" + }, + { + "bbox": [ + 231, + 375, + 249, + 385 + ], + "score": 0.89, + "content": "3 n _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "steps for two trajectories to converge to the linear structure. The", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "quantitative differences might be due to the absence of the decoder in the effective theory, which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 316, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 316, + 409 + ], + "score": 1.0, + "content": "assumes the decoder to take infinitesimal step sizes.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 342, + 506, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 304, + 425 + ], + "score": 1.0, + "content": "Dependence of grokking on data size Note that", + "type": "text" + }, + { + "bbox": [ + 305, + 413, + 320, + 424 + ], + "score": 0.9, + "content": "\\ell _ { \\mathrm { e f f } }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "involves averaging over parallelograms in the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 320, + 435 + ], + "score": 1.0, + "content": "training set, it is dependent on training data size, so is", + "type": "text" + }, + { + "bbox": [ + 321, + 424, + 332, + 434 + ], + "score": 0.88, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 424, + 505, + 435 + ], + "score": 1.0, + "content": ". In Figure 5 (a), we plot the dependence of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 118, + 445 + ], + "score": 0.86, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 435, + 424, + 446 + ], + "score": 1.0, + "content": "on training data fraction. There are many datasets with the same data size, so", + "type": "text" + }, + { + "bbox": [ + 425, + 435, + 436, + 445 + ], + "score": 0.87, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 435, + 505, + 446 + ], + "score": 1.0, + "content": "is a probabilistic", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 445, + 193, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 193, + 457 + ], + "score": 1.0, + "content": "function of data size.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 412, + 505, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "Two insights on grokking can be extracted from this plot: (i) When the data fraction is below some", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 202, + 485 + ], + "score": 1.0, + "content": "threshold (around 0.4),", + "type": "text" + }, + { + "bbox": [ + 203, + 473, + 214, + 484 + ], + "score": 0.87, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "is zero with high probability, corresponding to no generalization. This", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 442, + 497 + ], + "score": 1.0, + "content": "again verifies our critical point in Figure 4. (ii) When data size is above the threshold,", + "type": "text" + }, + { + "bbox": [ + 442, + 484, + 453, + 495 + ], + "score": 0.86, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "(on average)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 493, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 381, + 509 + ], + "score": 1.0, + "content": "is an increasing function of data size. This implies that grokking time", + "type": "text" + }, + { + "bbox": [ + 382, + 494, + 420, + 506 + ], + "score": 0.93, + "content": "t \\sim 1 / \\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 493, + 506, + 509 + ], + "score": 1.0, + "content": "decreases as training", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 505, + 350, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 350, + 517 + ], + "score": 1.0, + "content": "data size becomes larger, an important observation from [1].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 461, + 506, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "To verify our effective theory, we compare the grokking steps obtained from real neural network", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 527, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 223, + 551 + ], + "score": 1.0, + "content": "training (defined as steps to", + "type": "text" + }, + { + "bbox": [ + 223, + 533, + 276, + 544 + ], + "score": 0.87, + "content": "\\mathrm { R Q I } > 0 . 9 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 527, + 423, + 551 + ], + "score": 1.0, + "content": ", and those predicted by our theory", + "type": "text" + }, + { + "bbox": [ + 424, + 532, + 467, + 547 + ], + "score": 0.92, + "content": "\\begin{array} { r } { t _ { \\mathrm { t h } } \\sim \\frac { 1 } { \\lambda _ { 3 } \\eta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 534, + 479, + 544 + ], + "score": 0.57, + "content": "\\dot { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 527, + 505, + 551 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 544, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 507, + 559 + ], + "score": 1.0, + "content": "embedding learning rate), shown in Figure 5 (b). The theory agrees qualitatively with neural networks,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "showing the trend of decreasing grokking steps as increasing data size. The quantitative differences", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 567, + 397, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 397, + 579 + ], + "score": 1.0, + "content": "might be explained as the gap between our effective loss and actual loss.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 521, + 507, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "Limitations of the effective theory While our theory defines an effective loss based on the Euclidean", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 226, + 608 + ], + "score": 1.0, + "content": "distance between embeddings", + "type": "text" + }, + { + "bbox": [ + 227, + 595, + 263, + 606 + ], + "score": 0.94, + "content": "\\mathbf { E } _ { i } + \\mathbf { E } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 592, + 281, + 608 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 594, + 322, + 606 + ], + "score": 0.92, + "content": " { \\mathbf { E } } _ { n } + { \\mathbf { E } } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 592, + 506, + 608 + ], + "score": 1.0, + "content": ", one could imagine generalizing the theory to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "define a broader notion of parallogram given by some other metric on the representation space. For", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "instance, if we have a decoder like in Figure 2 (d) then the distance between distinct representations", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "within the same “pizza slice” is low, meaning that representations arranged not in parallelograms", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 639, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 506, + 650 + ], + "score": 1.0, + "content": "w.r.t. the Euclidean metric may be parallelograms with respect to the metric defined by the decoder.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 583, + 506, + 650 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 664, + 343, + 678 + ], + "lines": [ + { + "bbox": [ + 104, + 663, + 345, + 682 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 345, + 682 + ], + "score": 1.0, + "content": "4 Delayed Generalization: A Phase Diagram", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 703 + ], + "score": 1.0, + "content": "So far, we have (1) observed empirically that generalization on algorithmic datasets corresponds with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "the emergence of well-structured representations, (2) defined a notion of representation quality in a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "toy setting and shown that it predicts generalization, and (3) developed an effective theory to describe", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "the learning dynamics of the representations in the same toy setting. We now study how optimizer", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "hyperparameters affect high-level learning performance. In particular, we develop phase diagrams for", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "how learning performance depends on the representation learning rate, decoder learning rate and the", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "decoder weight decay. 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(a) Dependence of", + "type": "text" + }, + { + "bbox": [ + 206, + 227, + 218, + 238 + ], + "score": 0.87, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "on training data fraction. Above the critical data fraction (around 0.4),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 219, + 251 + ], + "score": 1.0, + "content": "as data size becomes larger,", + "type": "text" + }, + { + "bbox": [ + 220, + 238, + 231, + 249 + ], + "score": 0.87, + "content": "\\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 237, + 357, + 251 + ], + "score": 1.0, + "content": "increases hence grokking time", + "type": "text" + }, + { + "bbox": [ + 357, + 237, + 395, + 249 + ], + "score": 0.92, + "content": "t \\sim 1 / \\lambda _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "(predicted by our effective", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 248, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 347, + 262 + ], + "score": 1.0, + "content": "theory) decreases. (b) Comparing grokking steps (defined as", + "type": "text" + }, + { + "bbox": [ + 348, + 249, + 399, + 260 + ], + "score": 0.87, + "content": "\\mathrm { R Q I } > 0 . 9 5 ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 248, + 506, + 262 + ], + "score": 1.0, + "content": "predicted by the effective", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 257, + 463, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 266, + 273 + ], + "score": 1.0, + "content": "theory with real neural network results.", + "type": "text" + }, + { + "bbox": [ + 266, + 259, + 306, + 271 + ], + "score": 0.92, + "content": "\\eta = 1 \\bar { 0 } ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 257, + 463, + 273 + ], + "score": 1.0, + "content": "is the learning rate of the embeddings.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "the learning dynamics of the representations in the same toy setting. We now study how optimizer", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "hyperparameters affect high-level learning performance. In particular, we develop phase diagrams for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "how learning performance depends on the representation learning rate, decoder learning rate and the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "decoder weight decay. These parameters are of interest since they most explicitly regulate a kind of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 344, + 385, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 385, + 356 + ], + "score": 1.0, + "content": "competition between the encoder and decoder, as we elaborate below.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 376, + 256, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 258, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 258, + 390 + ], + "score": 1.0, + "content": "4.1 Phase diagram of a toy model", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "Training details We update the representation and the decoder with different optimizers. For the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 408, + 507, + 425 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 359, + 425 + ], + "score": 1.0, + "content": "1D embeddings, we use the Adam optimizer with learning rate", + "type": "text" + }, + { + "bbox": [ + 360, + 410, + 411, + 422 + ], + "score": 0.91, + "content": "[ 1 0 ^ { - 5 } , 1 0 ^ { - 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 408, + 507, + 425 + ], + "score": 1.0, + "content": "and zero weight decay.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 390, + 435 + ], + "score": 1.0, + "content": "For the decoder, we use an AdamW optimizer with the learning rate in", + "type": "text" + }, + { + "bbox": [ + 390, + 421, + 442, + 433 + ], + "score": 0.91, + "content": "[ 1 0 ^ { - 5 } , 1 0 ^ { - 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "and the weight", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 235, + 445 + ], + "score": 1.0, + "content": "decay in [0, 10] (regression) or", + "type": "text" + }, + { + "bbox": [ + 235, + 433, + 261, + 444 + ], + "score": 0.36, + "content": "[ 0 , 2 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "(classification). 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The ideal phase (comprehension) requires representation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 640, + 333, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 333, + 652 + ], + "score": 1.0, + "content": "learning to be faster, but not too much, than the decoder.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 563, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "Drawing from an analogy to physical systems, one can think of embedding vectors as a group of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "particles. In our effective theory from Section 3.2, the dynamics of the particles are described only", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "by their relative positions, in that sense, structure forms mainly due to inter-particle interactions (in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "reality, these interactions are mediated by the decoder and the loss). The decoder plays the role of an", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "environment exerting external forces on the embeddings. 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(a) shows the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 393, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 507, + 406 + ], + "score": 1.0, + "content": "competition between representation and decoder. (b)(c)(d): each phase diagram contains four phases:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "comprehension, grokking, memorization and confusion, defined in Table 1. In (b)(c)(d), grokking is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 416, + 332, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 332, + 427 + ], + "score": 1.0, + "content": "sandwiched between comprehension and memorization.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + } + ], + "index": 17.75 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 406, + 461 + ], + "score": 1.0, + "content": "Universality of phase diagrams We fix the embedding learning rate to be", + "type": "text" + }, + { + "bbox": [ + 407, + 447, + 428, + 459 + ], + "score": 0.91, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "and sweep instead", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "decoder weight decay in Figure 6 (b)(c)(d). The phase diagrams correspond to addition regression (b),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 469, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 484 + ], + "score": 1.0, + "content": "addition classification (c) and permutation regression (d), respectively. Common phenomena emerge", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "from these different tasks: (i) they all include four phases; (ii) The top right corner (a fast and capable", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "decoder) is the memorization phase; (iii) the bottom right corner (a fast and simple decoder) is the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "confusion phase; (iv) grokking is sandwiched between comprehension and memorization, which", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 497, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 497, + 526 + ], + "score": 1.0, + "content": "seems to imply that it is an undesirable phase that stems from improperly tuned hyperparameters.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 222, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 224, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 224, + 552 + ], + "score": 1.0, + "content": "4.2 Beyond the toy model", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "We conjecture that many of the principles which we saw dictate the training dynamics in the toy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "model also apply more generally. Below, we will see how our framework generalizes to transformer", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 295, + 592 + ], + "score": 1.0, + "content": "architectures for the task of addition modulo", + "type": "text" + }, + { + "bbox": [ + 295, + 582, + 301, + 591 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 580, + 505, + 592 + ], + "score": 1.0, + "content": ", a minimal reproducible example of the original", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 591, + 186, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 186, + 604 + ], + "score": 1.0, + "content": "grokking paper [1].", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 168, + 619 + ], + "score": 1.0, + "content": "We first encode", + "type": "text" + }, + { + "bbox": [ + 169, + 607, + 199, + 618 + ], + "score": 0.9, + "content": "p = 5 3", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 606, + 507, + 619 + ], + "score": 1.0, + "content": "integers into 256D learnable embeddings, then pass two integers to a decoder-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 618, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 504, + 630 + ], + "score": 1.0, + "content": "only transformer architecture. For simplicity, we do not encode the operation symbols here. The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 627, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 506, + 643 + ], + "score": 1.0, + "content": "outputs from the last layer are concatenated and passed to a linear layer for classification. Training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "both the encoder and the decoder with the same optimizer (i.e., with the same hyperparameters)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "leads to the grokking phenomenon. Generalization appears much earlier once we lower the effective", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 661, + 383, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 383, + 675 + ], + "score": 1.0, + "content": "decoder capacity with weight decay (full phase diagram in Figure 7).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "Early on, the model is able to perfectly fit the training set while having no generalization. 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(a) shows the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 393, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 507, + 406 + ], + "score": 1.0, + "content": "competition between representation and decoder. (b)(c)(d): each phase diagram contains four phases:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "comprehension, grokking, memorization and confusion, defined in Table 1. In (b)(c)(d), grokking is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 416, + 332, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 332, + 427 + ], + "score": 1.0, + "content": "sandwiched between comprehension and memorization.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + } + ], + "index": 17.75 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 406, + 461 + ], + "score": 1.0, + "content": "Universality of phase diagrams We fix the embedding learning rate to be", + "type": "text" + }, + { + "bbox": [ + 407, + 447, + 428, + 459 + ], + "score": 0.91, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "and sweep instead", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "decoder weight decay in Figure 6 (b)(c)(d). The phase diagrams correspond to addition regression (b),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 469, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 484 + ], + "score": 1.0, + "content": "addition classification (c) and permutation regression (d), respectively. Common phenomena emerge", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "from these different tasks: (i) they all include four phases; (ii) The top right corner (a fast and capable", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "decoder) is the memorization phase; (iii) the bottom right corner (a fast and simple decoder) is the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "confusion phase; (iv) grokking is sandwiched between comprehension and memorization, which", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 497, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 497, + 526 + ], + "score": 1.0, + "content": "seems to imply that it is an undesirable phase that stems from improperly tuned hyperparameters.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 447, + 506, + 526 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 222, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 224, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 224, + 552 + ], + "score": 1.0, + "content": "4.2 Beyond the toy model", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "We conjecture that many of the principles which we saw dictate the training dynamics in the toy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "model also apply more generally. Below, we will see how our framework generalizes to transformer", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 295, + 592 + ], + "score": 1.0, + "content": "architectures for the task of addition modulo", + "type": "text" + }, + { + "bbox": [ + 295, + 582, + 301, + 591 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 580, + 505, + 592 + ], + "score": 1.0, + "content": ", a minimal reproducible example of the original", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 591, + 186, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 186, + 604 + ], + "score": 1.0, + "content": "grokking paper [1].", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 557, + 506, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 168, + 619 + ], + "score": 1.0, + "content": "We first encode", + "type": "text" + }, + { + "bbox": [ + 169, + 607, + 199, + 618 + ], + "score": 0.9, + "content": "p = 5 3", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 606, + 507, + 619 + ], + "score": 1.0, + "content": "integers into 256D learnable embeddings, then pass two integers to a decoder-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 618, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 504, + 630 + ], + "score": 1.0, + "content": "only transformer architecture. For simplicity, we do not encode the operation symbols here. The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 627, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 506, + 643 + ], + "score": 1.0, + "content": "outputs from the last layer are concatenated and passed to a linear layer for classification. Training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "both the encoder and the decoder with the same optimizer (i.e., with the same hyperparameters)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "leads to the grokking phenomenon. Generalization appears much earlier once we lower the effective", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 661, + 383, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 383, + 675 + ], + "score": 1.0, + "content": "decoder capacity with weight decay (full phase diagram in Figure 7).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 606, + 507, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "Early on, the model is able to perfectly fit the training set while having no generalization. We study", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "the embeddings at different training times and find that neither PCA (shown in Figure 1) nor t-SNE", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "(not shown here) reveal any structure. Eventually, validation accuracy starts to increase, and perfect", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "generalization coincides with the PCA projecting the embeddings into a circle in 2D. Of course, no", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 267, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 283 + ], + "score": 1.0, + "content": "choice of dimensionality reduction is guaranteed to find any structure, and thus, it is challenging", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 280, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 505, + 291 + ], + "score": 1.0, + "content": "to show explicitly that generalization only occurs when a structure exists. Nevertheless, the fact", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "that, when coupled with the implicit regularization of the optimizer for sparse solutions, such a clear", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "structure appears in a simple PCA so quickly at generalization time suggests that our analysis in", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "the toy setting is applicable here as well. This is also seen in the evolution of the entropy of the", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 322, + 507, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 375, + 338 + ], + "score": 1.0, + "content": "explained variance ratio in the PCA of the embeddings (defined as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 376, + 323, + 456, + 336 + ], + "score": 0.92, + "content": "\\begin{array} { r } { S = - \\sum _ { i } \\sigma _ { i } \\log \\sigma _ { i } } \\end{array}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 456, + 322, + 484, + 338 + ], + "score": 1.0, + "content": "where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 484, + 325, + 495, + 335 + ], + "score": 0.85, + "content": "\\sigma _ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 495, + 322, + 507, + 338 + ], + "score": 1.0, + "content": "is", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "the fractional variance explained by the ith principal component). As seen in Figure 7, the entropy", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "increases up to generalization time then decreases drastically afterwards which would be consistent", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "with the conjecture that generalization occurs when a low-dimensional structure is discovered. The", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "decoder then primarily relies on the information in this low-dimensional manifold and essentially", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "“prunes” the rest of the high-dimensional embedding space. Another interesting insight appears when", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "we project the embeddings at initialization onto the principal axes at the end of training. Some of the", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "structure required for generalization exists before training hinting at a connection with the Lottery", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 411, + 321, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 321, + 423 + ], + "score": 1.0, + "content": "Ticket Hypothesis. See Appendix K for more details.", + "type": "text", + "cross_page": true + } + ], + "index": 21 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 126, + 75, + 485, + 185 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 75, + 485, + 185 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 75, + 485, + 185 + ], + "spans": [ + { + "bbox": [ + 126, + 75, + 485, + 185 + ], + "score": 0.973, + "type": "image", + "image_path": "f97ba3b50d927fef5d6356bddf07766187a7aa148dc09b2096966b4a90acf17f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 126, + 75, + 485, + 111.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 126, + 111.66666666666666, + 485, + 148.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 126, + 148.33333333333331, + 485, + 184.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 191, + 506, + 246 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 203 + ], + "score": 1.0, + "content": "Figure 7: Left: Evolution of the effective dimension of the embeddings (defined as the exponential of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "score": 1.0, + "content": "the entropy) during training and evaluated over 100 seeds. Center: Effect of dropout on speeding up", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "generalization. Right: Phase diagram of the transformer architecture. 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As seen in Figure 7, the entropy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "increases up to generalization time then decreases drastically afterwards which would be consistent", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "with the conjecture that generalization occurs when a low-dimensional structure is discovered. 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Center: Effect of dropout on speeding up", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "generalization. Right: Phase diagram of the transformer architecture. 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[2] describe the circuit that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "transformers use to perform modular addition, track its formation over training, and broadly suggest", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "that grokking is related to the phenomenon of “phase changes” in neural network training. [3, 4]", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "provided earlier speculative, informal conjectures on grokking [3, 4]. 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We developed an effective theory of representation learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "dynamics (in a toy setting) which predicts the critical dependence of learning on the training data", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "fraction. 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While we have mostly focused on a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 640, + 456, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 456, + 653 + ], + "score": 1.0, + "content": "toy model, we find preliminary evidence that our results generalize to the setting of [1].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Our work can be viewed as a step towards a statistical physics of deep learning, connecting the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "“microphysics” of low-level network dynamics with the “thermodynamics” of high-level model", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "behavior. 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In a recent work, researchers at Anthropic [23], connect a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 500, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 507, + 513 + ], + "score": 1.0, + "content": "sudden decrease in loss during training with the emergence of induction heads within their models.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "They analogize their work to statistical physics, since it bridges a “microscopic”, mechanistic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 456, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 456, + 535 + ], + "score": 1.0, + "content": "understanding of networks with “macroscopic” facts about overall model performance.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 444, + 507, + 535 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 549, + 183, + 562 + ], + "lines": [ + { + "bbox": [ + 104, + 547, + 185, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 185, + 565 + ], + "score": 1.0, + "content": "6 Conclusion", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 588 + ], + "score": 1.0, + "content": "We have shown how, in both toy models and general settings, that representation enables generalization", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "score": 1.0, + "content": "when it reflects structure in the data. 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While we have mostly focused on a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 640, + 456, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 456, + 653 + ], + "score": 1.0, + "content": "toy model, we find preliminary evidence that our results generalize to the setting of [1].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 573, + 507, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Our work can be viewed as a step towards a statistical physics of deep learning, connecting the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "“microphysics” of low-level network dynamics with the “thermodynamics” of high-level model", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "behavior. 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/dev/null +++ b/parse/dev/AXDNM76T1nc/AXDNM76T1nc_content_list.json @@ -0,0 +1,1793 @@ +[ + { + "type": "text", + "text": "Video PreTraining (VPT): Learning to Act by Watching Unlabeled Online Videos ", + "text_level": 1, + "bbox": [ + 225, + 122, + 774, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Bowen Baker⇤† bowen@openai.com ", + "bbox": [ + 235, + 226, + 372, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ilge Akkaya⇤† ilge@openai.com ", + "bbox": [ + 441, + 226, + 560, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Peter Zhokhov⇤† peterz@openai.com ", + "bbox": [ + 627, + 226, + 761, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Joost Huizinga⇤† joost@openai.com ", + "bbox": [ + 233, + 275, + 359, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jie Tang⇤† jietang@openai.com ", + "bbox": [ + 424, + 275, + 562, + 304 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Adrien Ecoffet⇤† adrien@openai.com ", + "bbox": [ + 629, + 275, + 764, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Brandon Houghton⇤† brandon@openai.com ", + "bbox": [ + 228, + 324, + 379, + 353 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Raul Sampedro⇤† raulsamg@gmail.com ", + "bbox": [ + 434, + 324, + 581, + 353 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jeff Clune⇤†‡ jclune@gmail.com ", + "bbox": [ + 642, + 324, + 769, + 353 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 388, + 535, + 405 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Pretraining on noisy, internet-scale datasets has been heavily studied as a technique for training models with broad, general capabilities for text, images, and other modalities. 1–6 However, for many sequential decision domains such as robotics, video games, and computer use, publicly available data does not contain the labels required to train behavioral priors in the same way. We extend the internet-scale pretraining paradigm to sequential decision domains through semi-supervised imitation learning wherein agents learn to act by watching online unlabeled videos. Specifically, we show that with a small amount of labeled data we can train an inverse dynamics model accurate enough to label a huge unlabeled source of online data – here, online videos of people playing Minecraft – from which we can then train a general behavioral prior. Despite using the native human interface (mouse and keyboard at $2 0 \\mathrm { H z }$ ), we show that this behavioral prior has nontrivial zeroshot capabilities and that it can be fine-tuned, with both imitation learning and reinforcement learning, to hard-exploration tasks that are impossible to learn from scratch via reinforcement learning. For many tasks our models exhibit humanlevel performance, and we are the first to report computer agents that can craft diamond tools, which can take proficient humans upwards of 20 minutes (24,000 environment actions) of gameplay to accomplish. ", + "bbox": [ + 233, + 420, + 766, + 669 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 695, + 310, + 713 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Work in recent years has demonstrated the efficacy of pretraining large and general foundation models 7 on noisy internet-scale datasets for use in downstream tasks in natural language 1–4, computer vision, 5,6,8 and multi-task models. 9 For sequential decision domains (e.g. robotics, game playing, and computer usage) where agents must repeatedly act within an environment, a wealth of data also exists on the web; however, most of this data is in the form of unlabeled video (i.e. without the actions taken at each frame), making it much less straightforward to train a behavioral prior in these domains than it is in e.g. natural language. In a few rare settings, such as Chess, Go, and StarCraft, there already exist large datasets with action labels from various online platforms that researchers have used for imitation learning. 10,11 When large labeled datasets do not exist, the canonical strategy for training capable agents is reinforcement learning (RL), 12 which can be sample inefficient and expensive for hard-exploration problems. 13–19 Many virtual tasks, e.g. navigating websites, using Photoshop, booking flights, etc., can be very hard to learn with RL and do not have large, commonly available sources of labeled data. 20,21 In this paper, we seek to extend the paradigm of training large, general-purpose foundation models to sequential decision domains by utilizing freely available internet-scale unlabeled video datasets with a simple semi-supervised imitation learning method. We call this method Video PreTraining (VPT) and demonstrate its efficacy in the domain of Minecraft. ", + "bbox": [ + 174, + 727, + 825, + 810 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 229 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Existing semi-supervised imitation learning methods aim to learn with few or no explicit action labels; however, they generally rely on the policy’s ability to explore the environment throughout training, making them susceptible to exploration bottlenecks. 22–26 Furthermore, most prior semi-supervised imitation learning work was tested in the relatively low data regime; because we experiment with far more data ( $\\mathord { \\sim } 7 0 \\mathrm { k }$ hours of unlabeled video), we hypothesize that we can achieve good performance with a much simpler method, a trend that has proven true for pretraining in other modalities such as text. 1 In particular, given a large but unlabeled dataset, we propose generating pseudo-labels by gathering a small amount of labeled data to train an inverse dynamics model (IDM) that predicts the action taken at each timestep in a video. Behavioral cloning (BC) can require a large amount of data because the model must learn to infer intent and the distribution over future behaviors from only past observations. In contrast, the inverse dynamics modeling task is simpler because it is non-causal, meaning it can look at both past and future frames to infer actions. In most settings, environment mechanics are far simpler than the breadth of human behavior that can take place within the environment, suggesting that non-causal IDMs could require far less data to train than causal BC models. Using pseudo-labels generated from the IDM, we then train a model to mimic the distribution of behavior in the previously unlabeled dataset with standard behavioral cloning at scale, which does not require any model rollouts and thus does not suffer from any potential exploration bottlenecks in the environment. Finally, we show we can fine-tune this model to downstream tasks with either behavioral cloning or reinforcement learning. ", + "bbox": [ + 174, + 234, + 825, + 498 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We chose to test our method in Minecraft because it (a) is one of the most actively played games in the world27 and thus has a wealth of online video data, (b) is an open-ended sandbox game with an extremely wide variety of potential things to do, build, and collect, making our results more applicable to real-world applications such as computer usage, which also tends to be varied and open-ended, and (c) has already garnered interest by the RL community as a research domain due to its complexity and correspondingly difficult exploration challenges. 28–32 In this work we use the native human interface for Minecraft so that we can (1) most accurately model the human behavior distribution and reduce domain shift between video data and the environment, (2) make data collection easier by allowing our human contractors to play the game without modification, and (3) eliminate the need to hand-engineer a custom interface for models to interact with the environment. This choice means that our models play at 20 frames per second and must use a mouse and keyboard interface to interact with human GUIs for crafting, smelting, trading, etc., including dragging items to specific slots or navigating the recipe book with the mouse cursor (Fig. 1). Compared to prior work in Minecraft that uses a lower frame rate and constructs crafting and attacking macros, 31,33–35 using the native human interface drastically increases the environment’s exploration difficulty, making most simple tasks near impossible with RL from scratch. Even the simple task of gathering a single wooden log while already facing a tree takes 60 consecutive attack actions with the human interface, meaning the chance for a naive random policy to succeed is $1 / 2 ^ { 6 0 }$ . While this paper shows results in Minecraft only, the VPT method is general and could be applied to any domain. ", + "bbox": [ + 174, + 505, + 594, + 683 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/9ecaae8777654a20f20ae479e79fd1eb563c824f512fc612e394f000c310a23c.jpg", + "image_caption": [ + "Figure 1: Example Minecraft crafting GUI. Agents use the mouse and keyboard to navigate menus and drag and drop items. " + ], + "image_footnote": [], + "bbox": [ + 607, + 502, + 821, + 604 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 684, + 825, + 835 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In Section 4 we show that the VPT foundation model has nontrivial zero-shot performance, accomplishing tasks impossible to learn with RL alone, such as crafting planks and crafting tables (tasks requiring a human proficient in Minecraft a median of 50 seconds or ${ \\sim } 9 7 0 $ consecutive actions). Through fine-tuning with behavioral cloning to smaller datasets that target more specific behavior distributions, our agent is able to push even further into the technology tree, crafting stone tools (taking a human a median of 2.3 minutes or ${ \\sim } 2 7 9 0$ actions). Finally, fine-tuning via RL produces the most dramatic improvements: our agent is able to craft diamond tools, an unprecedented result in Minecraft made even more challenging by using the native human interface. This task requires a proficient human a median upwards of 20 minutes or ${ \\sim } 2 4 0 0 0$ actions. The main contributions of this work are (1) we are the first to show promising results applying semi-supervised imitation learning to extremely large, noisy, and freely available video datasets for sequential decision domains, (2) we show that such pretraining plus fine-tuning enables agents to solve tasks that were otherwise impossible to learn, (3) we show that labeled contractor data is far more efficiently used within the VPT method than it would be by directly training a foundation model from it and (4) we open source our contractor data, trained model weights, and Minecraft environment for future research into learning to act via semi-supervised imitation learning at scale. ", + "bbox": [ + 174, + 842, + 825, + 897 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 257 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Preliminaries and Related Work ", + "text_level": 1, + "bbox": [ + 176, + 276, + 475, + 294 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Imitation learning methods 36–39 seek to construct a policy that accurately models the distribution of behavior in some dataset $D = \\{ ( o _ { i } , a _ { i } ) \\}$ , $i \\in \\{ 1 . . . N \\}$ of action-observation pairs. In order to roll out these policies in an environment, they must be causal, meaning they condition on observations from the current timestep $t$ and past timesteps only, i.e. $\\pi \\sim p ( \\bar { a } _ { t } | o _ { 1 } . . . o _ { t } )$ . Imitation learning is simplest when demonstrations are labeled with corresponding actions. Imitating labeled trajectories has seen success in aerial vehicles, 40,41 self-driving cars, 42,43 board games, 10,44 and video games. 11,45 ", + "bbox": [ + 174, + 306, + 825, + 391 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "When labeled demonstrations are not available, standard behavioral cloning will not work; however, there is a large body of work in imitating behavior from unlabeled demonstrations. 23 For instance, GAIL24 constructs an adversarial objective incentivizing the trained policy to exhibit behaviors indistinguishable from those in the target dataset. Edwards et al. 46 propose to first learn a latent policy using unlabeled demonstrations and then map the learned latent actions to real actions using environment interaction. Peng et al. 47 use motion-capture methods to track agent positions in videos and then train RL agents to match these waypoints. Similarly, Behbahani et al. 48 and Aytar et al. 49 task a RL agent to match waypoints; however, their waypoints are embeddings from unsupervised feature learning models. Pathak et al. 50 and Nair et al. 51 train goal conditioned policies to take actions that move towards expert-provided goal states expressed as high dimensional visual waypoints. Most similar to our own work, Torabi et al. 25 simultaneously train (1) an inverse dynamics model (IDM), 52 which aims to uncover the underlying action between timesteps given observations of past and future timesteps, e.g. $p _ { \\mathrm { I D M } } ( a _ { t } | o _ { t } , o _ { t + 1 } )$ , and (2) a behavioral cloning (BC) model on trajectories of observations labeled with the IDM. Data to train the IDM is collected by rolling out the BC model in the target environment such that both models improve in tandem. However, at any point in training if there are sequences in the dataset that the IDM performs poorly on, it requires that the BC model perform those or similar sequences in order for the IDM to improve and correctly label them. Therefore, if the BC model does not explore efficiently, it could severely slow down learning. In order to avoid this potential issue we opted for a simpler two-stage approach: we first train an IDM on a small number of labeled trajectories collected from human contractors (they play the game as would normally as we record their keypresses and mouse movements). Because human contractors reach most relevant parts of the state space, we can hold the IDM fixed throughout BC training. ", + "bbox": [ + 173, + 396, + 826, + 702 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Compared to most previous work in semi-supervised imitation learning, we experiment in the much more complex and open-ended environment of Minecraft. Minecraft is a voxel-based 3D video game that, due its popularity and wide variety of mechanics, has attracted a vast amount of RL research. 28,29,31–35,53–61 A large body of work focuses on small, custom-made Minecraft worlds with tasks such as navigation, 54,61 block placing, 55,56 instruction following, 59,60 combat, 57 and others. 29,32,58 Work operating in the massive, randomly generated environments of Minecraft itself has included hill climbing, 53 automated curriculum learning31 and, most closely related to the RL experiments presented in Sec. 4.4, diamond mining. 28,33–35 However, to the best of our knowledge, there is no published work that operates in the full, unmodified human action space, which includes drag-and-drop inventory management and item crafting. ", + "bbox": [ + 174, + 707, + 825, + 845 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Methods ", + "text_level": 1, + "bbox": [ + 174, + 858, + 279, + 875 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Inverse Dynamics Models (IDM) VPT, illustrated in Figure 2, requires we first collect a small amount of labeled contractor data with which to train an inverse dynamics model $p _ { \\mathrm { I D M } } ( a _ { t } | o _ { 1 \\ldots T } )$ ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/064a1527f9fa1ec271241672e70e73bab60b52c51f73a35ec94e5f1f718ad9c6.jpg", + "image_caption": [ + "Figure 2: Video Pretraining (VPT) Method Overview. " + ], + "image_footnote": [], + "bbox": [ + 173, + 87, + 821, + 236 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "which seeks to minimize the negative log-likelihood of an action at timestep $t$ given a trajectory of $T$ observations $o _ { t } ~ : ~ t \\in [ 1 . . . T ]$ . In contrast to an imitation learning policy, the IDM can be non-causal, meaning its prediction for $a _ { t }$ can be a function of both past and future events, i.e. $o _ { t ^ { \\prime } > t }$ . Compared to the behavioral cloning objective of modeling the distribution of human intent given past frames only, we hypothesize that inverting environment dynamics is easier and more data efficient to learn. Indeed, Sec. 4.1 will show that the IDM objective is much easier to learn, and furthermore Sec. 4.6 will show that with very little labeled data (as few as 100 hours) we can train a fairly accurate IDM. This IDM can be used to label online videos, providing the large amount of data required for the harder task of behavioral cloning. See appendices D and B for IDM training and data collection details. ", + "bbox": [ + 173, + 266, + 826, + 392 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Data Filtering We gather a large dataset of Minecraft videos by searching the web for related keywords (Appendix A). Online videos often (1) include overlaid artifacts, such as a video feed of the player’s face, channel logos, watermarks, etc., (2) are collected from platforms other than a computer with different gameplay, or (3) are from different game modes, e.g. in Minecraft we only want \"survival mode\" where players start from scratch and must gather or craft all their items. We call data “clean” if it does not contain visual artifacts and is from survival mode, and call all other data “unclean.” With enough data, a large enough model, and enough training compute, a BC model trained on both unclean and clean videos would likely still perform well in a clean Minecraft environment. However, for simplicity and training compute efficiency, we choose to filter out unclean segments of video (note that a video may contain both clean and unclean segments). We do this by training a model to filter out unclean segments using a small dataset (8800) of images sampled from online videos labeled by contractors as clean or unclean. We did not tune this process as it is fairly standard; see Appendix A.2 for more details and ablations showing data cleaning is beneficial. ", + "bbox": [ + 173, + 398, + 825, + 579 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "VPT Foundation Model We train a foundation model with standard behavioral cloning, i.e. minimizing the negative log-likelihood of actions predicted by the IDM on clean data. For a particular trajectory of length $T$ we minimize ", + "bbox": [ + 176, + 593, + 825, + 636 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/cfbfd741ea4af0f1d32c152b95117fc0595c9b22619cd87951e0a632b5a552d2.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\sum _ { t \\in [ 1 \\ldots T ] } - \\log \\pi _ { \\theta } ( a _ { t } | o _ { 1 } , \\ldots , o _ { t } ) , { \\mathrm { w h e r e ~ } } a _ { t } \\sim p _ { \\mathrm { I D M } } ( a _ { t } | o _ { 1 } , \\ldots , o _ { t } , \\ldots , o _ { T } )\n$$", + "text_format": "latex", + "bbox": [ + 240, + 643, + 756, + 680 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As we will see in the following sections, this model exhibits nontrivial zero-shot behavior and can be fine-tuned with both imitation learning and RL to perform even more complex skills. ", + "bbox": [ + 174, + 694, + 825, + 723 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Results ", + "text_level": 1, + "bbox": [ + 174, + 742, + 266, + 760 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 Performance of the Inverse Dynamics Model ", + "text_level": 1, + "bbox": [ + 174, + 773, + 522, + 790 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The IDM architecture is comprised primarily of a temporal convolution layer, a ResNet 63 image processing stack, and residual unmasked attention layers, from which the IDM simultaneously predicts keypresses and mouse movements (see Appendix D for IDM architecture and training details). A key hypothesis behind our work is that IDMs can be trained with a relatively small amount of labeled data. While more data improves both mouse movement and keypress predictions, our best IDM trains on only 1962 hours of data (compared to the $\\mathrm { \\sim } 7 0 \\mathrm { k }$ hours of clean data we collected from the internet) and achieves $9 0 . 6 \\%$ keypress accuracy and a $0 . 9 7 ~ R ^ { 2 }$ for mouse movements evaluated on a held-out validation set of contractor-labeled data (Figure 3 left). ", + "bbox": [ + 173, + 799, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/4b3ea22386575829558b307187db5f20ebab1ca9a8165026d52849f357087422.jpg", + "image_caption": [ + "Figure 3: (Left) IDM keypress accuracy and mouse movement $R ^ { 2 }$ (explained variance 62) as a function of dataset size. (Right) IDM vs. behavioral cloning data efficiency. " + ], + "image_footnote": [], + "bbox": [ + 174, + 89, + 823, + 181 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 3 (right) validates our hypothesis that IDMs are far more data efficient than BC models, likely because inverting environment mechanics is far easier than modeling the entire distribution of human behavior. The IDM is two orders of magnitude more data efficient than a BC model trained on the same data and improves more quickly with more data. This evidence supports our hypothesis that it is more effective to use contractor data within the VPT pipeline by training an IDM than it is to train a foundation model from contractor data directly (Sections 4.5 and 4.6 provide additional evidence). Due to their data efficiency, training an IDM uses a negligible fraction of the overall compute needed to train a VPT model. ", + "bbox": [ + 173, + 253, + 825, + 364 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 VPT Foundation Model Training and Zero-Shot Performance ", + "text_level": 1, + "bbox": [ + 173, + 382, + 640, + 397 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/34a95ee247846f0d6edcf6bdccdb1ad4fe749757d92015a2c7b77655f520b49f.jpg", + "image_caption": [ + "Figure 4: (Left) Training and validation loss on the web_clean internet dataset with IDM pseudolabels, and loss on the main IDM contractor dataset, which has ground-truth labels but is out-ofdistribution (see text). (Right) Amount a given item was collected per episode averaged over 2500 60-minute survival episodes as a function of training epoch, shaded with the standard error of the mean. Basic mining refers to collection of dirt, gravel, or sand (all materials that can be gathered without tools). Logs are obtained by repeatedly hitting trees for three seconds, a difficult feat for an RL agent to achieve as we show in Sec. 4.4. Planks can be crafted from logs, and crafting tables crafted from planks. Crafting requires using in-game crafting GUIs, and proficient humans take a median of 50 seconds (970 consecutive actions) to make a crafting table. " + ], + "image_footnote": [], + "bbox": [ + 176, + 417, + 823, + 506 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now explore the emergent behavior learned by a behavioral cloning policy trained on an extremely large, but noisy, internet dataset labeled with our IDM. To collect the unlabeled internet dataset, we searched for publicly available videos of Minecraft play with search terms such as “minecraft survival for beginners.” These searches resulted in ${ \\sim } 2 7 0 \\mathrm { k }$ hours of video, which we filtered down to “clean” video segments yielding an unlabeled dataset of $\\mathrm { \\sim } 7 0 \\mathrm { k }$ hours, which we refer to as web_clean (Appendix A has further details on data scraping and filtering). We then generated pseudo-labels for web_clean with our best IDM (Section 3) and then trained the VPT foundation model with behavioral cloning. Preliminary model scaling experiments suggested that our model could benefit from 30 epochs of training and that a 0.5 billion parameter model was required to stay in the efficient learning regime 64 for that training duration (Appendix H shows results comparing model size and the benefit of scaling to 0.5B parameters), which took ${ \\sim } 9$ days on 720 V100 GPUs. ", + "bbox": [ + 173, + 656, + 825, + 808 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We evaluate our models by measuring validation loss (Fig. 4, left) and rolling them out in the Minecraft environment. Unless otherwise noted, in all environment evaluations we spawn agents in a standard survival mode game where they play for 60 minutes, i.e. 72000 consecutive actions, and we plot the mean and shade the standard error of the mean for various game statistics such as crafting and collection rates (Fig. 4, right). The VPT foundation model quickly learns to chop down trees to collect logs, a task we found near impossible for an RL agent to achieve with the native human interface (Sec. 4.4). It also learns to craft those logs into wooden planks and then use those planks to craft a crafting table, which are required to unlock most other technology in the game and take a human proficient in Minecraft approximately 50 seconds (970 consecutive actions) to collect. While these behaviors are fairly complex in the native human action space, the VPT foundation model crafts these items at a rate far below that of our proficient contractors, e.g. on average our contractors craft 5.44 crafting tables in 60 minutes of play versus 0.19 for the foundation model. The model also crafts a non-negligible amount of wooden sticks, which are required to make wooden tools; collects various flowers and crafts dyes from them; kills zombies that appear during the night; hunts wild animals; collects various berries and mushrooms and eats them; and finds game-generated villages from which to collect various rare items from chests. The model also learned to navigate uneven terrain, swim, and pillar jump, which involves the agent repeatedly jumping and quickly placing a block below itself such that it climbs upward by making a pillar.(iv) ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 243 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "While training and validation loss decrease healthily over training (Fig. 4, left), loss on our contractor dataset (which the VPT model does not train on) begins increasing after 7 epochs. Contractor data could be out-of-distribution because our contractors may have a different distribution of play or because there is some impactful visual domain shift compared to videos from the web, and we provide some evidence for this phenomenon in Appendix H. While one could have expected this would be predictive of declining evaluation performance, we do not see notable game statistics from the VPT foundation model rollouts (Figure 4, right) decrease over training, and in the next section we show that BC fine-tuning performance continually improves as the VPT foundation model trains. ", + "bbox": [ + 173, + 250, + 825, + 361 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 Fine-Tuning with Behavioral Cloning ", + "text_level": 1, + "bbox": [ + 176, + 381, + 472, + 396 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Foundation models are designed to have a broad behavior profile and be generally capable across a wide variety of tasks. To incorporate new knowledge or allow them to specialize on a narrower task distribution, it is common practice to fine-tune these models to smaller, more specific datasets. 1 The VPT foundation model trained on the broad web_clean dataset had nontrivial zero-shot performance; it was able to craft a crafting table yet unable to go past this in the technology tree. As a case study into BC fine-tuning, we attempt to improve the VPT foundation model’s ability to collect and craft these “early game” items by fine-tuning to two narrower datasets targeted at Minecraft behavior within the first few minutes of players starting in a fresh world. In the first dataset, contractor_house, contractors have 10 minutes to build a basic house from scratch using primarily wood, sand, and dirt. Collecting contractor data can be difficult and expensive, so we also construct a dataset earlygame_keyword by searching for videos online with descriptions that match keywords such as “new world”, “let’s play episode 1”, etc.; this is a subset of web_clean and is labeled with the IDM. See Appendix B.4 and A.3 for full descriptions of both datasets. ", + "bbox": [ + 174, + 409, + 825, + 589 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/6de54e50401bab8c4182be82806eb8a59bcf1d9c74a1f831f1c8e532f5d5450c.jpg", + "image_caption": [ + "Figure 5: (Left) Collection and crafting rates for three policies: the zero-shot VPT foundation model, and the VPT foundation model BC fine-tuned to the earlygame_keyword or contractor_house datasets. BC fine-tuning to either dataset improves performance, including (for the contractor_house dataset) yielding wooden and stone tools. Proficient Minecraft players take a median of 1.2 minutes (1390 actions) to construct wooden tools and 2.3 minutes (2790 actions) to construct stone tools. (Right) Collection and crafting rates for VPT foundation model snapshots throughout training after they are BC fine-tuned to the contractor_house dataset. In general, crafting-related behaviors increase throughout foundation model training. Fig. 4 defines the other task terms (logs, planks, crafting tables, and total crafting). " + ], + "image_footnote": [], + "bbox": [ + 173, + 606, + 821, + 738 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Fine-tuning to earlygame_keyword results in a large boost compared to the zero-shot foundation model: $2 . 5 \\mathrm { x }$ more crafting tables, 6.1x more planks, $4 . 3 \\mathbf { x }$ more logs, and $5 . 5 \\mathrm { x }$ more crafting overall (Fig. 5). However, when fine-tuning to this dataset we did not see any new behaviors emerge, only a refinement of existing skills. We saw an even bigger improvement when fine-tuning to the contractor_house dataset: $2 1 3 \\mathrm { x }$ more crafting tables, $5 9 \\mathrm { x }$ more wooden planks, $7 \\mathbf { x }$ more logs, and $5 9 \\mathrm { x }$ more crafting over all. In addition, we saw the emergence of crafting wooden tools, which requires placing a crafting table on the ground, opening it to reveal a new crafting interface, and then using it to craft wooden tools. This entire sequence takes a proficient human player a median of 1.2 minutes (1390 consecutive actions) to accomplish. The model goes further and collects cobblestone, which requires a wooden pickaxe to mine, and crafts stone tools, requiring it to again use a crafting table; this takes a proficient human player a median of 2.3 minutes (2790 consecutive actions). We also saw this model more frequently raiding villages that randomly spawn in the game, hunting animals for food, in addition to many behaviors we saw performed by the foundation model.(v) ", + "bbox": [ + 173, + 92, + 825, + 270 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Despite the foundation model’s zero-shot rollout performance plateauing 1/3 into training (Fig. 4, right), fine-tuning performance does continue to increase throughout foundation model training (Fig. 5, right). Additionally, there is a stark difference in performance when training from scratch vs. fine-tuning from the VPT foundation model (Fig. 5 right, comparing the left and rightmost points). ", + "bbox": [ + 174, + 277, + 825, + 333 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.4 Fine-Tuning with Reinforcement Learning ", + "text_level": 1, + "bbox": [ + 173, + 348, + 509, + 363 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/cba0e6b9fd67795bb3b4d12861ce77a9904cf7d4c4cee89a406ab396ae3a0429.jpg", + "image_caption": [ + "Figure 6: Typical sequence of items for obtaining a diamond pickaxe. Below each item is the median time and number of actions contractors required to obtain that item and the percentage of contractors that got the item within 10 minutes. The median time to obtain a diamond pickaxe is unknown (except that it is $> 2 0 \\mathrm { m } ,$ ) because contractors obtained this item in less than $5 0 \\%$ of 20-minute episodes. " + ], + "image_footnote": [], + "bbox": [ + 174, + 378, + 825, + 439 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To demonstrate the efficacy of RL fine-tuning, we chose the challenging goal of obtaining a diamond pickaxe within 10 minutes starting from a fresh Minecraft survival world. Doing so involves acquiring a sequence of difficult-to-obtain items that require complex skills like mining, inventory management, crafting with and without a crafting table, tool use, operating a furnace, and mining at the lowest depths, where many hazards like enemies and lava exist (Fig. 6). Adding to the difficulty, progress can be easily lost by dropping items, destroying items, or dying. Obtaining a diamond pickaxe more often than not takes a proficient human over 20 minutes (24,000 actions). ", + "bbox": [ + 174, + 522, + 825, + 619 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Agents are rewarded for each item obtained in the sequence, with lower rewards for items that have to be collected in bulk and higher rewards for items near the end of the sequence. Agents are optimized with the phasic policy gradient 65 RL algorithm for ${ \\sim } 1 . 3$ million episodes (roughly $1 . 4 \\times 1 0 ^ { 1 0 }$ frames). Episodes last for 10 minutes. See Appendix G.1 for reward function and RL training details. Due to computational constraints, RL experiments use a $\\sim 2 4 8$ million parameter VPT model (Appendix H). ", + "bbox": [ + 174, + 626, + 825, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A major problem when fine-tuning with RL is catastrophic forgetting66,67 because previously learned skills can be lost before their value is realized. For instance, while our VPT foundation model never exhibits the entire sequence of behaviors required to smelt iron zero-shot, it did train on examples of players smelting with furnaces. It therefore may have some latent ability to smelt iron once the many prerequisites to do so have been performed. To combat the catastrophic forgetting of latent skills such that they can continually improve exploration throughout RL fine-tuning, we add an auxiliary Kullback-Leibler (KL) divergence loss between the RL model and the frozen pretrained policy. 11 ", + "bbox": [ + 174, + 700, + 825, + 799 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Training from a randomly initialized policy fails to achieve almost any reward, underscoring how hard an exploration challenge the diamond pickaxe task is for RL in the native human action space (Fig. 7a). The model never learns to reliably collect logs, typically the first of many steps to obtaining a diamond pickaxe (Fig. 7b). RL fine-tuning from the VPT foundation model does substantially better (Fig. 7a), learning everything up to mining iron ore and crafting furnaces. (Fig. 7c). However, this agent fails at smelting an iron ingot, the next item required to get further into the tech tree, likely because the zero-shot probability that the VPT foundation model smelts an iron ingot is too low, even when given the prerequisite materials. ", + "bbox": [ + 174, + 804, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/3465129888555f408ab610b7e6a3dbdf535575989fd849fe8c5dda363f075cfd.jpg", + "image_caption": [ + "Figure 7: RL Fine-tuning results. (a) RL from a randomly initialized model fails to get almost any reward, RL fine-tuning from the VPT foundation model performs substantially better with a reward near 13, and RL fine-tuning from the early-game model performs best with a reward of 25. When training the early-game model without a KL loss to the original policy (No KL-loss) progress stalls after 100,000 episodes, suggesting that the skills necessary to make further progress have been catastrophically forgotten. (b) RL from a randomly initialized model occasionally collects sticks by breaking leaves (an easy but inefficient method of getting sticks that does not require logs or planks) and never learns to reliably collect logs. (c) RL fine-tuning from the VPT Foundation model learns everything in the curriculum up to iron ore and making furnaces, but fails to learn to use the furnace to smelt iron ingots. (d) RL fine-tuning from the early-game model learns to obtain (at human-level) all items in the sequence towards a diamond pickaxe and crafts a diamond pickaxe in $2 . 5 \\%$ of episodes. " + ], + "image_footnote": [], + "bbox": [ + 178, + 87, + 816, + 255 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 450, + 823, + 478 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results further improve by first BC fine-tuning the VPT Foundation Model to the earlygame_keyword dataset (the early-game model, Sec. 4.3) and then fine-tuning with RL (Fig. 7a), which in preliminary experiments we found to perform better than first fine-tuning to contractor_house followed by fine-tuning with RL (Appendix G.2). The three-phase training (pretraining, BC fine-tuning, and then RL fine-tuning) succeeds in learning extremely difficult tasks: it achieves over $8 0 \\%$ reliability on iron pickaxes, almost $2 0 \\%$ reliability on collecting diamonds, and $2 . 5 \\%$ reliability on obtaining a diamond pickaxe (Fig. 7d). For comparison, human players given the objective of obtaining a diamond pickaxe collect these items in $5 7 \\%$ , $1 5 \\%$ , and $1 2 \\%$ of episodes, respectively, meaning our model is human-level for crafting iron pickaxes and mining diamonds. Others have managed to obtain diamonds with $\\sim 0 . 1 \\%$ reliability in 15 minutes 33,34 but always with a simplified action space designed to ease exploration. To the best of our knowledge, we are the first to report non-zero success rates on crafting a diamond pickaxe. Qualitatively, the model developed useful skills for diamond mining, such as efficient mining patterns, cave exploration, returning to previously placed objects like crafting tables, and advanced techniques like using wooden pickaxes as fuel when moving on to iron tools.(vi) ", + "bbox": [ + 173, + 484, + 825, + 691 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, we validated the importance of the KL loss to the pretrained model during RL fine-tuning. The treatment without a KL loss obtains only items early in the sequence (logs, planks, sticks, and crafting tables) limiting its reward (Fig. 7a). This failure to progress further into the sequence is likely because, while the initial skills of chopping logs and crafting planks are being learned with RL, subsequent skills like crafting a wooden pickaxe are lost due to catastrophic forgetting. ", + "bbox": [ + 174, + 699, + 825, + 768 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.5 Data Scaling Properties of the Foundation Model ", + "text_level": 1, + "bbox": [ + 174, + 786, + 553, + 801 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section we validate a core hypothesis behind this work: that it is far more effective to use labeled contractor data to train an IDM within the VPT method than it is to directly train a BC foundation model from that same small contractor dataset. If we could cheaply collect a labeled contractor dataset of a similar order of magnitude as web_clean, then this would not be important; however, collecting that scale of data would have cost millions of dollars. Figure 8 compares foundation models trained on increasing orders of magnitude of data from 1 hour up to the full ${ \\sim } 7 0 \\mathrm { k }$ web_clean dataset. Foundation models trained up to and including 1k hours are trained on the IDM contractor data, and those trained on 5k hours and above are trained on subsets of web_clean, which does not contain any IDM contractor data. Scaling training data increases log collection, mining, and crafting capabilities. The zero-shot model only begins to start crafting crafting tables at over 5000 hours of training data. When fine-tuning each foundation model to contractor_house, we see that crafting rates for crafting tables and wooden tools increase by orders of magnitude when using the entire ${ \\sim } 7 0 \\mathrm { k }$ hour web_clean dataset. We furthermore only see the emergence of crafting stone tools at the largest data scale. ", + "bbox": [ + 176, + 813, + 823, + 882 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/a4e776f01b9e9a1ad25c549eef4298ddfee753b2cc183af6b0d9ea8fde1fc6bf.jpg", + "image_caption": [ + "Figure 8: (Left) Zero-shot rollout performance of foundation models trained on varying amounts of data. Models to the left of the dashed black line (points $\\leq 1 \\mathrm { k }$ hours) were trained on contractor data (ground-truth labels), and models to the right were trained on IDM pseudo-labeled subsets of web_clean. Due to compute limitations, this analysis was performed with smaller (71 million parameter) models except for the final point, which is the 0.5 billion parameter VPT foundation model. (Right) The corresponding performance of each model after BC fine-tuning each model to the contractor_house dataset. " + ], + "image_footnote": [], + "bbox": [ + 174, + 89, + 825, + 184 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 327, + 825, + 452 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.6 Effect of Inverse Dynamics Model Quality on Behavioral Cloning ", + "text_level": 1, + "bbox": [ + 176, + 472, + 665, + 487 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This section investigates how downstream BC performance is affected by IDM quality. We train IDMs on increasingly larger datasets and use each to independently label the earlygame_keyword dataset (this smaller dataset was chosen due to a limited compute budget). We then train a BC model from scratch on each dataset and report game statistics for each model as a function of IDM contractor dataset size (Fig. 9). ", + "bbox": [ + 174, + 500, + 485, + 638 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "IDMs trained on at least 10 hours of data are required for any crafting, and the crafting rate increases quickly up until 100 hours of data, ", + "bbox": [ + 174, + 645, + 483, + 686 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/555a4d47272fd3d0f24896ab4bc7825e8002ccc73861b6de1f09a96cbfe40ed5.jpg", + "image_caption": [ + "Figure 9: Zero-shot performance of BC models trained from scratch on the earlygame_keyword dataset labeled with IDMs that were trained on increasing amounts of contractor data. " + ], + "image_footnote": [], + "bbox": [ + 498, + 500, + 820, + 594 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "after which there are few to no gains and differences are likely due to noise. Similarly, crafting tables are only crafted after 50 or more hours of IDM data, and again gains plateau after 100 hours. While in all previous experiments we use our best IDM trained on 1962 hours of data, these results suggest we could reduce that number to as low as 100 hours. ", + "bbox": [ + 174, + 686, + 826, + 741 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Discussion and Conclusion ", + "text_level": 1, + "bbox": [ + 176, + 766, + 428, + 784 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The results presented in this paper help pave the path to utilizing the wealth of unlabeled data on the web for many sequential decision domains. Compared to representation learning methods, e.g. generative video modeling, VPT offers the exciting possibility of directly learning to act during pretraining and using these learned behavioral priors as extremely effective exploration priors for RL. VPT could even be an effective representation learning method for downstream tasks that do not require acting, e.g. video captioning, because arguably the most important information in any given scene would be present in features trained to correctly predict the distribution over future human actions. We leave this intriguing direction to future work. ", + "bbox": [ + 173, + 800, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Future work could improve results with more data (we estimate we could collect ${ \\bf \\Lambda } > 1 { \\bf M }$ hours) and larger, better-tuned models. Our internet data was fairly noisy and varied (players choose their own graphics settings); we hope future work will investigate even noisier sources of data, as well as how to use both first and third person demonstrations. Furthermore, all models in this work condition on past observations only; we cannot ask the model to perform specific tasks. Appendix I presents preliminary experiments on conditioning our models on closed captions (text transcripts of speech in videos), showing they become weakly steerable; we believe this a rich direction for future research. By definition behavioral priors must predict actions, and in this work we found this objective sufficient to train capable agents; however, a fruitful direction could be incorporating auxiliary representation learning objectives (e.g. contrastive losses, environment dynamics modeling, etc.) to reduce the sample complexity of both the IDM and foundation models. Similarly, it would be interesting to see if VPT could benefit from pretraining its attention layers with a language modeling task as in Li et al. 68 and Reid et al. 69 Loss was not consistently correlated with downstream evaluation metrics (Sec. 4.2), which often made progress slow. Another worthwhile future direction would be to investigate the correlation between various training metrics and downstream evaluations. ", + "bbox": [ + 174, + 92, + 825, + 297 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "For RL fine-tuning we only experimented with a standard policy gradient based RL algorithm (PPG); an interesting future direction would be to investigate how well VPT can be combined with other RL algorithms, e.g. off-policy or model based. Furthermore, we showed the efficacy of fine-tuning VPT with RL using a very difficult, albeit handcrafted, reward function aimed at crafting diamond tools. We hope future work will combine VPT with methods that can generate more generic reward functions, e.g. natural language based reward functions as proposed in MineDojo70 (released after this paper). Finally, while we do not anticipate any direct negative societal impacts from the models trained in this work, as VPT improves and expands to other domains it will be important to assess and mitigate harms that emerge with other forms of pretraining on internet datasets, such as emulating inappropriate behavior. 71 ", + "bbox": [ + 174, + 305, + 825, + 443 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In conclusion, VPT extends the paradigm of training large and general purpose behavioral priors to sequential decision domains that have commonly available unlabeled internet data. Our models exhibited impressive zero-shot behavior and, when fine-tuned with RL, achieved an unprecedented result of crafting a diamond pickaxe in Minecraft (all the more difficult given the human interface). We further showed that contractor data is far better used within the VPT pipeline than to train a foundation model directly and that only a small amount of contractor data (about $\\$ 2000$ USD) was required to unlock massive amounts of unlabeled online data for use in BC. Finally, learning with the human keyboard and mouse interface is highly general and allows losslessly modeling the entire distribution of human behavior. While we only experiment in Minecraft, we believe that VPT provides a general recipe for training behavioral priors in hard, yet generic, action spaces in any domain that has a large amount of freely available unlabeled data, such as computer usage. ", + "bbox": [ + 174, + 449, + 825, + 602 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements ", + "text_level": 1, + "bbox": [ + 176, + 623, + 338, + 640 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank the following people for helpful discussions and support: Bob McGrew, Ken Stanley, Joel Lehman, Ilya Sutskever, Wojciech Zaremba, Ingmar Kanitscheider, David Farhi, Glenn Powell, Jonathan Gordon, and the OpenAI supercomputing team, especially Christian Gibson, Ben Chess, and Christopher Berner. ", + "bbox": [ + 174, + 656, + 826, + 710 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 733, + 266, + 750 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. 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For many tasks our models exhibit human-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 497, + 470, + 509 + ], + "spans": [ + { + "bbox": [ + 141, + 497, + 470, + 509 + ], + "score": 1.0, + "content": "level performance, and we are the first to report computer agents that can craft", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 508, + 469, + 520 + ], + "spans": [ + { + "bbox": [ + 142, + 508, + 469, + 520 + ], + "score": 1.0, + "content": "diamond tools, which can take proficient humans upwards of 20 minutes (24,000", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 520, + 342, + 532 + ], + "spans": [ + { + "bbox": [ + 142, + 520, + 342, + 532 + ], + "score": 1.0, + "content": "environment actions) of gameplay to accomplish.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 29.5, + "bbox_fs": [ + 140, + 333, + 471, + 532 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 551, + 190, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 192, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 192, + 567 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "Work in recent years has demonstrated the efficacy of pretraining large and general foundation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 585, + 507, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 507, + 601 + ], + "score": 1.0, + "content": "models 7 on noisy internet-scale datasets for use in downstream tasks in natural language 1–4, computer", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 103, + 594, + 508, + 613 + ], + "spans": [ + { + "bbox": [ + 103, + 594, + 508, + 613 + ], + "score": 1.0, + "content": "vision, 5,6,8 and multi-task models. 9 For sequential decision domains (e.g. robotics, game playing,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "and computer usage) where agents must repeatedly act within an environment, a wealth of data also", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 620, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 631 + ], + "score": 1.0, + "content": "exists on the web; however, most of this data is in the form of unlabeled video (i.e. without the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "actions taken at each frame), making it much less straightforward to train a behavioral prior in these", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "domains than it is in e.g. natural language. In a few rare settings, such as Chess, Go, and StarCraft,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "there already exist large datasets with action labels from various online platforms that researchers", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 91, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 507, + 109 + ], + "score": 1.0, + "content": "have used for imitation learning. 10,11 When large labeled datasets do not exist, the canonical strategy", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "for training capable agents is reinforcement learning (RL), 12 which can be sample inefficient and", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 103, + 113, + 507, + 132 + ], + "spans": [ + { + "bbox": [ + 103, + 113, + 507, + 132 + ], + "score": 1.0, + "content": "expensive for hard-exploration problems. 13–19 Many virtual tasks, e.g. navigating websites, using", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "score": 1.0, + "content": "Photoshop, booking flights, etc., can be very hard to learn with RL and do not have large, commonly", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 103, + 135, + 507, + 153 + ], + "spans": [ + { + "bbox": [ + 103, + 135, + 507, + 153 + ], + "score": 1.0, + "content": "available sources of labeled data. 20,21 In this paper, we seek to extend the paradigm of training", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "large, general-purpose foundation models to sequential decision domains by utilizing freely available", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "internet-scale unlabeled video datasets with a simple semi-supervised imitation learning method. We", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 503, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 503, + 183 + ], + "score": 1.0, + "content": "call this method Video PreTraining (VPT) and demonstrate its efficacy in the domain of Minecraft.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 42.5, + "bbox_fs": [ + 103, + 576, + 508, + 643 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "domains than it is in e.g. natural language. In a few rare settings, such as Chess, Go, and StarCraft,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "there already exist large datasets with action labels from various online platforms that researchers", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 91, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 507, + 109 + ], + "score": 1.0, + "content": "have used for imitation learning. 10,11 When large labeled datasets do not exist, the canonical strategy", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "for training capable agents is reinforcement learning (RL), 12 which can be sample inefficient and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 103, + 113, + 507, + 132 + ], + "spans": [ + { + "bbox": [ + 103, + 113, + 507, + 132 + ], + "score": 1.0, + "content": "expensive for hard-exploration problems. 13–19 Many virtual tasks, e.g. navigating websites, using", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 141 + ], + "score": 1.0, + "content": "Photoshop, booking flights, etc., can be very hard to learn with RL and do not have large, commonly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 103, + 135, + 507, + 153 + ], + "spans": [ + { + "bbox": [ + 103, + 135, + 507, + 153 + ], + "score": 1.0, + "content": "available sources of labeled data. 20,21 In this paper, we seek to extend the paradigm of training", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "large, general-purpose foundation models to sequential decision domains by utilizing freely available", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "internet-scale unlabeled video datasets with a simple semi-supervised imitation learning method. We", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 171, + 503, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 503, + 183 + ], + "score": 1.0, + "content": "call this method Video PreTraining (VPT) and demonstrate its efficacy in the domain of Minecraft.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "Existing semi-supervised imitation learning methods aim to learn with few or no explicit action labels;", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 507, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 507, + 212 + ], + "score": 1.0, + "content": "however, they generally rely on the policy’s ability to explore the environment throughout training,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 206, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 104, + 206, + 506, + 223 + ], + "score": 1.0, + "content": "making them susceptible to exploration bottlenecks. 22–26 Furthermore, most prior semi-supervised", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "imitation learning work was tested in the relatively low data regime; because we experiment with far", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 152, + 244 + ], + "score": 1.0, + "content": "more data (", + "type": "text" + }, + { + "bbox": [ + 152, + 231, + 176, + 241 + ], + "score": 0.84, + "content": "\\mathord { \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "hours of unlabeled video), we hypothesize that we can achieve good performance", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "with a much simpler method, a trend that has proven true for pretraining in other modalities such", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 252, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 506, + 267 + ], + "score": 1.0, + "content": "as text. 1 In particular, given a large but unlabeled dataset, we propose generating pseudo-labels by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "gathering a small amount of labeled data to train an inverse dynamics model (IDM) that predicts", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "the action taken at each timestep in a video. Behavioral cloning (BC) can require a large amount", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "of data because the model must learn to infer intent and the distribution over future behaviors from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "only past observations. In contrast, the inverse dynamics modeling task is simpler because it is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 307, + 507, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 307, + 507, + 321 + ], + "score": 1.0, + "content": "non-causal, meaning it can look at both past and future frames to infer actions. In most settings,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "environment mechanics are far simpler than the breadth of human behavior that can take place within", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "the environment, suggesting that non-causal IDMs could require far less data to train than causal BC", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "models. 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Finally, we show we can fine-tune this model to downstream tasks with either", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 383, + 289, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 289, + 397 + ], + "score": 1.0, + "content": "behavioral cloning or reinforcement learning.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 400, + 364, + 541 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 365, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 365, + 412 + ], + "score": 1.0, + "content": "We chose to test our method in Minecraft because it (a) is one", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 410, + 365, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 365, + 423 + ], + "score": 1.0, + "content": "of the most actively played games in the world27 and thus has a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 422, + 365, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 365, + 434 + ], + "score": 1.0, + "content": "wealth of online video data, (b) is an open-ended sandbox game", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 433, + 365, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 365, + 444 + ], + "score": 1.0, + "content": "with an extremely wide variety of potential things to do, build,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 443, + 365, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 365, + 456 + ], + "score": 1.0, + "content": "and collect, making our results more applicable to real-world", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 455, + 365, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 365, + 466 + ], + "score": 1.0, + "content": "applications such as computer usage, which also tends to be", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 465, + 365, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 365, + 478 + ], + "score": 1.0, + "content": "varied and open-ended, and (c) has already garnered interest by", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 476, + 365, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 365, + 489 + ], + "score": 1.0, + "content": "the RL community as a research domain due to its complexity", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 485, + 366, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 366, + 500 + ], + "score": 1.0, + "content": "and correspondingly difficult exploration challenges. 28–32 In this", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 498, + 364, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 364, + 510 + ], + "score": 1.0, + "content": "work we use the native human interface for Minecraft so that we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 509, + 364, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 364, + 520 + ], + "score": 1.0, + "content": "can (1) most accurately model the human behavior distribution", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 519, + 365, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 365, + 532 + ], + "score": 1.0, + "content": "and reduce domain shift between video data and the environment,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 531, + 365, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 365, + 543 + ], + "score": 1.0, + "content": "(2) make data collection easier by allowing our human contractors", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35 + }, + { + "type": "image", + "bbox": [ + 372, + 398, + 503, + 479 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 372, + 398, + 503, + 479 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 372, + 398, + 503, + 479 + ], + "spans": [ + { + "bbox": [ + 372, + 398, + 503, + 479 + ], + "score": 0.962, + "type": "image", + "image_path": "9ecaae8777654a20f20ae479e79fd1eb563c824f512fc612e394f000c310a23c.jpg" + } + ] + } + ], + "index": 42.5, + "virtual_lines": [ + { + "bbox": [ + 372, + 398, + 503, + 438.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 372, + 438.5, + 503, + 479.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 371, + 489, + 505, + 533 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 371, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 371, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "Figure 1: Example Minecraft", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 371, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 371, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "crafting GUI. 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While this paper shows results in Minecraft only, the VPT method is general and", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 651, + 237, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 237, + 664 + ], + "score": 1.0, + "content": "could be applied to any domain.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "In Section 4 we show that the VPT foundation model has nontrivial zero-shot performance, accom-", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "plishing tasks impossible to learn with RL alone, such as crafting planks and crafting tables (tasks", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 689, + 507, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 392, + 701 + ], + "score": 1.0, + "content": "requiring a human proficient in Minecraft a median of 50 seconds or", + "type": "text" + }, + { + "bbox": [ + 393, + 689, + 417, + 700 + ], + "score": 0.83, + "content": "{ \\sim } 9 7 0 ", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 690, + 507, + 701 + ], + "score": 1.0, + "content": "consecutive actions).", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "Through fine-tuning with behavioral cloning to smaller datasets that target more specific behavior", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 60.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 182 + ], + "lines": [], + "index": 4.5, + "bbox_fs": [ + 103, + 73, + 507, + 183 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "Existing semi-supervised imitation learning methods aim to learn with few or no explicit action labels;", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 507, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 507, + 212 + ], + "score": 1.0, + "content": "however, they generally rely on the policy’s ability to explore the environment throughout training,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 206, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 104, + 206, + 506, + 223 + ], + "score": 1.0, + "content": "making them susceptible to exploration bottlenecks. 22–26 Furthermore, most prior semi-supervised", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "imitation learning work was tested in the relatively low data regime; because we experiment with far", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 152, + 244 + ], + "score": 1.0, + "content": "more data (", + "type": "text" + }, + { + "bbox": [ + 152, + 231, + 176, + 241 + ], + "score": 0.84, + "content": "\\mathord { \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "hours of unlabeled video), we hypothesize that we can achieve good performance", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "with a much simpler method, a trend that has proven true for pretraining in other modalities such", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 252, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 506, + 267 + ], + "score": 1.0, + "content": "as text. 1 In particular, given a large but unlabeled dataset, we propose generating pseudo-labels by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "gathering a small amount of labeled data to train an inverse dynamics model (IDM) that predicts", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "the action taken at each timestep in a video. Behavioral cloning (BC) can require a large amount", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "of data because the model must learn to infer intent and the distribution over future behaviors from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "only past observations. In contrast, the inverse dynamics modeling task is simpler because it is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 307, + 507, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 307, + 507, + 321 + ], + "score": 1.0, + "content": "non-causal, meaning it can look at both past and future frames to infer actions. In most settings,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "environment mechanics are far simpler than the breadth of human behavior that can take place within", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "the environment, suggesting that non-causal IDMs could require far less data to train than causal BC", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "models. 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This choice means that our models play at 20 frames", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "per second and must use a mouse and keyboard interface to interact with human GUIs for crafting,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "smelting, trading, etc., including dragging items to specific slots or navigating the recipe book with", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "the mouse cursor (Fig. 1). 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This task requires", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 335, + 128 + ], + "score": 1.0, + "content": "a proficient human a median upwards of 20 minutes or", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 335, + 117, + 370, + 127 + ], + "score": 0.86, + "content": "{ \\sim } 2 4 0 0 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 370, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "actions. 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In order to roll", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "out these policies in an environment, they must be causal, meaning they condition on observations", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 214, + 290 + ], + "score": 1.0, + "content": "from the current timestep", + "type": "text" + }, + { + "bbox": [ + 214, + 278, + 219, + 286 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 276, + 340, + 290 + ], + "score": 1.0, + "content": "and past timesteps only, i.e.", + "type": "text" + }, + { + "bbox": [ + 340, + 276, + 412, + 289 + ], + "score": 0.93, + "content": "\\pi \\sim p ( \\bar { a } _ { t } | o _ { 1 } . . . o _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 276, + 505, + 290 + ], + "score": 1.0, + "content": ". 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Imitating labeled trajectories", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 296, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 506, + 312 + ], + "score": 1.0, + "content": "has seen success in aerial vehicles, 40,41 self-driving cars, 42,43 board games, 10,44 and video games. 11,45", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 328 + ], + "score": 1.0, + "content": "When labeled demonstrations are not available, standard behavioral cloning will not work; however,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 324, + 508, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 508, + 340 + ], + "score": 1.0, + "content": "there is a large body of work in imitating behavior from unlabeled demonstrations. 23 For instance,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 335, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 506, + 350 + ], + "score": 1.0, + "content": "GAIL24 constructs an adversarial objective incentivizing the trained policy to exhibit behaviors", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "indistinguishable from those in the target dataset. Edwards et al. 46 propose to first learn a latent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "score": 1.0, + "content": "policy using unlabeled demonstrations and then map the learned latent actions to real actions using", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 383 + ], + "score": 1.0, + "content": "environment interaction. Peng et al. 47 use motion-capture methods to track agent positions in videos", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "and then train RL agents to match these waypoints. Similarly, Behbahani et al. 48 and Aytar et al. 49", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "task a RL agent to match waypoints; however, their waypoints are embeddings from unsupervised", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "feature learning models. Pathak et al. 50 and Nair et al. 51 train goal conditioned policies to take", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 414, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 425 + ], + "score": 1.0, + "content": "actions that move towards expert-provided goal states expressed as high dimensional visual waypoints.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 506, + 437 + ], + "score": 1.0, + "content": "Most similar to our own work, Torabi et al. 25 simultaneously train (1) an inverse dynamics model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "(IDM), 52 which aims to uncover the underlying action between timesteps given observations of past", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 445, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 208, + 460 + ], + "score": 1.0, + "content": "and future timesteps, e.g.", + "type": "text" + }, + { + "bbox": [ + 208, + 446, + 279, + 458 + ], + "score": 0.92, + "content": "p _ { \\mathrm { I D M } } ( a _ { t } | o _ { t } , o _ { t + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 445, + 506, + 460 + ], + "score": 1.0, + "content": ", and (2) a behavioral cloning (BC) model on trajectories", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "of observations labeled with the IDM. Data to train the IDM is collected by rolling out the BC", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "model in the target environment such that both models improve in tandem. However, at any point in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "training if there are sequences in the dataset that the IDM performs poorly on, it requires that the BC", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 490, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 506, + 501 + ], + "score": 1.0, + "content": "model perform those or similar sequences in order for the IDM to improve and correctly label them.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Therefore, if the BC model does not explore efficiently, it could severely slow down learning. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "order to avoid this potential issue we opted for a simpler two-stage approach: we first train an IDM", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "on a small number of labeled trajectories collected from human contractors (they play the game as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "would normally as we record their keypresses and mouse movements). Because human contractors", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 543, + 489, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 489, + 557 + ], + "score": 1.0, + "content": "reach most relevant parts of the state space, we can hold the IDM fixed throughout BC training.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 670 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "Compared to most previous work in semi-supervised imitation learning, we experiment in the much", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "more complex and open-ended environment of Minecraft. Minecraft is a voxel-based 3D video", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 582, + 507, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 507, + 595 + ], + "score": 1.0, + "content": "game that, due its popularity and wide variety of mechanics, has attracted a vast amount of RL", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 103, + 588, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 103, + 588, + 507, + 607 + ], + "score": 1.0, + "content": "research. 28,29,31–35,53–61 A large body of work focuses on small, custom-made Minecraft worlds", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 600, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 505, + 619 + ], + "score": 1.0, + "content": "with tasks such as navigation, 54,61 block placing, 55,56 instruction following, 59,60 combat, 57 and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 611, + 507, + 630 + ], + "spans": [ + { + 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In order to roll", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "out these policies in an environment, they must be causal, meaning they condition on observations", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 214, + 290 + ], + "score": 1.0, + "content": "from the current timestep", + "type": "text" + }, + { + "bbox": [ + 214, + 278, + 219, + 286 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 276, + 340, + 290 + ], + "score": 1.0, + "content": "and past timesteps only, i.e.", + "type": "text" + }, + { + "bbox": [ + 340, + 276, + 412, + 289 + ], + "score": 0.93, + "content": "\\pi \\sim p ( \\bar { a } _ { t } | o _ { 1 } . . . o _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 276, + 505, + 290 + ], + "score": 1.0, + "content": ". 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Imitating labeled trajectories", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 296, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 104, + 296, + 506, + 312 + ], + "score": 1.0, + "content": "has seen success in aerial vehicles, 40,41 self-driving cars, 42,43 board games, 10,44 and video games. 11,45", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 240, + 507, + 312 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 506, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 328 + ], + "score": 1.0, + "content": "When labeled demonstrations are not available, standard behavioral cloning will not work; however,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 324, + 508, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 508, + 340 + ], + "score": 1.0, + "content": "there is a large body of work in imitating behavior from unlabeled demonstrations. 23 For instance,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 335, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 506, + 350 + ], + "score": 1.0, + "content": "GAIL24 constructs an adversarial objective incentivizing the trained policy to exhibit behaviors", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "indistinguishable from those in the target dataset. Edwards et al. 46 propose to first learn a latent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "score": 1.0, + "content": "policy using unlabeled demonstrations and then map the learned latent actions to real actions using", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 383 + ], + "score": 1.0, + "content": "environment interaction. Peng et al. 47 use motion-capture methods to track agent positions in videos", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "and then train RL agents to match these waypoints. Similarly, Behbahani et al. 48 and Aytar et al. 49", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "task a RL agent to match waypoints; however, their waypoints are embeddings from unsupervised", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "feature learning models. Pathak et al. 50 and Nair et al. 51 train goal conditioned policies to take", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 414, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 425 + ], + "score": 1.0, + "content": "actions that move towards expert-provided goal states expressed as high dimensional visual waypoints.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 506, + 437 + ], + "score": 1.0, + "content": "Most similar to our own work, Torabi et al. 25 simultaneously train (1) an inverse dynamics model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "(IDM), 52 which aims to uncover the underlying action between timesteps given observations of past", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 445, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 445, + 208, + 460 + ], + "score": 1.0, + "content": "and future timesteps, e.g.", + "type": "text" + }, + { + "bbox": [ + 208, + 446, + 279, + 458 + ], + "score": 0.92, + "content": "p _ { \\mathrm { I D M } } ( a _ { t } | o _ { t } , o _ { t + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 445, + 506, + 460 + ], + "score": 1.0, + "content": ", and (2) a behavioral cloning (BC) model on trajectories", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "of observations labeled with the IDM. Data to train the IDM is collected by rolling out the BC", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "model in the target environment such that both models improve in tandem. However, at any point in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "training if there are sequences in the dataset that the IDM performs poorly on, it requires that the BC", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 490, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 506, + 501 + ], + "score": 1.0, + "content": "model perform those or similar sequences in order for the IDM to improve and correctly label them.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Therefore, if the BC model does not explore efficiently, it could severely slow down learning. In", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "order to avoid this potential issue we opted for a simpler two-stage approach: we first train an IDM", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "on a small number of labeled trajectories collected from human contractors (they play the game as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "would normally as we record their keypresses and mouse movements). Because human contractors", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 543, + 489, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 489, + 557 + ], + "score": 1.0, + "content": "reach most relevant parts of the state space, we can hold the IDM fixed throughout BC training.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 313, + 508, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 670 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "Compared to most previous work in semi-supervised imitation learning, we experiment in the much", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "more complex and open-ended environment of Minecraft. Minecraft is a voxel-based 3D video", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 582, + 507, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 507, + 595 + ], + "score": 1.0, + "content": "game that, due its popularity and wide variety of mechanics, has attracted a vast amount of RL", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 103, + 588, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 103, + 588, + 507, + 607 + ], + "score": 1.0, + "content": "research. 28,29,31–35,53–61 A large body of work focuses on small, custom-made Minecraft worlds", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 600, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 505, + 619 + ], + "score": 1.0, + "content": "with tasks such as navigation, 54,61 block placing, 55,56 instruction following, 59,60 combat, 57 and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 611, + 507, + 630 + ], + "spans": [ + { + 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T ]", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 222, + 506, + 236 + ], + "score": 1.0, + "content": ". In contrast to an imitation learning policy, the IDM can be non-causal,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 210, + 247 + ], + "score": 1.0, + "content": "meaning its prediction for", + "type": "text" + }, + { + "bbox": [ + 210, + 236, + 220, + 245 + ], + "score": 0.85, + "content": "a _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 234, + 425, + 247 + ], + "score": 1.0, + "content": "can be a function of both past and future events, i.e.", + "type": "text" + }, + { + "bbox": [ + 426, + 235, + 447, + 245 + ], + "score": 0.88, + "content": "o _ { t ^ { \\prime } > t }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 234, + 506, + 247 + ], + "score": 1.0, + "content": ". Compared to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "score": 1.0, + "content": "the behavioral cloning objective of modeling the distribution of human intent given past frames only,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 255, + 507, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 507, + 268 + ], + "score": 1.0, + "content": "we hypothesize that inverting environment dynamics is easier and more data efficient to learn. Indeed,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "Sec. 4.1 will show that the IDM objective is much easier to learn, and furthermore Sec. 4.6 will show", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 277, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 289 + ], + "score": 1.0, + "content": "that with very little labeled data (as few as 100 hours) we can train a fairly accurate IDM. This IDM", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "can be used to label online videos, providing the large amount of data required for the harder task of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 300, + 466, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 466, + 312 + ], + "score": 1.0, + "content": "behavioral cloning. See appendices D and B for IDM training and data collection details.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "Data Filtering We gather a large dataset of Minecraft videos by searching the web for related", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "keywords (Appendix A). Online videos often (1) include overlaid artifacts, such as a video feed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "of the player’s face, channel logos, watermarks, etc., (2) are collected from platforms other than", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "a computer with different gameplay, or (3) are from different game modes, e.g. in Minecraft we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "only want \"survival mode\" where players start from scratch and must gather or craft all their items.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "We call data “clean” if it does not contain visual artifacts and is from survival mode, and call all", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "other data “unclean.” With enough data, a large enough model, and enough training compute, a BC", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "score": 1.0, + "content": "model trained on both unclean and clean videos would likely still perform well in a clean Minecraft", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "environment. However, for simplicity and training compute efficiency, we choose to filter out unclean", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 414, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 506, + 429 + ], + "score": 1.0, + "content": "segments of video (note that a video may contain both clean and unclean segments). We do this by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "training a model to filter out unclean segments using a small dataset (8800) of images sampled from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "online videos labeled by contractors as clean or unclean. 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While more data improves both mouse movement and keypress predictions, our best", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 329, + 702 + ], + "score": 1.0, + "content": "IDM trains on only 1962 hours of data (compared to the", + "type": "text" + }, + { + "bbox": [ + 330, + 689, + 353, + 699 + ], + "score": 0.85, + "content": "\\mathrm { \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "hours of clean data we collected from", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 211, + 712 + ], + "score": 1.0, + "content": "the internet) and achieves", + "type": "text" + }, + { + "bbox": [ + 211, + 700, + 239, + 711 + ], + "score": 0.84, + "content": "9 0 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 699, + 338, + 712 + ], + "score": 1.0, + "content": "keypress accuracy and a", + "type": "text" + }, + { + "bbox": [ + 339, + 699, + 372, + 710 + ], + "score": 0.39, + "content": "0 . 9 7 ~ R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "for mouse movements evaluated", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 711, + 382, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 382, + 722 + ], + "score": 1.0, + "content": "on a held-out validation set of contractor-labeled data (Figure 3 left).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 632, + 506, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 71, + 504, + 144 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 71, + 504, + 144 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 71, + 504, + 144 + ], + "spans": [ + { + "bbox": [ + 107, + 71, + 504, + 144 + ], + "score": 0.959, + "type": "image", + "image_path": "4b3ea22386575829558b307187db5f20ebab1ca9a8165026d52849f357087422.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 71, + 504, + 95.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 95.33333333333333, + 504, + 119.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 119.66666666666666, + 504, + 144.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 155, + 506, + 178 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 153, + 507, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 374, + 170 + ], + "score": 1.0, + "content": "Figure 3: (Left) IDM keypress accuracy and mouse movement", + "type": "text" + }, + { + "bbox": [ + 375, + 155, + 388, + 166 + ], + "score": 0.87, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 153, + 507, + 170 + ], + "score": 1.0, + "content": "(explained variance 62) as a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 165, + 412, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 412, + 180 + ], + "score": 1.0, + "content": "function of dataset size. (Right) IDM vs. behavioral cloning data efficiency.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "Figure 3 (right) validates our hypothesis that IDMs are far more data efficient than BC models, likely", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 212, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 506, + 224 + ], + "score": 1.0, + "content": "because inverting environment mechanics is far easier than modeling the entire distribution of human", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "behavior. The IDM is two orders of magnitude more data efficient than a BC model trained on the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "same data and improves more quickly with more data. 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(Right) Amount a given item was collected per episode averaged over 2500", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "60-minute survival episodes as a function of training epoch, shaded with the standard error of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "mean. Basic mining refers to collection of dirt, gravel, or sand (all materials that can be gathered", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "without tools). Logs are obtained by repeatedly hitting trees for three seconds, a difficult feat for an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "RL agent to achieve as we show in Sec. 4.4. Planks can be crafted from logs, and crafting tables", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 506, + 497 + ], + "score": 1.0, + "content": "crafted from planks. Crafting requires using in-game crafting GUIs, and proficient humans take a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 494, + 398, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 398, + 507 + ], + "score": 1.0, + "content": "median of 50 seconds (970 consecutive actions) to make a crafting table.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + } + ], + "index": 18.0 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "We now explore the emergent behavior learned by a behavioral cloning policy trained on an extremely", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "large, but noisy, internet dataset labeled with our IDM. To collect the unlabeled internet dataset,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "we searched for publicly available videos of Minecraft play with search terms such as “minecraft", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 307, + 565 + ], + "score": 1.0, + "content": "survival for beginners.” These searches resulted in", + "type": "text" + }, + { + "bbox": [ + 308, + 553, + 336, + 563 + ], + "score": 0.88, + "content": "{ \\sim } 2 7 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "hours of video, which we filtered down to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 324, + 576 + ], + "score": 1.0, + "content": "“clean” video segments yielding an unlabeled dataset of", + "type": "text" + }, + { + "bbox": [ + 325, + 564, + 349, + 574 + ], + "score": 0.88, + "content": "\\mathrm { \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "hours, which we refer to as web_clean", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "(Appendix A has further details on data scraping and filtering). We then generated pseudo-labels", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "for web_clean with our best IDM (Section 3) and then trained the VPT foundation model with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "behavioral cloning. Preliminary model scaling experiments suggested that our model could benefit", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "from 30 epochs of training and that a 0.5 billion parameter model was required to stay in the efficient", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "learning regime 64 for that training duration (Appendix H shows results comparing model size and the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 629, + 428, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 308, + 641 + ], + "score": 1.0, + "content": "benefit of scaling to 0.5B parameters), which took", + "type": "text" + }, + { + "bbox": [ + 308, + 630, + 323, + 639 + ], + "score": 0.79, + "content": "{ \\sim } 9", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 629, + 428, + 641 + ], + "score": 1.0, + "content": "days on 720 V100 GPUs.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "We evaluate our models by measuring validation loss (Fig. 4, left) and rolling them out in the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "Minecraft environment. Unless otherwise noted, in all environment evaluations we spawn agents in a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "standard survival mode game where they play for 60 minutes, i.e. 72000 consecutive actions, and we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "plot the mean and shade the standard error of the mean for various game statistics such as crafting", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "and collection rates (Fig. 4, right). The VPT foundation model quickly learns to chop down trees", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 701, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 505, + 712 + ], + "score": 1.0, + "content": "to collect logs, a task we found near impossible for an RL agent to achieve with the native human", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "interface (Sec. 4.4). It also learns to craft those logs into wooden planks and then use those planks", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 71, + 504, + 144 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 71, + 504, + 144 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 71, + 504, + 144 + ], + "spans": [ + { + "bbox": [ + 107, + 71, + 504, + 144 + ], + "score": 0.959, + "type": "image", + "image_path": "4b3ea22386575829558b307187db5f20ebab1ca9a8165026d52849f357087422.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 71, + 504, + 95.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 95.33333333333333, + 504, + 119.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 119.66666666666666, + 504, + 144.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 155, + 506, + 178 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 153, + 507, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 374, + 170 + ], + "score": 1.0, + "content": "Figure 3: (Left) IDM keypress accuracy and mouse movement", + "type": "text" + }, + { + "bbox": [ + 375, + 155, + 388, + 166 + ], + "score": 0.87, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 153, + 507, + 170 + ], + "score": 1.0, + "content": "(explained variance 62) as a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 165, + 412, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 412, + 180 + ], + "score": 1.0, + "content": "function of dataset size. (Right) IDM vs. behavioral cloning data efficiency.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 201, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "Figure 3 (right) validates our hypothesis that IDMs are far more data efficient than BC models, likely", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 212, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 506, + 224 + ], + "score": 1.0, + "content": "because inverting environment mechanics is far easier than modeling the entire distribution of human", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "behavior. The IDM is two orders of magnitude more data efficient than a BC model trained on the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "same data and improves more quickly with more data. This evidence supports our hypothesis that it", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "is more effective to use contractor data within the VPT pipeline by training an IDM than it is to train", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 256, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 268 + ], + "score": 1.0, + "content": "a foundation model from contractor data directly (Sections 4.5 and 4.6 provide additional evidence).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 280 + ], + "score": 1.0, + "content": "Due to their data efficiency, training an IDM uses a negligible fraction of the overall compute needed", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 277, + 195, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 195, + 289 + ], + "score": 1.0, + "content": "to train a VPT model.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 201, + 506, + 289 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 303, + 392, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 393, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 393, + 317 + ], + "score": 1.0, + "content": "4.2 VPT Foundation Model Training and Zero-Shot Performance", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "image", + "bbox": [ + 108, + 331, + 504, + 401 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 331, + 504, + 401 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 331, + 504, + 401 + ], + "spans": [ + { + "bbox": [ + 108, + 331, + 504, + 401 + ], + "score": 0.965, + "type": "image", + "image_path": "34a95ee247846f0d6edcf6bdccdb1ad4fe749757d92015a2c7b77655f520b49f.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 108, + 331, + 504, + 354.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 108, + 354.3333333333333, + 504, + 377.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 108, + 377.66666666666663, + 504, + 400.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 407, + 506, + 506 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "Figure 4: (Left) Training and validation loss on the web_clean internet dataset with IDM pseudo-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "labels, and loss on the main IDM contractor dataset, which has ground-truth labels but is out-of-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "distribution (see text). (Right) Amount a given item was collected per episode averaged over 2500", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "60-minute survival episodes as a function of training epoch, shaded with the standard error of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "mean. Basic mining refers to collection of dirt, gravel, or sand (all materials that can be gathered", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "without tools). Logs are obtained by repeatedly hitting trees for three seconds, a difficult feat for an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "RL agent to achieve as we show in Sec. 4.4. Planks can be crafted from logs, and crafting tables", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 506, + 497 + ], + "score": 1.0, + "content": "crafted from planks. Crafting requires using in-game crafting GUIs, and proficient humans take a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 494, + 398, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 398, + 507 + ], + "score": 1.0, + "content": "median of 50 seconds (970 consecutive actions) to make a crafting table.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + } + ], + "index": 18.0 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "We now explore the emergent behavior learned by a behavioral cloning policy trained on an extremely", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "large, but noisy, internet dataset labeled with our IDM. To collect the unlabeled internet dataset,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "we searched for publicly available videos of Minecraft play with search terms such as “minecraft", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 307, + 565 + ], + "score": 1.0, + "content": "survival for beginners.” These searches resulted in", + "type": "text" + }, + { + "bbox": [ + 308, + 553, + 336, + 563 + ], + "score": 0.88, + "content": "{ \\sim } 2 7 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "hours of video, which we filtered down to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 324, + 576 + ], + "score": 1.0, + "content": "“clean” video segments yielding an unlabeled dataset of", + "type": "text" + }, + { + "bbox": [ + 325, + 564, + 349, + 574 + ], + "score": 0.88, + "content": "\\mathrm { \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "hours, which we refer to as web_clean", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "(Appendix A has further details on data scraping and filtering). We then generated pseudo-labels", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "for web_clean with our best IDM (Section 3) and then trained the VPT foundation model with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "behavioral cloning. Preliminary model scaling experiments suggested that our model could benefit", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "from 30 epochs of training and that a 0.5 billion parameter model was required to stay in the efficient", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "learning regime 64 for that training duration (Appendix H shows results comparing model size and the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 629, + 428, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 308, + 641 + ], + "score": 1.0, + "content": "benefit of scaling to 0.5B parameters), which took", + "type": "text" + }, + { + "bbox": [ + 308, + 630, + 323, + 639 + ], + "score": 0.79, + "content": "{ \\sim } 9", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 629, + 428, + 641 + ], + "score": 1.0, + "content": "days on 720 V100 GPUs.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 519, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "We evaluate our models by measuring validation loss (Fig. 4, left) and rolling them out in the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "Minecraft environment. Unless otherwise noted, in all environment evaluations we spawn agents in a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "standard survival mode game where they play for 60 minutes, i.e. 72000 consecutive actions, and we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "plot the mean and shade the standard error of the mean for various game statistics such as crafting", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "and collection rates (Fig. 4, right). The VPT foundation model quickly learns to chop down trees", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 701, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 505, + 712 + ], + "score": 1.0, + "content": "to collect logs, a task we found near impossible for an RL agent to achieve with the native human", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "interface (Sec. 4.4). It also learns to craft those logs into wooden planks and then use those planks", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "to craft a crafting table, which are required to unlock most other technology in the game and take a", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "human proficient in Minecraft approximately 50 seconds (970 consecutive actions) to collect. While", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "these behaviors are fairly complex in the native human action space, the VPT foundation model crafts", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "these items at a rate far below that of our proficient contractors, e.g. on average our contractors craft", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "5.44 crafting tables in 60 minutes of play versus 0.19 for the foundation model. The model also crafts", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "a non-negligible amount of wooden sticks, which are required to make wooden tools; collects various", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "flowers and crafts dyes from them; kills zombies that appear during the night; hunts wild animals;", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "collects various berries and mushrooms and eats them; and finds game-generated villages from which", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "to collect various rare items from chests. The model also learned to navigate uneven terrain, swim,", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 170, + 507, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 507, + 185 + ], + "score": 1.0, + "content": "and pillar jump, which involves the agent repeatedly jumping and quickly placing a block below itself", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 181, + 303, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 303, + 194 + ], + "score": 1.0, + "content": "such that it climbs upward by making a pillar.(iv)", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 645, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "to craft a crafting table, which are required to unlock most other technology in the game and take a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "human proficient in Minecraft approximately 50 seconds (970 consecutive actions) to collect. 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The model also crafts", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "a non-negligible amount of wooden sticks, which are required to make wooden tools; collects various", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "flowers and crafts dyes from them; kills zombies that appear during the night; hunts wild animals;", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "collects various berries and mushrooms and eats them; and finds game-generated villages from which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "to collect various rare items from chests. 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BC fine-tuning to either dataset improves performance, including (for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "the contractor_house dataset) yielding wooden and stone tools. Proficient Minecraft players take", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "a median of 1.2 minutes (1390 actions) to construct wooden tools and 2.3 minutes (2790 actions)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "to construct stone tools. (Right) Collection and crafting rates for VPT foundation model snapshots", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 662, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 506, + 675 + ], + "score": 1.0, + "content": "throughout training after they are BC fine-tuned to the contractor_house dataset. In general,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "crafting-related behaviors increase throughout foundation model training. Fig. 4 defines the other", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 684, + 343, + 697 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 343, + 697 + ], + "score": 1.0, + "content": "task terms (logs, planks, crafting tables, and total crafting).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + } + ], + "index": 37.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "Fine-tuning to earlygame_keyword results in a large boost compared to the zero-shot foundation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 137, + 96 + ], + "score": 1.0, + "content": "model:", + "type": "text" + }, + { + "bbox": [ + 137, + 84, + 156, + 94 + ], + "score": 0.61, + "content": "2 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 84, + 316, + 96 + ], + "score": 1.0, + "content": "more crafting tables, 6.1x more planks,", + "type": "text" + }, + { + "bbox": [ + 317, + 84, + 336, + 95 + ], + "score": 0.62, + "content": "4 . 3 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 84, + 398, + 96 + ], + "score": 1.0, + "content": "more logs, and", + "type": "text" + }, + { + "bbox": [ + 398, + 84, + 417, + 94 + ], + "score": 0.67, + "content": "5 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "more crafting overall", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "(Fig. 5). However, when fine-tuning to this dataset we did not see any new behaviors emerge,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "only a refinement of existing skills. We saw an even bigger improvement when fine-tuning to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 227, + 129 + ], + "score": 1.0, + "content": "contractor_house dataset:", + "type": "text" + }, + { + "bbox": [ + 228, + 117, + 250, + 127 + ], + "score": 0.44, + "content": "2 1 3 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 116, + 338, + 129 + ], + "score": 1.0, + "content": "more crafting tables,", + "type": "text" + }, + { + "bbox": [ + 339, + 117, + 356, + 127 + ], + "score": 0.44, + "content": "5 9 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 116, + 447, + 129 + ], + "score": 1.0, + "content": "more wooden planks,", + "type": "text" + }, + { + "bbox": [ + 447, + 117, + 459, + 127 + ], + "score": 0.75, + "content": "7 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "more logs,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 123, + 139 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 128, + 140, + 138 + ], + "score": 0.76, + "content": "5 9 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 128, + 505, + 139 + ], + "score": 1.0, + "content": "more crafting over all. In addition, we saw the emergence of crafting wooden tools, which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "requires placing a crafting table on the ground, opening it to reveal a new crafting interface, and then", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "using it to craft wooden tools. This entire sequence takes a proficient human player a median of 1.2", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 161, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 506, + 172 + ], + "score": 1.0, + "content": "minutes (1390 consecutive actions) to accomplish. The model goes further and collects cobblestone,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 169, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 506, + 185 + ], + "score": 1.0, + "content": "which requires a wooden pickaxe to mine, and crafts stone tools, requiring it to again use a crafting", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "table; this takes a proficient human player a median of 2.3 minutes (2790 consecutive actions). We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 191, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 207 + ], + "score": 1.0, + "content": "also saw this model more frequently raiding villages that randomly spawn in the game, hunting", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 486, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 486, + 216 + ], + "score": 1.0, + "content": "animals for food, in addition to many behaviors we saw performed by the foundation model.(v)", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 220, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "Despite the foundation model’s zero-shot rollout performance plateauing 1/3 into training (Fig. 4,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 229, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 506, + 245 + ], + "score": 1.0, + "content": "right), fine-tuning performance does continue to increase throughout foundation model training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "(Fig. 5, right). Additionally, there is a stark difference in performance when training from scratch vs.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 502, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 502, + 265 + ], + "score": 1.0, + "content": "fine-tuning from the VPT foundation model (Fig. 5 right, comparing the left and rightmost points).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 106, + 276, + 312, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 313, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 313, + 292 + ], + "score": 1.0, + "content": "4.4 Fine-Tuning with Reinforcement Learning", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "image", + "bbox": [ + 107, + 300, + 505, + 348 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 300, + 505, + 348 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 300, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 505, + 348 + ], + "score": 0.955, + "type": "image", + "image_path": "cba0e6b9fd67795bb3b4d12861ce77a9904cf7d4c4cee89a406ab396ae3a0429.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 107, + 300, + 505, + 316.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 107, + 316.0, + 505, + 332.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 107, + 332.0, + 505, + 348.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 357, + 505, + 401 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "Figure 6: Typical sequence of items for obtaining a diamond pickaxe. Below each item is the median", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "time and number of actions contractors required to obtain that item and the percentage of contractors", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "that got the item within 10 minutes. The median time to obtain a diamond pickaxe is unknown (except", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 390, + 491, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 141, + 402 + ], + "score": 1.0, + "content": "that it is", + "type": "text" + }, + { + "bbox": [ + 141, + 391, + 172, + 401 + ], + "score": 0.83, + "content": "> 2 0 \\mathrm { m } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 390, + 377, + 402 + ], + "score": 1.0, + "content": ") because contractors obtained this item in less than", + "type": "text" + }, + { + "bbox": [ + 377, + 390, + 397, + 401 + ], + "score": 0.87, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 390, + 491, + 402 + ], + "score": 1.0, + "content": "of 20-minute episodes.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + } + ], + "index": 20.75 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "To demonstrate the efficacy of RL fine-tuning, we chose the challenging goal of obtaining a diamond", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "score": 1.0, + "content": "pickaxe within 10 minutes starting from a fresh Minecraft survival world. Doing so involves acquiring", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "a sequence of difficult-to-obtain items that require complex skills like mining, inventory management,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "crafting with and without a crafting table, tool use, operating a furnace, and mining at the lowest", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "depths, where many hazards like enemies and lava exist (Fig. 6). Adding to the difficulty, progress", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "can be easily lost by dropping items, destroying items, or dying. Obtaining a diamond pickaxe more", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 480, + 401, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 401, + 492 + ], + "score": 1.0, + "content": "often than not takes a proficient human over 20 minutes (24,000 actions).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "Agents are rewarded for each item obtained in the sequence, with lower rewards for items that have to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "be collected in bulk and higher rewards for items near the end of the sequence. Agents are optimized", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 515, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 304, + 531 + ], + "score": 1.0, + "content": "with the phasic policy gradient 65 RL algorithm for", + "type": "text" + }, + { + "bbox": [ + 304, + 518, + 326, + 528 + ], + "score": 0.84, + "content": "{ \\sim } 1 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 515, + 429, + 531 + ], + "score": 1.0, + "content": "million episodes (roughly", + "type": "text" + }, + { + "bbox": [ + 429, + 517, + 471, + 528 + ], + "score": 0.91, + "content": "1 . 4 \\times 1 0 ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 515, + 506, + 531 + ], + "score": 1.0, + "content": "frames).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Episodes last for 10 minutes. See Appendix G.1 for reward function and RL training details. Due to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 299, + 552 + ], + "score": 1.0, + "content": "computational constraints, RL experiments use a", + "type": "text" + }, + { + "bbox": [ + 299, + 540, + 326, + 550 + ], + "score": 0.82, + "content": "\\sim 2 4 8", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "million parameter VPT model (Appendix H).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 569 + ], + "score": 1.0, + "content": "A major problem when fine-tuning with RL is catastrophic forgetting66,67 because previously learned", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "score": 1.0, + "content": "skills can be lost before their value is realized. For instance, while our VPT foundation model never", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "exhibits the entire sequence of behaviors required to smelt iron zero-shot, it did train on examples of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "players smelting with furnaces. It therefore may have some latent ability to smelt iron once the many", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "prerequisites to do so have been performed. To combat the catastrophic forgetting of latent skills", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "such that they can continually improve exploration throughout RL fine-tuning, we add an auxiliary", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 497, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 497, + 634 + ], + "score": 1.0, + "content": "Kullback-Leibler (KL) divergence loss between the RL model and the frozen pretrained policy. 11", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 649 + ], + "score": 1.0, + "content": "Training from a randomly initialized policy fails to achieve almost any reward, underscoring how", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "hard an exploration challenge the diamond pickaxe task is for RL in the native human action space", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 658, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 673 + ], + "score": 1.0, + "content": "(Fig. 7a). The model never learns to reliably collect logs, typically the first of many steps to obtaining", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "a diamond pickaxe (Fig. 7b). RL fine-tuning from the VPT foundation model does substantially", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 680, + 507, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 507, + 694 + ], + "score": 1.0, + "content": "better (Fig. 7a), learning everything up to mining iron ore and crafting furnaces. (Fig. 7c). However,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "score": 1.0, + "content": "this agent fails at smelting an iron ingot, the next item required to get further into the tech tree, likely", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 112, + 711, + 490, + 722 + ], + "lines": [ + { + "bbox": [ + 113, + 707, + 491, + 726 + ], + "spans": [ + { + "bbox": [ + 113, + 707, + 491, + 726 + ], + "score": 1.0, + "content": "(v)Sample Videos: https://www.youtube.com/playlist?list=PLNAOIb_agjf2yDSs4AqcoyPv4z_eWUiKm", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "Fine-tuning to earlygame_keyword results in a large boost compared to the zero-shot foundation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 137, + 96 + ], + "score": 1.0, + "content": "model:", + "type": "text" + }, + { + "bbox": [ + 137, + 84, + 156, + 94 + ], + "score": 0.61, + "content": "2 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 84, + 316, + 96 + ], + "score": 1.0, + "content": "more crafting tables, 6.1x more planks,", + "type": "text" + }, + { + "bbox": [ + 317, + 84, + 336, + 95 + ], + "score": 0.62, + "content": "4 . 3 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 84, + 398, + 96 + ], + "score": 1.0, + "content": "more logs, and", + "type": "text" + }, + { + "bbox": [ + 398, + 84, + 417, + 94 + ], + "score": 0.67, + "content": "5 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "more crafting overall", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "(Fig. 5). However, when fine-tuning to this dataset we did not see any new behaviors emerge,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "only a refinement of existing skills. We saw an even bigger improvement when fine-tuning to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 227, + 129 + ], + "score": 1.0, + "content": "contractor_house dataset:", + "type": "text" + }, + { + "bbox": [ + 228, + 117, + 250, + 127 + ], + "score": 0.44, + "content": "2 1 3 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 116, + 338, + 129 + ], + "score": 1.0, + "content": "more crafting tables,", + "type": "text" + }, + { + "bbox": [ + 339, + 117, + 356, + 127 + ], + "score": 0.44, + "content": "5 9 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 116, + 447, + 129 + ], + "score": 1.0, + "content": "more wooden planks,", + "type": "text" + }, + { + "bbox": [ + 447, + 117, + 459, + 127 + ], + "score": 0.75, + "content": "7 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "more logs,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 123, + 139 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 128, + 140, + 138 + ], + "score": 0.76, + "content": "5 9 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 128, + 505, + 139 + ], + "score": 1.0, + "content": "more crafting over all. In addition, we saw the emergence of crafting wooden tools, which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "requires placing a crafting table on the ground, opening it to reveal a new crafting interface, and then", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "using it to craft wooden tools. This entire sequence takes a proficient human player a median of 1.2", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 161, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 506, + 172 + ], + "score": 1.0, + "content": "minutes (1390 consecutive actions) to accomplish. The model goes further and collects cobblestone,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 169, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 506, + 185 + ], + "score": 1.0, + "content": "which requires a wooden pickaxe to mine, and crafts stone tools, requiring it to again use a crafting", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "table; this takes a proficient human player a median of 2.3 minutes (2790 consecutive actions). We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 191, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 207 + ], + "score": 1.0, + "content": "also saw this model more frequently raiding villages that randomly spawn in the game, hunting", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 203, + 486, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 486, + 216 + ], + "score": 1.0, + "content": "animals for food, in addition to many behaviors we saw performed by the foundation model.(v)", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 73, + 506, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 220, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "Despite the foundation model’s zero-shot rollout performance plateauing 1/3 into training (Fig. 4,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 229, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 506, + 245 + ], + "score": 1.0, + "content": "right), fine-tuning performance does continue to increase throughout foundation model training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "(Fig. 5, right). Additionally, there is a stark difference in performance when training from scratch vs.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 502, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 502, + 265 + ], + "score": 1.0, + "content": "fine-tuning from the VPT foundation model (Fig. 5 right, comparing the left and rightmost points).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 219, + 506, + 265 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 276, + 312, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 313, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 313, + 292 + ], + "score": 1.0, + "content": "4.4 Fine-Tuning with Reinforcement Learning", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "image", + "bbox": [ + 107, + 300, + 505, + 348 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 300, + 505, + 348 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 300, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 505, + 348 + ], + "score": 0.955, + "type": "image", + "image_path": "cba0e6b9fd67795bb3b4d12861ce77a9904cf7d4c4cee89a406ab396ae3a0429.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 107, + 300, + 505, + 316.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 107, + 316.0, + 505, + 332.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 107, + 332.0, + 505, + 348.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 357, + 505, + 401 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "Figure 6: Typical sequence of items for obtaining a diamond pickaxe. Below each item is the median", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "time and number of actions contractors required to obtain that item and the percentage of contractors", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "that got the item within 10 minutes. The median time to obtain a diamond pickaxe is unknown (except", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 390, + 491, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 141, + 402 + ], + "score": 1.0, + "content": "that it is", + "type": "text" + }, + { + "bbox": [ + 141, + 391, + 172, + 401 + ], + "score": 0.83, + "content": "> 2 0 \\mathrm { m } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 390, + 377, + 402 + ], + "score": 1.0, + "content": ") because contractors obtained this item in less than", + "type": "text" + }, + { + "bbox": [ + 377, + 390, + 397, + 401 + ], + "score": 0.87, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 390, + 491, + 402 + ], + "score": 1.0, + "content": "of 20-minute episodes.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + } + ], + "index": 20.75 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "To demonstrate the efficacy of RL fine-tuning, we chose the challenging goal of obtaining a diamond", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "score": 1.0, + "content": "pickaxe within 10 minutes starting from a fresh Minecraft survival world. Doing so involves acquiring", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "a sequence of difficult-to-obtain items that require complex skills like mining, inventory management,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "crafting with and without a crafting table, tool use, operating a furnace, and mining at the lowest", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "depths, where many hazards like enemies and lava exist (Fig. 6). Adding to the difficulty, progress", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "can be easily lost by dropping items, destroying items, or dying. Obtaining a diamond pickaxe more", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 480, + 401, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 401, + 492 + ], + "score": 1.0, + "content": "often than not takes a proficient human over 20 minutes (24,000 actions).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 414, + 506, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 496, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "Agents are rewarded for each item obtained in the sequence, with lower rewards for items that have to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "be collected in bulk and higher rewards for items near the end of the sequence. Agents are optimized", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 515, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 104, + 515, + 304, + 531 + ], + "score": 1.0, + "content": "with the phasic policy gradient 65 RL algorithm for", + "type": "text" + }, + { + "bbox": [ + 304, + 518, + 326, + 528 + ], + "score": 0.84, + "content": "{ \\sim } 1 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 515, + 429, + 531 + ], + "score": 1.0, + "content": "million episodes (roughly", + "type": "text" + }, + { + "bbox": [ + 429, + 517, + 471, + 528 + ], + "score": 0.91, + "content": "1 . 4 \\times 1 0 ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 515, + 506, + 531 + ], + "score": 1.0, + "content": "frames).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Episodes last for 10 minutes. See Appendix G.1 for reward function and RL training details. Due to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 299, + 552 + ], + "score": 1.0, + "content": "computational constraints, RL experiments use a", + "type": "text" + }, + { + "bbox": [ + 299, + 540, + 326, + 550 + ], + "score": 0.82, + "content": "\\sim 2 4 8", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "million parameter VPT model (Appendix H).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 496, + 506, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 569 + ], + "score": 1.0, + "content": "A major problem when fine-tuning with RL is catastrophic forgetting66,67 because previously learned", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "score": 1.0, + "content": "skills can be lost before their value is realized. For instance, while our VPT foundation model never", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "exhibits the entire sequence of behaviors required to smelt iron zero-shot, it did train on examples of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "players smelting with furnaces. It therefore may have some latent ability to smelt iron once the many", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "prerequisites to do so have been performed. To combat the catastrophic forgetting of latent skills", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "such that they can continually improve exploration throughout RL fine-tuning, we add an auxiliary", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 497, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 497, + 634 + ], + "score": 1.0, + "content": "Kullback-Leibler (KL) divergence loss between the RL model and the frozen pretrained policy. 11", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 554, + 506, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 649 + ], + "score": 1.0, + "content": "Training from a randomly initialized policy fails to achieve almost any reward, underscoring how", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "hard an exploration challenge the diamond pickaxe task is for RL in the native human action space", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 658, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 673 + ], + "score": 1.0, + "content": "(Fig. 7a). The model never learns to reliably collect logs, typically the first of many steps to obtaining", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "a diamond pickaxe (Fig. 7b). RL fine-tuning from the VPT foundation model does substantially", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 680, + 507, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 507, + 694 + ], + "score": 1.0, + "content": "better (Fig. 7a), learning everything up to mining iron ore and crafting furnaces. (Fig. 7c). However,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "score": 1.0, + "content": "this agent fails at smelting an iron ingot, the next item required to get further into the tech tree, likely", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "because the zero-shot probability that the VPT foundation model smelts an iron ingot is too low, even", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 368, + 260, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 260, + 381 + ], + "score": 1.0, + "content": "when given the prerequisite materials.", + "type": "text", + "cross_page": true + } + ], + "index": 15 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 638, + 507, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 69, + 500, + 202 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 69, + 500, + 202 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 69, + 500, + 202 + ], + "spans": [ + { + "bbox": [ + 109, + 69, + 500, + 202 + ], + "score": 0.972, + "type": "image", + "image_path": "3465129888555f408ab610b7e6a3dbdf535575989fd849fe8c5dda363f075cfd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 69, + 500, + 113.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 113.33333333333334, + 500, + 157.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 157.66666666666669, + 500, + 202.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 213, + 505, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "Figure 7: RL Fine-tuning results. (a) RL from a randomly initialized model fails to get almost", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 224, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 506, + 236 + ], + "score": 1.0, + "content": "any reward, RL fine-tuning from the VPT foundation model performs substantially better with a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "reward near 13, and RL fine-tuning from the early-game model performs best with a reward of 25.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 260 + ], + "score": 1.0, + "content": "When training the early-game model without a KL loss to the original policy (No KL-loss) progress", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 256, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 271 + ], + "score": 1.0, + "content": "stalls after 100,000 episodes, suggesting that the skills necessary to make further progress have been", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "catastrophically forgotten. (b) RL from a randomly initialized model occasionally collects sticks by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "breaking leaves (an easy but inefficient method of getting sticks that does not require logs or planks)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "and never learns to reliably collect logs. (c) RL fine-tuning from the VPT Foundation model learns", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "everything in the curriculum up to iron ore and making furnaces, but fails to learn to use the furnace to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "smelt iron ingots. (d) RL fine-tuning from the early-game model learns to obtain (at human-level) all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 432, + 335 + ], + "score": 1.0, + "content": "items in the sequence towards a diamond pickaxe and crafts a diamond pickaxe in", + "type": "text" + }, + { + "bbox": [ + 432, + 322, + 455, + 333 + ], + "score": 0.86, + "content": "2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "of episodes.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 108, + 357, + 504, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "because the zero-shot probability that the VPT foundation model smelts an iron ingot is too low, even", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 368, + 260, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 260, + 381 + ], + "score": 1.0, + "content": "when given the prerequisite materials.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "Results further improve by first BC fine-tuning the VPT Foundation Model to the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "earlygame_keyword dataset (the early-game model, Sec. 4.3) and then fine-tuning with RL", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "score": 1.0, + "content": "(Fig. 7a), which in preliminary experiments we found to perform better than first fine-tuning to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 415, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 506, + 432 + ], + "score": 1.0, + "content": "contractor_house followed by fine-tuning with RL (Appendix G.2). The three-phase training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "(pretraining, BC fine-tuning, and then RL fine-tuning) succeeds in learning extremely difficult tasks:", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 169, + 452 + ], + "score": 1.0, + "content": "it achieves over", + "type": "text" + }, + { + "bbox": [ + 170, + 439, + 190, + 450 + ], + "score": 0.89, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 438, + 330, + 452 + ], + "score": 1.0, + "content": "reliability on iron pickaxes, almost", + "type": "text" + }, + { + "bbox": [ + 330, + 439, + 350, + 450 + ], + "score": 0.89, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "reliability on collecting diamonds, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 129, + 461 + ], + "score": 0.86, + "content": "2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "reliability on obtaining a diamond pickaxe (Fig. 7d). For comparison, human players given", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 372, + 474 + ], + "score": 1.0, + "content": "the objective of obtaining a diamond pickaxe collect these items in", + "type": "text" + }, + { + "bbox": [ + 372, + 461, + 392, + 472 + ], + "score": 0.88, + "content": "5 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 460, + 395, + 474 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 396, + 461, + 415, + 472 + ], + "score": 0.86, + "content": "1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 460, + 435, + 474 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 436, + 461, + 456, + 471 + ], + "score": 0.9, + "content": "1 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "of episodes,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "respectively, meaning our model is human-level for crafting iron pickaxes and mining diamonds.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 289, + 497 + ], + "score": 1.0, + "content": "Others have managed to obtain diamonds with", + "type": "text" + }, + { + "bbox": [ + 289, + 483, + 322, + 493 + ], + "score": 0.91, + "content": "\\sim 0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 480, + 507, + 497 + ], + "score": 1.0, + "content": "reliability in 15 minutes 33,34 but always with a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "simplified action space designed to ease exploration. To the best of our knowledge, we are the first to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "report non-zero success rates on crafting a diamond pickaxe. Qualitatively, the model developed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "useful skills for diamond mining, such as efficient mining patterns, cave exploration, returning to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "previously placed objects like crafting tables, and advanced techniques like using wooden pickaxes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 537, + 269, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 269, + 549 + ], + "score": 1.0, + "content": "as fuel when moving on to iron tools.(vi)", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "Finally, we validated the importance of the KL loss to the pretrained model during RL fine-tuning.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "The treatment without a KL loss obtains only items early in the sequence (logs, planks, sticks, and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 576, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 589 + ], + "score": 1.0, + "content": "crafting tables) limiting its reward (Fig. 7a). This failure to progress further into the sequence is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 585, + 507, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 507, + 600 + ], + "score": 1.0, + "content": "likely because, while the initial skills of chopping logs and crafting planks are being learned with RL,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 596, + 455, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 455, + 611 + ], + "score": 1.0, + "content": "subsequent skills like crafting a wooden pickaxe are lost due to catastrophic forgetting.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 339, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 340, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 340, + 637 + ], + "score": 1.0, + "content": "4.5 Data Scaling Properties of the Foundation Model", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 644, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "In this section we validate a core hypothesis behind this work: that it is far more effective to use", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "labeled contractor data to train an IDM within the VPT method than it is to directly train a BC", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 667, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "foundation model from that same small contractor dataset. If we could cheaply collect a labeled", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "contractor dataset of a similar order of magnitude as web_clean, then this would not be important;", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "however, collecting that scale of data would have cost millions of dollars. Figure 8 compares", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 109, + 712, + 491, + 722 + ], + "lines": [ + { + "bbox": [ + 111, + 707, + 492, + 725 + ], + "spans": [ + { + "bbox": [ + 111, + 707, + 492, + 725 + ], + "score": 1.0, + "content": "(vi)Videos found at https://www.youtube.com/playlist?list=PLNAOIb_agjf3e_UKweM5pQUSfTw8r-Wfc", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 69, + 500, + 202 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 69, + 500, + 202 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 69, + 500, + 202 + ], + "spans": [ + { + "bbox": [ + 109, + 69, + 500, + 202 + ], + "score": 0.972, + "type": "image", + "image_path": "3465129888555f408ab610b7e6a3dbdf535575989fd849fe8c5dda363f075cfd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 69, + 500, + 113.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 113.33333333333334, + 500, + 157.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 157.66666666666669, + 500, + 202.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 213, + 505, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "Figure 7: RL Fine-tuning results. (a) RL from a randomly initialized model fails to get almost", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 224, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 506, + 236 + ], + "score": 1.0, + "content": "any reward, RL fine-tuning from the VPT foundation model performs substantially better with a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "reward near 13, and RL fine-tuning from the early-game model performs best with a reward of 25.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 260 + ], + "score": 1.0, + "content": "When training the early-game model without a KL loss to the original policy (No KL-loss) progress", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 256, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 271 + ], + "score": 1.0, + "content": "stalls after 100,000 episodes, suggesting that the skills necessary to make further progress have been", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "catastrophically forgotten. (b) RL from a randomly initialized model occasionally collects sticks by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "breaking leaves (an easy but inefficient method of getting sticks that does not require logs or planks)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "and never learns to reliably collect logs. (c) RL fine-tuning from the VPT Foundation model learns", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "everything in the curriculum up to iron ore and making furnaces, but fails to learn to use the furnace to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "smelt iron ingots. (d) RL fine-tuning from the early-game model learns to obtain (at human-level) all", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 432, + 335 + ], + "score": 1.0, + "content": "items in the sequence towards a diamond pickaxe and crafts a diamond pickaxe in", + "type": "text" + }, + { + "bbox": [ + 432, + 322, + 455, + 333 + ], + "score": 0.86, + "content": "2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "of episodes.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 108, + 357, + 504, + 379 + ], + "lines": [], + "index": 14.5, + "bbox_fs": [ + 105, + 356, + 505, + 381 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "Results further improve by first BC fine-tuning the VPT Foundation Model to the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "earlygame_keyword dataset (the early-game model, Sec. 4.3) and then fine-tuning with RL", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 420 + ], + "score": 1.0, + "content": "(Fig. 7a), which in preliminary experiments we found to perform better than first fine-tuning to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 415, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 506, + 432 + ], + "score": 1.0, + "content": "contractor_house followed by fine-tuning with RL (Appendix G.2). The three-phase training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "(pretraining, BC fine-tuning, and then RL fine-tuning) succeeds in learning extremely difficult tasks:", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 169, + 452 + ], + "score": 1.0, + "content": "it achieves over", + "type": "text" + }, + { + "bbox": [ + 170, + 439, + 190, + 450 + ], + "score": 0.89, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 438, + 330, + 452 + ], + "score": 1.0, + "content": "reliability on iron pickaxes, almost", + "type": "text" + }, + { + "bbox": [ + 330, + 439, + 350, + 450 + ], + "score": 0.89, + "content": "2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "reliability on collecting diamonds, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 129, + 461 + ], + "score": 0.86, + "content": "2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "reliability on obtaining a diamond pickaxe (Fig. 7d). For comparison, human players given", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 372, + 474 + ], + "score": 1.0, + "content": "the objective of obtaining a diamond pickaxe collect these items in", + "type": "text" + }, + { + "bbox": [ + 372, + 461, + 392, + 472 + ], + "score": 0.88, + "content": "5 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 460, + 395, + 474 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 396, + 461, + 415, + 472 + ], + "score": 0.86, + "content": "1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 460, + 435, + 474 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 436, + 461, + 456, + 471 + ], + "score": 0.9, + "content": "1 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "of episodes,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "respectively, meaning our model is human-level for crafting iron pickaxes and mining diamonds.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 289, + 497 + ], + "score": 1.0, + "content": "Others have managed to obtain diamonds with", + "type": "text" + }, + { + "bbox": [ + 289, + 483, + 322, + 493 + ], + "score": 0.91, + "content": "\\sim 0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 480, + 507, + 497 + ], + "score": 1.0, + "content": "reliability in 15 minutes 33,34 but always with a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "simplified action space designed to ease exploration. To the best of our knowledge, we are the first to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "report non-zero success rates on crafting a diamond pickaxe. Qualitatively, the model developed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "useful skills for diamond mining, such as efficient mining patterns, cave exploration, returning to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 540 + ], + "score": 1.0, + "content": "previously placed objects like crafting tables, and advanced techniques like using wooden pickaxes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 537, + 269, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 269, + 549 + ], + "score": 1.0, + "content": "as fuel when moving on to iron tools.(vi)", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 384, + 507, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 567 + ], + "score": 1.0, + "content": "Finally, we validated the importance of the KL loss to the pretrained model during RL fine-tuning.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "The treatment without a KL loss obtains only items early in the sequence (logs, planks, sticks, and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 576, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 589 + ], + "score": 1.0, + "content": "crafting tables) limiting its reward (Fig. 7a). This failure to progress further into the sequence is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 585, + 507, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 507, + 600 + ], + "score": 1.0, + "content": "likely because, while the initial skills of chopping logs and crafting planks are being learned with RL,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 596, + 455, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 455, + 611 + ], + "score": 1.0, + "content": "subsequent skills like crafting a wooden pickaxe are lost due to catastrophic forgetting.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 552, + 507, + 611 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 339, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 340, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 340, + 637 + ], + "score": 1.0, + "content": "4.5 Data Scaling Properties of the Foundation Model", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 644, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "In this section we validate a core hypothesis behind this work: that it is far more effective to use", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "labeled contractor data to train an IDM within the VPT method than it is to directly train a BC", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 667, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "foundation model from that same small contractor dataset. If we could cheaply collect a labeled", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "contractor dataset of a similar order of magnitude as web_clean, then this would not be important;", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "however, collecting that scale of data would have cost millions of dollars. Figure 8 compares", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 480, + 271 + ], + "score": 1.0, + "content": "foundation models trained on increasing orders of magnitude of data from 1 hour up to the full", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 480, + 259, + 505, + 270 + ], + "score": 0.8, + "content": "{ \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 271, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 281 + ], + "score": 1.0, + "content": "web_clean dataset. Foundation models trained up to and including 1k hours are trained on the IDM", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "contractor data, and those trained on 5k hours and above are trained on subsets of web_clean, which", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "does not contain any IDM contractor data. Scaling training data increases log collection, mining, and", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "crafting capabilities. The zero-shot model only begins to start crafting crafting tables at over 5000", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "hours of training data. When fine-tuning each foundation model to contractor_house, we see that", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "crafting rates for crafting tables and wooden tools increase by orders of magnitude when using the", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 131, + 348 + ], + "score": 1.0, + "content": "entire", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 131, + 336, + 155, + 346 + ], + "score": 0.86, + "content": "{ \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 155, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "hour web_clean dataset. 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Models to the left of the dashed black line (points", + "type": "text" + }, + { + "bbox": [ + 346, + 169, + 366, + 179 + ], + "score": 0.87, + "content": "\\leq 1 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "hours) were trained on contractor", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 179, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 191 + ], + "score": 1.0, + "content": "data (ground-truth labels), and models to the right were trained on IDM pseudo-labeled subsets", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 190, + 504, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 504, + 203 + ], + "score": 1.0, + "content": "of web_clean. Due to compute limitations, this analysis was performed with smaller (71 million", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "parameter) models except for the final point, which is the 0.5 billion parameter VPT foundation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "score": 1.0, + "content": "model. (Right) The corresponding performance of each model after BC fine-tuning each model to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 223, + 240, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 240, + 234 + ], + "score": 1.0, + "content": "the contractor_house dataset.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 480, + 271 + ], + "score": 1.0, + "content": "foundation models trained on increasing orders of magnitude of data from 1 hour up to the full", + "type": "text" + }, + { + "bbox": [ + 480, + 259, + 505, + 270 + ], + "score": 0.8, + "content": "{ \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 271, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 281 + ], + "score": 1.0, + "content": "web_clean dataset. Foundation models trained up to and including 1k hours are trained on the IDM", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "contractor data, and those trained on 5k hours and above are trained on subsets of web_clean, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "does not contain any IDM contractor data. Scaling training data increases log collection, mining, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "crafting capabilities. 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When fine-tuning each foundation model to contractor_house, we see that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "crafting rates for crafting tables and wooden tools increase by orders of magnitude when using the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 131, + 348 + ], + "score": 1.0, + "content": "entire", + "type": "text" + }, + { + "bbox": [ + 131, + 336, + 155, + 346 + ], + "score": 0.86, + "content": "{ \\sim } 7 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "hour web_clean dataset. We furthermore only see the emergence of crafting stone tools", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 347, + 203, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 203, + 359 + ], + "score": 1.0, + "content": "at the largest data scale.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 374, + 407, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 409, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 409, + 389 + ], + "score": 1.0, + "content": "4.6 Effect of Inverse Dynamics Model Quality on Behavioral Cloning", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 297, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 297, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 297, + 408 + ], + "score": 1.0, + "content": "This section investigates how downstream", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 407, + 298, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 298, + 420 + ], + "score": 1.0, + "content": "BC performance is affected by IDM qual-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 417, + 298, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 298, + 431 + ], + "score": 1.0, + "content": "ity. We train IDMs on increasingly larger", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 429, + 297, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 297, + 441 + ], + "score": 1.0, + "content": "datasets and use each to independently label", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 440, + 298, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 298, + 452 + ], + "score": 1.0, + "content": "the earlygame_keyword dataset (this smaller", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 451, + 298, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 298, + 463 + ], + "score": 1.0, + "content": "dataset was chosen due to a limited compute bud-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 298, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 298, + 473 + ], + "score": 1.0, + "content": "get). We then train a BC model from scratch on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 297, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 297, + 484 + ], + "score": 1.0, + "content": "each dataset and report game statistics for each", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 483, + 298, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 298, + 495 + ], + "score": 1.0, + "content": "model as a function of IDM contractor dataset", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 494, + 159, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 159, + 507 + ], + "score": 1.0, + "content": "size (Fig. 9).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 296, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 297, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 297, + 523 + ], + "score": 1.0, + "content": "IDMs trained on at least 10 hours of data are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 297, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 297, + 534 + ], + "score": 1.0, + "content": "required for any crafting, and the crafting rate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 533, + 299, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 299, + 545 + ], + "score": 1.0, + "content": "increases quickly up until 100 hours of data,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "image", + "bbox": [ + 305, + 396, + 502, + 471 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 305, + 396, + 502, + 471 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 305, + 396, + 502, + 471 + ], + "spans": [ + { + "bbox": [ + 305, + 396, + 502, + 471 + ], + "score": 0.967, + "type": "image", + "image_path": "555a4d47272fd3d0f24896ab4bc7825e8002ccc73861b6de1f09a96cbfe40ed5.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 305, + 396, + 502, + 408.5 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 305, + 408.5, + 502, + 421.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 305, + 421.0, + 502, + 433.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 305, + 433.5, + 502, + 446.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 305, + 446.0, + 502, + 458.5 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 305, + 458.5, + 502, + 471.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 482, + 505, + 527 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 303, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 303, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "Figure 9: Zero-shot performance of BC models", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 303, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 303, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "trained from scratch on the earlygame_keyword", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 303, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 303, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "dataset labeled with IDMs that were trained on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 304, + 515, + 459, + 527 + ], + "spans": [ + { + "bbox": [ + 304, + 515, + 459, + 527 + ], + "score": 1.0, + "content": "increasing amounts of contractor data.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + } + ], + "index": 38.0 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 506, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "after which there are few to no gains and differences are likely due to noise. Similarly, crafting tables", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "are only crafted after 50 or more hours of IDM data, and again gains plateau after 100 hours. While", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "in all previous experiments we use our best IDM trained on 1962 hours of data, these results suggest", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 576, + 318, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 318, + 588 + ], + "score": 1.0, + "content": "we could reduce that number to as low as 100 hours.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + }, + { + "type": "title", + "bbox": [ + 108, + 607, + 262, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 263, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 263, + 622 + ], + "score": 1.0, + "content": "5 Discussion and Conclusion", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "The results presented in this paper help pave the path to utilizing the wealth of unlabeled data on", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 644, + 507, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 507, + 659 + ], + "score": 1.0, + "content": "the web for many sequential decision domains. Compared to representation learning methods, e.g.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "generative video modeling, VPT offers the exciting possibility of directly learning to act during", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 668, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 679 + ], + "score": 1.0, + "content": "pretraining and using these learned behavioral priors as extremely effective exploration priors for", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "RL. VPT could even be an effective representation learning method for downstream tasks that do not", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "require acting, e.g. video captioning, because arguably the most important information in any given", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "scene would be present in features trained to correctly predict the distribution over future human", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 711, + 338, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 338, + 722 + ], + "score": 1.0, + "content": "actions. We leave this intriguing direction to future work.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 51.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 71, + 505, + 146 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 71, + 505, + 146 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 71, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 107, + 71, + 505, + 146 + ], + "score": 0.936, + "type": "image", + "image_path": "a4e776f01b9e9a1ad25c549eef4298ddfee753b2cc183af6b0d9ea8fde1fc6bf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 71, + 505, + 96.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 96.0, + 505, + 121.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 121.0, + 505, + 146.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 157, + 505, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 157, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 170 + ], + "score": 1.0, + "content": "Figure 8: (Left) Zero-shot rollout performance of foundation models trained on varying amounts", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 346, + 180 + ], + "score": 1.0, + "content": "of data. Models to the left of the dashed black line (points", + "type": "text" + }, + { + "bbox": [ + 346, + 169, + 366, + 179 + ], + "score": 0.87, + "content": "\\leq 1 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "hours) were trained on contractor", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 179, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 191 + ], + "score": 1.0, + "content": "data (ground-truth labels), and models to the right were trained on IDM pseudo-labeled subsets", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 190, + 504, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 504, + 203 + ], + "score": 1.0, + "content": "of web_clean. 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(Right) The corresponding performance of each model after BC fine-tuning each model to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 223, + 240, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 240, + 234 + ], + "score": 1.0, + "content": "the contractor_house dataset.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 505, + 358 + ], + "lines": [], + "index": 14, + "bbox_fs": [ + 105, + 259, + 505, + 359 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 374, + 407, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 409, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 409, + 389 + ], + "score": 1.0, + "content": "4.6 Effect of Inverse Dynamics Model Quality on Behavioral Cloning", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 297, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 297, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 297, + 408 + ], + "score": 1.0, + "content": "This section investigates how downstream", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 407, + 298, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 298, + 420 + ], + "score": 1.0, + "content": "BC performance is affected by IDM qual-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 417, + 298, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 298, + 431 + ], + "score": 1.0, + "content": "ity. 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We then train a BC model from scratch on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 297, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 297, + 484 + ], + "score": 1.0, + "content": "each dataset and report game statistics for each", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 483, + 298, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 298, + 495 + ], + "score": 1.0, + "content": "model as a function of IDM contractor dataset", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 494, + 159, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 159, + 507 + ], + "score": 1.0, + "content": "size (Fig. 9).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 395, + 298, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 296, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 297, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 297, + 523 + ], + "score": 1.0, + "content": "IDMs trained on at least 10 hours of data are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 297, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 297, + 534 + ], + "score": 1.0, + "content": "required for any crafting, and the crafting rate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 533, + 299, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 299, + 545 + ], + "score": 1.0, + "content": "increases quickly up until 100 hours of data,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 511, + 299, + 545 + ] + }, + { + "type": "image", + "bbox": [ + 305, + 396, + 502, + 471 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 305, + 396, + 502, + 471 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 305, + 396, + 502, + 471 + ], + "spans": [ + { + "bbox": [ + 305, + 396, + 502, + 471 + ], + "score": 0.967, + "type": "image", + "image_path": "555a4d47272fd3d0f24896ab4bc7825e8002ccc73861b6de1f09a96cbfe40ed5.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 305, + 396, + 502, + 408.5 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 305, + 408.5, + 502, + 421.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 305, + 421.0, + 502, + 433.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 305, + 433.5, + 502, + 446.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 305, + 446.0, + 502, + 458.5 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 305, + 458.5, + 502, + 471.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 482, + 505, + 527 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 303, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 303, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "Figure 9: Zero-shot performance of BC models", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 303, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 303, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "trained from scratch on the earlygame_keyword", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 303, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 303, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "dataset labeled with IDMs that were trained on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 304, + 515, + 459, + 527 + ], + "spans": [ + { + "bbox": [ + 304, + 515, + 459, + 527 + ], + "score": 1.0, + "content": "increasing amounts of contractor data.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + } + ], + "index": 38.0 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 506, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "after which there are few to no gains and differences are likely due to noise. Similarly, crafting tables", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "are only crafted after 50 or more hours of IDM data, and again gains plateau after 100 hours. While", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "in all previous experiments we use our best IDM trained on 1962 hours of data, these results suggest", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 576, + 318, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 318, + 588 + ], + "score": 1.0, + "content": "we could reduce that number to as low as 100 hours.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 543, + 506, + 588 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 607, + 262, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 263, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 263, + 622 + ], + "score": 1.0, + "content": "5 Discussion and Conclusion", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 106, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "The results presented in this paper help pave the path to utilizing the wealth of unlabeled data on", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 644, + 507, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 507, + 659 + ], + "score": 1.0, + "content": "the web for many sequential decision domains. Compared to representation learning methods, e.g.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 670 + ], + "score": 1.0, + "content": "generative video modeling, VPT offers the exciting possibility of directly learning to act during", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 668, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 679 + ], + "score": 1.0, + "content": "pretraining and using these learned behavioral priors as extremely effective exploration priors for", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "RL. VPT could even be an effective representation learning method for downstream tasks that do not", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "require acting, e.g. video captioning, because arguably the most important information in any given", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "scene would be present in features trained to correctly predict the distribution over future human", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 711, + 338, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 338, + 722 + ], + "score": 1.0, + "content": "actions. We leave this intriguing direction to future work.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 634, + 507, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 436, + 84 + ], + "score": 1.0, + "content": "Future work could improve results with more data (we estimate we could collect", + "type": "text" + }, + { + "bbox": [ + 436, + 73, + 459, + 83 + ], + "score": 0.75, + "content": "{ \\bf \\Lambda } > 1 { \\bf M }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "hours) and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "larger, better-tuned models. Our internet data was fairly noisy and varied (players choose their own", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "graphics settings); we hope future work will investigate even noisier sources of data, as well as how", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "to use both first and third person demonstrations. Furthermore, all models in this work condition", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "on past observations only; we cannot ask the model to perform specific tasks. Appendix I presents", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "preliminary experiments on conditioning our models on closed captions (text transcripts of speech in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "videos), showing they become weakly steerable; we believe this a rich direction for future research.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "By definition behavioral priors must predict actions, and in this work we found this objective sufficient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "to train capable agents; however, a fruitful direction could be incorporating auxiliary representation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "learning objectives (e.g. contrastive losses, environment dynamics modeling, etc.) to reduce the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "sample complexity of both the IDM and foundation models. Similarly, it would be interesting to see if", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 191, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 506, + 206 + ], + "score": 1.0, + "content": "VPT could benefit from pretraining its attention layers with a language modeling task as in Li et al. 68", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 202, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 506, + 216 + ], + "score": 1.0, + "content": "and Reid et al. 69 Loss was not consistently correlated with downstream evaluation metrics (Sec. 4.2),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "which often made progress slow. Another worthwhile future direction would be to investigate the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 226, + 402, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 402, + 237 + ], + "score": 1.0, + "content": "correlation between various training metrics and downstream evaluations.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "For RL fine-tuning we only experimented with a standard policy gradient based RL algorithm (PPG);", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "an interesting future direction would be to investigate how well VPT can be combined with other", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 262, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 506, + 277 + ], + "score": 1.0, + "content": "RL algorithms, e.g. off-policy or model based. Furthermore, we showed the efficacy of fine-tuning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "VPT with RL using a very difficult, albeit handcrafted, reward function aimed at crafting diamond", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "tools. We hope future work will combine VPT with methods that can generate more generic reward", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "score": 1.0, + "content": "functions, e.g. natural language based reward functions as proposed in MineDojo70 (released after", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "this paper). Finally, while we do not anticipate any direct negative societal impacts from the models", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "trained in this work, as VPT improves and expands to other domains it will be important to assess and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "mitigate harms that emerge with other forms of pretraining on internet datasets, such as emulating", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 338, + 210, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 210, + 353 + ], + "score": 1.0, + "content": "inappropriate behavior. 71", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "In conclusion, VPT extends the paradigm of training large and general purpose behavioral priors", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "to sequential decision domains that have commonly available unlabeled internet data. Our models", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 378, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 389 + ], + "score": 1.0, + "content": "exhibited impressive zero-shot behavior and, when fine-tuned with RL, achieved an unprecedented", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 390, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 401 + ], + "score": 1.0, + "content": "result of crafting a diamond pickaxe in Minecraft (all the more difficult given the human interface).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 399, + 507, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 507, + 412 + ], + "score": 1.0, + "content": "We further showed that contractor data is far better used within the VPT pipeline than to train a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 433, + 423 + ], + "score": 1.0, + "content": "foundation model directly and that only a small amount of contractor data (about", + "type": "text" + }, + { + "bbox": [ + 433, + 411, + 460, + 422 + ], + "score": 0.88, + "content": "\\$ 2000", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "USD) was", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "required to unlock massive amounts of unlabeled online data for use in BC. Finally, learning with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "the human keyboard and mouse interface is highly general and allows losslessly modeling the entire", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "distribution of human behavior. While we only experiment in Minecraft, we believe that VPT provides", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "a general recipe for training behavioral priors in hard, yet generic, action spaces in any domain that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 465, + 420, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 420, + 479 + ], + "score": 1.0, + "content": "has a large amount of freely available unlabeled data, such as computer usage.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 494, + 207, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 208, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 208, + 510 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 506, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "We thank the following people for helpful discussions and support: Bob McGrew, Ken Stanley,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "Joel Lehman, Ilya Sutskever, Wojciech Zaremba, Ingmar Kanitscheider, David Farhi, Glenn Powell,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 541, + 507, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 507, + 555 + ], + "score": 1.0, + "content": "Jonathan Gordon, and the OpenAI supercomputing team, especially Christian Gibson, Ben Chess,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 204, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 204, + 566 + ], + "score": 1.0, + "content": "and Christopher Berner.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 107, + 581, + 163, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 165, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 165, + 596 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 109, + 601, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 110, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "[1] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 127, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 127, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 127, + 623, + 500, + 636 + ], + "spans": [ + { + "bbox": [ + 127, + 623, + 500, + 636 + ], + "score": 1.0, + "content": "few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 111, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 111, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "[2] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 127, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 127, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 126, + 665, + 154, + 680 + ], + "spans": [ + { + "bbox": [ + 126, + 665, + 154, + 680 + ], + "score": 1.0, + "content": "2018.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 110, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 110, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "[3] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 126, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 126, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "Lewis, Luke Zettlemoyer, and Veselin Stoyanov. Roberta: A robustly optimized bert pretraining", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 127, + 712, + 333, + 723 + ], + "spans": [ + { + "bbox": [ + 127, + 712, + 333, + 723 + ], + "score": 1.0, + "content": "approach. arXiv preprint arXiv:1907.11692, 2019.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 755 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 755 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 436, + 84 + ], + "score": 1.0, + "content": "Future work could improve results with more data (we estimate we could collect", + "type": "text" + }, + { + "bbox": [ + 436, + 73, + 459, + 83 + ], + "score": 0.75, + "content": "{ \\bf \\Lambda } > 1 { \\bf M }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "hours) and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "larger, better-tuned models. Our internet data was fairly noisy and varied (players choose their own", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "graphics settings); we hope future work will investigate even noisier sources of data, as well as how", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "to use both first and third person demonstrations. Furthermore, all models in this work condition", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "on past observations only; we cannot ask the model to perform specific tasks. Appendix I presents", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "preliminary experiments on conditioning our models on closed captions (text transcripts of speech in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "videos), showing they become weakly steerable; we believe this a rich direction for future research.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "By definition behavioral priors must predict actions, and in this work we found this objective sufficient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "to train capable agents; however, a fruitful direction could be incorporating auxiliary representation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "learning objectives (e.g. contrastive losses, environment dynamics modeling, etc.) to reduce the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "sample complexity of both the IDM and foundation models. Similarly, it would be interesting to see if", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 191, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 506, + 206 + ], + "score": 1.0, + "content": "VPT could benefit from pretraining its attention layers with a language modeling task as in Li et al. 68", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 202, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 506, + 216 + ], + "score": 1.0, + "content": "and Reid et al. 69 Loss was not consistently correlated with downstream evaluation metrics (Sec. 4.2),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "which often made progress slow. Another worthwhile future direction would be to investigate the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 226, + 402, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 402, + 237 + ], + "score": 1.0, + "content": "correlation between various training metrics and downstream evaluations.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 73, + 506, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "For RL fine-tuning we only experimented with a standard policy gradient based RL algorithm (PPG);", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "an interesting future direction would be to investigate how well VPT can be combined with other", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 262, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 506, + 277 + ], + "score": 1.0, + "content": "RL algorithms, e.g. off-policy or model based. Furthermore, we showed the efficacy of fine-tuning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "VPT with RL using a very difficult, albeit handcrafted, reward function aimed at crafting diamond", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "tools. We hope future work will combine VPT with methods that can generate more generic reward", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 309 + ], + "score": 1.0, + "content": "functions, e.g. natural language based reward functions as proposed in MineDojo70 (released after", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "this paper). Finally, while we do not anticipate any direct negative societal impacts from the models", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "trained in this work, as VPT improves and expands to other domains it will be important to assess and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "mitigate harms that emerge with other forms of pretraining on internet datasets, such as emulating", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 338, + 210, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 210, + 353 + ], + "score": 1.0, + "content": "inappropriate behavior. 71", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 241, + 506, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "In conclusion, VPT extends the paradigm of training large and general purpose behavioral priors", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "to sequential decision domains that have commonly available unlabeled internet data. Our models", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 378, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 389 + ], + "score": 1.0, + "content": "exhibited impressive zero-shot behavior and, when fine-tuned with RL, achieved an unprecedented", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 390, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 401 + ], + "score": 1.0, + "content": "result of crafting a diamond pickaxe in Minecraft (all the more difficult given the human interface).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 399, + 507, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 507, + 412 + ], + "score": 1.0, + "content": "We further showed that contractor data is far better used within the VPT pipeline than to train a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 433, + 423 + ], + "score": 1.0, + "content": "foundation model directly and that only a small amount of contractor data (about", + "type": "text" + }, + { + "bbox": [ + 433, + 411, + 460, + 422 + ], + "score": 0.88, + "content": "\\$ 2000", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "USD) was", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "required to unlock massive amounts of unlabeled online data for use in BC. Finally, learning with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "the human keyboard and mouse interface is highly general and allows losslessly modeling the entire", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "distribution of human behavior. While we only experiment in Minecraft, we believe that VPT provides", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "a general recipe for training behavioral priors in hard, yet generic, action spaces in any domain that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 465, + 420, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 420, + 479 + ], + "score": 1.0, + "content": "has a large amount of freely available unlabeled data, such as computer usage.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 357, + 507, + 479 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 494, + 207, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 208, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 208, + 510 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 506, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "We thank the following people for helpful discussions and support: Bob McGrew, Ken Stanley,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "Joel Lehman, Ilya Sutskever, Wojciech Zaremba, Ingmar Kanitscheider, David Farhi, Glenn Powell,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 541, + 507, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 507, + 555 + ], + "score": 1.0, + "content": "Jonathan Gordon, and the OpenAI supercomputing team, especially Christian Gibson, Ben Chess,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 204, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 204, + 566 + ], + "score": 1.0, + "content": "and Christopher Berner.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 519, + 507, + 566 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 581, + 163, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 165, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 165, + 596 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "list", + "bbox": [ + 109, + 601, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 110, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 110, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "[1] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal,", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 127, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 127, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 127, + 623, + 500, + 636 + ], + "spans": [ + { + "bbox": [ + 127, + 623, + 500, + 636 + ], + "score": 1.0, + "content": "few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 111, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 111, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "[2] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 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Problem size# examplesGPT3 Cost
without scratchpadwith scratchpad
1x181$0.12$7.44
2x1810$1.28$74.4
2x28100$12.96$744
3x18100$12.96$744
3x281000$129.6$7440
3x3810000$1296$74,404
4x181000$129.6$7440
4x2810000$1296$74,404
4x38100000$12,960$744,040
4x481000000$129,600$7,440,400
5x1810000$1296$74,404
5x28100000$12,960$744,040
5x381000000$129,600$7,440,400
5x4810000000$1,296,000$70,440,400
5x58100000000$12,960,000$700,440,400
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Relative Information Gain
Input variableOutput variable2x23x34x45x5
xn22n0.2230.2230.2230.223
yn2m0.2230.2230.2230.223
X1210.1980.1990.1990.199
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Relative Information Gain for each problem size
Input variableOutput variable2345678910
a1020.150.130.140.140.140.140.140.140.14
a1010.640.710.690.690.690.690.690.690.69
a2020.530.420.450.440.450.440.440.450.44
a3030.640.490.530.520.520.520.520.52
a4040.600.460.500.490.490.490.49
a5050.620.470.510.500.500.50
a6060.610.470.510.490.50
a7070.610.470.510.50
a8080.610.470.51
a9090.610.47
a100100.61
an-1On-10.640.600.620.610.610.610.610.61
an-2On-20.460.470.470.470.470.47
an-30n-30.510.510.510.51
an-40n-40.490.50
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Relative Information Gain for each problem size
Input variableOutput variable345678910
an-3 an-2 an-10n-10.950.950.950.95
an-3an-2an-1On-20.870.870.870.87
an-3 an-2 an-1On-30.640.640.640.64
a1 a2 a3011.000.960.970.970.970.970.970.97
α1 a2 a3021.000.910.920.910.920.910.920.91
a2 a3 a4020.560.550.550.550.550.550.56
α1 a2 a3031.000.660.730.710.720.720.720.72
a2 a3 a4030.860.770.780.780.780.780.78
a3 a4 a5030.670.660.660.660.660.66
a2 a3 a4040.940.640.70.680.690.690.69
a3 a4 a5040.880.790.810.80.80.8
a4 a5 a6040.630.620.620.620.62
a3 a4 a5050.950.650.710.690.70.7
a4 a5 a6050.870.780.790.790.79
a5 a6 a7050.640.630.630.64
a4 a5 a6060.940.640.710.690.7
a5 a6 a7060.870.780.80.8
a6 a7 a8060.640.620.63
a5 a6 a7070.950.640.710.69
a6 a7 a8070.870.780.8
a6 a7 a8080.950.640.71
a1 a2 a30.120.10.110.110.110.110.11
a2 a3 a4040.10.090.10.090.10.1
a3 a4 a505 060.110.10.10.10.11
a4 a5 a6070.110.090.10.11
a5 a6 a7080.110.090.11
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is able to translate a single input view into consistent and sharp completions across many views. The core component of 3DiM is a pose-conditional image-to-image diffusion model, which is trained to take a source view and its pose as inputs, and generates a novel view for a target pose as output. 3DiM can then generate multiple views that are approximately 3D consistent using a novel technique called stochastic conditioning. At inference time, the output views are generated autoregressively. When generating each novel view, one selects a random conditioning view from the set of previously generated views at each denoising step. We demonstrate that stochastic conditioning significantly improves 3D consistency compared to a na¨ıve sampler for an image-to-image diffusion model, which involves conditioning on a single fixed view. We compare 3DiM to prior work on the SRN ShapeNet dataset, demonstrating that 3DiM’s generated completions from a single view achieve much higher fidelity, while being approximately 3D consistent. We also introduce a new evaluation methodology, 3D consistency scoring, to quantify the 3D consistency of a generated object by training a neural field on the model’s output views. 3DiM is geometry free, does not rely on hyper-networks or test-time optimization for novel view synthesis, and allows a single model to easily scale to a large number of scenes. + +![](images/476bebe18bd204ae2dce90f0a46a3eb7c0418fe2ef3bb88c505ec200a6c9160a.jpg) +Figure 1: Given a single input image on the left, 3DiM performs novel view synthesis and generates the four views on the right. We trained a single ${ \sim } 4 7 1 \mathrm { M }$ parameter 3DiM on all of ShapeNet (without classconditioning) and sample frames with 256 steps (512 score function evaluations with classifier-free guidance). See the Supplementary Website (https://3d-diffusion.github.io/) for video outputs. + +# 1 INTRODUCTION + +Diffusion Probabilistic Models (DPMs) (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020), also known as simply diffusion models, have recently emerged as a powerful family of generative models, achieving state-of-the-art performance on audio and image synthesis (Chen et al., 2020; Dhariwal & Nichol, 2021), while admitting better training stability over adversarial approaches (Goodfellow et al., 2014), as well as likelihood computation, which enables further applications such as compression and density estimation (Song et al., 2021; Kingma et al., 2021). Diffusion models have achieved impressive empirical results in a variety of image-to-image translation tasks not limited to text-to-image, super-resolution, inpainting, colorization, uncropping, and artifact removal (Song et al., 2020; Saharia et al., 2021a; Ramesh et al., 2022; Saharia et al., 2022). + +One particular image-to-image translation problem where diffusion models have not been investigated is novel view synthesis, where, given a set of images of a given 3D scene, the task is to infer how the scene looks from novel viewpoints. Before the recent emergence of Scene Representation Networks (SRN) (Sitzmann et al., 2019) and Neural Radiance Fields (NeRF) (Mildenhall et al., 2020), state-of-the-art approaches to novel view synthesis were typically built on generative models (Sun et al., 2018) or more classical techniques on interpolation or disparity estimation (Park et al., 2017; Zhou et al., 2018). Today, these models have been outperformed by NeRF-class models (Yu et al., 2021; Niemeyer et al., 2021; Jang & Agapito, 2021), where 3D consistency is guaranteed by construction, as images are generated by volume rendering of a single underlying 3D representation (a.k.a. “geometry-aware” models). + +Still, these approaches feature different limitations. Heavily regularized NeRFs for novel view synthesis with few images such as RegNeRF (Niemeyer et al., 2021) produce undesired artifacts when given very few images, and fail to leverage knowledge from multiple scenes (recall NeRFs are trained on a single scene, i.e., one model per scene), and given one or very few views of a novel scene, a reasonable model must extrapolate to complete the occluded parts of the scene. PixelNeRF (Yu et al., 2021) and VisionNeRF (Lin et al., 2022) address this by training NeRF-like models conditioned on feature maps that encode the novel input view(s). However, these approaches are regressive rather than generative, and as a result, they cannot yield different plausible modes and are prone to blurriness. This type of failure has also been previously observed in regression-based models (Saharia et al., 2021b). Other works such as CodeNeRF (Jang & Agapito, 2021) and LoLNeRF (Rebain et al., 2021) instead employ test-time optimization to handle novel scenes, but still have issues with sample quality. + +In recent literature, geometry-free approaches (i.e., methods without explicit geometric inductive biases like those introduced by volume rendering) such as Light Field Networks (LFN) (Sitzmann et al., 2021) and Scene Representation Transformers (SRT) (Sajjadi et al., 2021) have achieved results competitive with 3D-aware methods in the “few-shot” setting, where the number of conditioning views is limited (i.e., 1-10 images vs. dozens of images as in the usual NeRF setting). Similarly to our approach, EG3D (Chan et al., 2022) provides approximate 3D consistency by leveraging generative models. EG3D employs a StyleGAN (Karras et al., 2019) with volumetric rendering, followed by generative super-resolution (the latter being responsible for the approximation). In comparison to this complex setup, we do not only provide a significantly simpler architecture, but also a simpler hyper-parameter tuning experience compared GANs, which are well-known to be notoriously difficult to tune (Mescheder et al., 2018). + +Motivated by these observations and the success of diffusion models in image-to-image tasks, we introduce 3D Diffusion Models (3DiMs). 3DiMs are image-to-image diffusion models trained on pairs of images of the same scene, where we assume the poses of the two images are known. Drawing inspiration from Scene Representation Transformers (Sajjadi et al., 2021), 3DiMs are trained to build a conditional generative model of one view given another view and their poses. Our key discovery is that we can turn this image-to-image model into a model that can produce an entire set of 3D-consistent frames through autoregressive generation, which we enable with our novel stochastic conditioning sampling algorithm. We cover stochastic conditioning in more detail in Section 2.2 and provide an illustration in Figure 3. Compared to prior work, 3DiMs are generative (vs. regressive) geometry free models, they allow training to scale to a large number of scenes, and offer a simple end-to-end approach. + +We now summarize our core contributions: + +1. We introduce 3DiM, a geometry-free image-to-image diffusion model for novel view synthesis. +2. We introduce the stochastic conditioning sampling algorithm, which encourages 3DiM to generate 3D-consistent outputs. +3. We introduce $X$ -UNet, a new UNet architecture (Ronneberger et al., 2015) variant for 3D novel view synthesis, demonstrating that changes in architecture are critical for high fidelity results. +4. We introduce an evaluation scheme for geometry-free view synthesis models, $3 D$ consistency scoring, that can numerically capture 3D consistency by training neural fields on model outputs. + +![](images/4f81f91c0027649513cafd37cab9de41c29642105305357d283cc0991c4575ab.jpg) +Figure 2: Pose-conditional image-to-image training – Example training inputs and outputs for pose-conditional image-to-image diffusion models, presented in Section 2.1. Given two frames from a common scene and their poses $( R , t )$ , the training task is to undo the noise added to one of the two frames. $( ^ { * } )$ In practice, our neural network is trained to predict the Gaussian noise $\epsilon$ used to corrupt the original view – the predicted view is still just a linear combination of the noisy input and the predicted $\epsilon$ . + +# 2 POSE-CONDITIONAL DIFFUSION MODELS + +To motivate 3DiMs, let us consider the problem of novel view synthesis given few images from a probabilistic perspective. Given a complete description of a 3D scene $s$ , for any pose $\pmb { p }$ , the view $\scriptstyle { \pmb { x } } ^ { ( { \pmb { p } } ) }$ at pose $\pmb { p }$ is fully determined from $s$ , i.e., views are conditionally independent given $s$ . However, we are interested in modeling distributions of the form $q ( \pmb { x } _ { 1 } , . . . , \pmb { x } _ { m } | \pmb { x } _ { m + 1 } , . . . , \pmb { x } _ { n } )$ without $s$ , where views are no longer conditionally independent. A concrete example is the following: given the back of a person’s head, there are multiple plausible views for the front. An image-to-image model sampling front views given only the back should indeed yield different outputs for each front view – with no guarantees that they will be consistent with each other – especially if it learns the data distribution perfectly. Similarly, given a single view of an object that appears small, there is ambiguity on the pose itself: is it small and close, or simply far away? Thus, given the inherent ambiguity in the few-shot setting, we need a sampling scheme where generated views can depend on each other in order to achieve 3D consistency. This contrasts NeRF approaches, where query rays are conditionally independent given a 3D representation $s$ – an even stronger condition than imposing conditional independence among frames. Such approaches try to learn the richest possible representation for a single scene $s$ , while 3DiM avoids the difficulty of learning a generative model for $s$ altogether. + +# 2.1 IMAGE-TO-IMAGE DIFFUSION MODELS WITH POSE CONDITIONING + +Given a data distribution $q ( \pmb { x } _ { 1 } , \pmb { x } _ { 2 } )$ of pairs of views from a common scene at poses $p _ { 1 } , p _ { 2 } \in \mathrm { S E } ( 3 )$ , we define an isotropic Gaussian process that adds increasing amounts of noise to data samples as the signal-to-noise-ratio $\lambda$ decreases, following Salimans & Ho (2022): + +$$ +q ( z _ { k } ^ { ( \lambda ) } | \pmb { x } _ { k } ) : = \mathcal { N } ( z _ { k } ^ { ( \lambda ) } ; \sigma ( \lambda ) ^ { \frac { 1 } { 2 } } \pmb { x } _ { k } , \sigma ( - \lambda ) \mathbf { I } ) +$$ + +where $\sigma ( \cdot )$ is the sigmoid function. We can apply the reparametrization trick (Kingma & Welling, 2013) and sample from these marginal distributions via + +$$ +z _ { k } ^ { ( \lambda ) } = \sigma ( \lambda ) ^ { \frac { 1 } { 2 } } x _ { k } + \sigma ( - \lambda ) ^ { \frac { 1 } { 2 } } \epsilon , \quad \epsilon \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } ) +$$ + +Then, given a pair of views, we learn to reverse this process in one of the two frames by minimizing the objective proposed by $\mathrm { H o }$ et al. (2020), which has been shown to yield much better sample quality than maximizing the true evidence lower bound (ELBO): + +$$ +L ( \theta ) = \mathbb { E } _ { q ( \pmb { x } _ { 1 } , \pmb { x } _ { 2 } ) } ~ \mathbb { E } _ { \lambda , \epsilon } ~ \| \epsilon _ { \theta } ( z _ { 2 } ^ { ( \lambda ) } , \pmb { x } _ { 1 } , \lambda , \pmb { p } _ { 1 } , \pmb { p } _ { 2 } ) - \epsilon \| _ { 2 } ^ { 2 } +$$ + +where θ is a neural network whose task is to denoise the frame z(λ)2 given a different (clean) frame $\scriptstyle { \mathbf { { \mathscr { x } } } } _ { 1 }$ , and $\lambda$ is the log signal-to-noise-ratio. To make our notation more legible, we slightly abuse notation and from now on we will simply write $\epsilon _ { \theta } ( z _ { 2 } ^ { ( \lambda ) } , x _ { 1 } )$ . We illustrate training in Figure 2. + +![](images/385024aab77ac1d8f5ad70135ef69214f2f8ac11db03cb3e9024e3b64386e72b.jpg) +Figure 3: Stochastic conditioning sampler – We illustrate our proposed inference procedure for 3DiM, outlined in Section 2.2. There are two main components to our sampling procedure: (1) the autoregressive generation of multiple frames (illustrated vertically as “step $1 ^ { \circ }$ , “step $2 ^ { \circ }$ , etc.), and (2) the denoising process to generate each individual frame (illustrated horizontally). When generating a new frame, we select a previous frame as the conditioning frame randomly at each denoising step (illustrated with the dice). Note that this is not part of 3DiM training; also, we omit the pose inputs in the diagram to avoid overloading the figure. + +# 2.2 3D CONSISTENCY VIA STOCHASTIC CONDITIONING + +Motivation. We begin this section by motivating the need of our stochastic conditioning sampler. In the ideal situation, we would model our 3D scene frames using the chain rule decomposition: + +$$ +p ( \pmb { x } ) = \prod _ { i } p ( \pmb { x } _ { i } | \pmb { x } _ { < i } ) +$$ + +This factorization is ideal, as it models the distribution exactly without making any conditional independence assumptions. Each frame is generated autoregressively, conditioned on all the previous frames. However, we found this solution to perform poorly. Due to memory limitations, we can only condition on a limited number of frames in practice, (i.e., a $k$ -Markovian model). We also find that, as we increase the maximum number of input frames $k$ , the worse the sample quality becomes. In order to achieve the best possible sample quality, we thus opt for the bare minimum of $k = 2$ (i.e., an image-to-image model). Our key discovery is that, with $k = 2$ , we can still achieve approximate 3D consistency. Instead of using a sampler that is Markovian over frames, we leverage the iterative nature of diffusion sampling by varying the conditioning frame at each denoising step. + +Stochastic Conditioning. We now detail our novel stochastic conditioning sampling procedure that allows us to generate 3D-consistent samples from a 3DiM. We start with a set of conditioning views $\mathcal { X } = \{ \pmb { x } _ { 1 } , . . . , \pmb { x } _ { k } \}$ of a static scene, where typically $k = 1$ or is very small. We then generate a new frame by running a modified version of the standard denoising diffusion reverse process for steps $\lambda _ { \operatorname* { m i n } } = \lambda _ { T } < \lambda _ { T - 1 } < . . . < \lambda _ { 0 } = \lambda _ { \operatorname* { m a x } }$ : + +$$ +\begin{array} { r c l } { { \hat { \pmb x } _ { k + 1 } } } & { { = } } & { { \displaystyle \frac { 1 } { \sigma ( \lambda _ { k } ) ^ { \frac { 1 } { 2 } } } \left( \pmb z _ { k + 1 } ^ { ( \lambda _ { t } ) } - \sigma ( - \lambda _ { t } ) ^ { \frac { 1 } { 2 } } \pmb \epsilon _ { \theta } ( \pmb z _ { k + 1 } ^ { ( \lambda _ { t } ) } , \pmb x _ { i } ) \right) } } \\ { { \pmb z _ { k + 1 } ^ { ( \lambda _ { t - 1 } ) } } } & { { \sim } } & { { q \left( \pmb z _ { k + 1 } ^ { ( \lambda _ { t - 1 } ) } | \pmb z _ { k + 1 } ^ { ( \lambda _ { t } ) } , \hat { \pmb x } _ { k + 1 } \right) } } \end{array} +$$ + +where, crucially, $i \sim \operatorname { U n i f o r m } ( \{ 1 , . . . , k \} )$ is re-sampled at each denoising step. In other words, each individual denoising step is conditioned on a different random view from $\mathcal { X }$ (the set that contains the input view(s) and the previously generated samples). Once we finish running this sampling chain and produce a final $\mathbf { \boldsymbol { x } } _ { k + 1 }$ , we simply add it to $\mathcal { X }$ and repeat this procedure if we want to sample more frames. Given sufficient denoising steps, stochastic conditioning allows each generated frame to be guided by all previous frames. See Figure 3 for an illustration. In practice, we use 256 denoising steps, which we find to be sufficient to achieve both high sample quality and approximate 3D consistency. As usual in the literature, the first (noisiest sample) is just a Gaussian, i.e., $z _ { i } ^ { ( \lambda _ { T } ) } \sim$ $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ , and at the last step $\lambda _ { 0 }$ , we sample noiselessly. + +We can interpret stochastic conditioning as a na¨ıve approximation to true autoregressive sampling that works well in practice. True autoregressive sampling would require a score model of the form $\nabla _ { z _ { k + 1 } ^ { ( \lambda ) } } \log q ( z _ { k + 1 } ^ { ( \lambda ) } | \pmb { x } _ { 1 } , . . . , \pmb { x } _ { k } )$ , but this would strictly require multi-view training data, while we are ultimately interested in enabling novel view synthesis with as few as two training views per scene. + +![](images/ebe08a760122f6e8d1f7271054f456b935b0433fc3bd38c58fcdc2c918b6d1a0.jpg) +Figure 4: X-UNet Architecture – We modify the typical UNet architecture used by recent work on diffusion models to accomodate 3D novel view synthesis. We share the same UNet weights among the two input frames, the clean conditioning view and the denoising target view. We add cross attention layers to mix information between the input and output view, illustrated in yellow. + +# 2.3 X-UNET + +The 3DiM model needs a neural network architecture that takes both the conditioning frame and the noisy frame as inputs. One natural way to do this is simply to concatenate the two images along the channel dimensions, and use the standard UNet architecture (Ronneberger et al., 2015; Ho et al., 2020). This “Concat-UNet” has found significant success in prior work of image-to-image diffusion models (Saharia et al., 2021b;a). However, in our early experiments, we found that the Concat-UNet yields very poor results – there were severe 3D inconsistencies and lack of alignment to the conditioning image. We hypothesize that, given limited model capacity and training data, it is difficult to learn complex, nonlinear image transformations that only rely on self-attention. We thus introduce our $X$ -UNet, whose core changes are (1) sharing parameters to process each of the two views, and (2) using cross attention between the two views. We find our X-UNet architecture to be very effective for 3D novel view synthesis. + +We now describe X-UNet in detail. We follow Ho et al. (2020); Song et al. (2020), and use the UNet (Ronneberger et al., 2015) with residual blocks and self-attention.We also take inspiration from Video Diffusion Models (Ho et al., 2022) by sharing weights over the two input frames for all the convolutional and self-attention layers, but with several key differences: + +1. We let each frame have its own noise level (recall that the inputs to a DDPM residual block are feature maps as well as a positional encoding for the noise level). We use a positional encoding of $\lambda _ { \mathrm { m a x } }$ for the clean frame. Ho et al. (2022) conversely denoise multiple frames simultaneously, each at the same noise level. 2. Alike Ho et al. (2020), we modulate each UNet block via FiLM (Dumoulin et al., 2018), but we use the sum of pose and noise-level positional encodings, as opposed to the noise-level embedding alone. Our pose encoding additionally differs in that they are of the same dimensionality as frames– they are camera rays, identical to those used by Sajjadi et al. (2021). 3. Instead of attending over “time” after each self-attention layer like Ho et al. (2022), which in our case would entail only two attention weights, we define a cross-attention layer and let each frame’s feature maps call this layer to query the other frame’s feature maps. + +For more details on our proposed architecture, we refer the reader to the Supplementary Material (Sec.6). We also provide a comparison to the “Concat-UNet” architecture in Section 3.2. + +![](images/8d2bd63b699f0034354bdfb4e444b444cc03acff6739f74814c081e8aee2e157.jpg) +Figure 5: State-of-the-art comparisons – Example input and output views of a 3DiM trained on the SRN cars dataset at the 128x128 resolution, compared to existing geometry-aware methods. Results are best appreciated in the Supplementary Website (https://3d-diffusion.github.io/). + +Table 2: State-of-the-art comparisons – Results on the SRN ShapeNet benchmark comparing 3DiMs to prior work on novel view synthesis from a single image. $( ^ { * } )$ SSIM scores may not necessarily be computed with a Gaussian kernel following Wang et al. (2004) – in practice we observe this can lead to differences of up to 0.02. We report these marked numbers directly from prior work. + +
SRN carsSRN chairs
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Geometry-aware
SRN22.250.8841.2122.890.8926.51
PixelNeRF23.170.8959.2423.720.9038.49
VisionNeRF22.880.9021.3124.480.9210.05
CodeNeRF23.80*0.9123.66*0.901
Geometry-free
LFN22.42*0.8922.26*0.90
ENR22.2622.831
3DiM (ours)21.010.578.9917.050.536.57
+ +# 3 EXPERIMENTS + +We benchmark 3DiMs on the SRN ShapeNet dataset (Sitzmann et al., 2019) to allow comparisons with prior work on novel view synthesis from a single image. This dataset consists of views and poses of car and chair ShapeNet (Chang et al., 2015) assets, rendered at the $1 2 8 \mathrm { x } 1 2 8$ resolution. + +We compare 3DiMs with Light Field Networks (LFN) (Sitzmann et al., 2021) and Equivariant Neural Rendering (ENR) (Dupont et al., 2020), two competitive geometry-free approaches, as well as geometry-aware approaches including Scene Representation Networks (SRN) (Sitzmann et al., 2019), PixelNeRF (Yu et al., 2021), CodeNeRF (Jang & Agapito, 2021) and the recent VisionNeRF (Lin et al., 2022). We include standard metrics in the literature: peak signal-to-noiseratio (PSNR), structural similarity (SSIM) (Wang et al., 2004), and the Frechet Inception Distance ´ (FID) (Heusel et al., 2017). To remain consistent with prior work, we evaluate over all scenes in the held-out dataset (each has 251 views), conditioning our models on a single view (view #64) to produce all other 250 views via a single call to our sampler outlined in Section 2.2, and then report the average PSNR and SSIM. Importantly, FID scores were computed with this same number of views, comparing the set of all generated views against the set of all the ground truth views. Because prior work did not include FID scores, we acquire the evaluation image outputs of SRN, PixelNeRF and VisionNeRF and compute the scores ourselves. We reproduce PSNR scores, and carefully note that some of the models do not follow Wang et al. (2004) on their SSIM computation (they use a uniform rather than Gaussian kernel), so we also recompute SSIM scores following (Wang et al., 2004). For more details, including hyperparameter choices, see Supplementary Material (Sec.7). + +Out-of-distribution poses on SRN chairs. We note that the test split of the SRN ShapeNet chairs was unintentionally released with out-of-distribution poses compared to the train split: most test views are at a much larger distance to the object than those in the training dataset; we confirmed this via correspondence with the authors of Sitzmann et al. (2019). Because 3DiMs are geometry-free, we (unsurprisingly) observe that they do not perform well on this out-of-distribution evaluation task, as all the poses used at test-time are of scale never seen during training. However, simply merging, shuffling, and re-splitting the dataset completely fixes the issue. To maintain comparability with prior work, results on SRN chairs in Table 2 are those on the original dataset, and we denote the re-split dataset as “SRN chairs\*” (note the \*) in all subsequent tables to avoid confusion. + +![](images/7d1b0d0366e8db892efbd41eaf393921e833c5e8045fe86d2a6b9b3edbd37994.jpg) +Figure 6: 3DiM samples on OOD synthetic images – We qualitatively demonstrate the text-to-3D capabilities of 3DiM by synthethizing an input view with Imagen, a text-to-image model, and then create more views with 3DiM. + +State-of-the-art comparisons. While 3DiM does not necessarily achieve superior reconstruction errors (PSNR and SSIM, see Table 2), qualitatively, we find that the fidelity of our generated videos can be strikingly better. We include an example in Figure 5. The use of diffusion models allows us to produce sharp samples, as opposed to regression models that are well-known to be prone to blurriness (Saharia et al., 2021b) in spite of high PSNR and SSIM scores. This is why we introduce and evaluate FID scores. In fact, we do not expect to achieve very low reconstruction errors due to the inherent ambiguity of novel view synthesis with a single image: constructing a consistent object with the given frame(s) is acceptable, but will still be punished by said reconstruction metrics if it differs from the ground truth views (even if achieving consistency). We additionally refer the reader to Section 3.1, where we include simple ablations demonstrating how worse models can improve the different standardized metrics in the literature. E.g., we show that with regression models, similarly to the baseline samples, PSNR and SSIM do not capture sharp modes well – they scores these models as much better despite the samples looking significantly more blurry (with PixelNeRF, which qualitatively seems the blurriest, achieving the best scores). + +Results on out-of-distribution images.. We additionally show qualitative results attempting to lift single images in the wild (without known poses) into 3D. To this end, we train a 471M parameter 3DiM on all of ShapeNet (except for 10 objects per class which we leave out for testing). This model has identical parameters to the 3DiM used for SRN cars. However, it differs in that we poses with relative camera extrinsics (i.e., the camera position and rotation corresponding to the noisy input frame is always the same) as we do not know the poses of images in the wild. Our qualitative results on single-image-to-3D generation include: + +1. ShapeNet objects held out from the training data +2. Images in the wild we directly took from the internet (with a white background, minimal shadow, and corresponding to any ShapeNet class) +3. Images synthesized by Imagen (Saharia et al., 2022), a text-to-image diffusion model + +In order to make Imagen synthesize objects with white backgrounds and minimal shadows at the $1 2 8 \mathrm { x } 1 2 8$ resolution, in addition to natural language prompting, we inpaint a 5px white border at each denoising step when generating a $6 4 \mathrm { x } 6 4$ image, and then upsample it using the second model on the Imagen cascade, i.e., a text-conditional super-resolution model. We find that this strategy is much more reliable than only relying on prompt engineering to create images with white backgrounds: this leads the text-to-image model to make the rest of background consistent with the fully white border, so the generated image results in an out-of-distribution object while maintaining an in-distribution background. We present example novel views given a ShapeNet test object in Figure 1, and given a synthetic image from Imagen in Figure 6. We include several video samples for all the cases described above in the Supplementary Website (https://3d-diffusion.github.io/). + +SRN cars +SRN chairs\* + +
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
3DiM21.010.578.9914.290.483.30
+ no stochastic conditioning23.820.591.8715.730.511.05
+regression22.550.8517.4516.850.7620.70
+ +Table 3: Ablation – study removing stochastic conditioning and diffusion altogether from 3DiM. +Results are included for novel view synthesis from a single image on the SRN ShapeNet benchmark. +$( ^ { * } )$ Re-split chairs dataset as detailed in Section 3 – not comparable to numbers in Table 2. + +![](images/d6dbef759ea682169ceebc121b6ec9861cd4dafcac248279f348c36b4c1adb3b.jpg) +Figure 7: Ablations – Example input and output views of 3DiM ablations on the SRN chairs dataset at the $1 2 8 \mathrm { x } 1 2 8$ resolution. The second “Image-to-Image” column corresponds to removing our stochastic conditioning sampler from Section 2.2. The third “Concat-UNet” column corresponds to removing our proposed architecture in Section 2.3, resorting to a UNet that concatenates both images along the channel axis instead of weight sharing over frames. The fourth “Regression” column is one-step diffusion model trained from scratch. + +We now present some ablation studies on 3DiM. First, we remove our proposed sampler and use a na¨ıve image-to-image model as discussed at the start of Section 2.2. We additionally remove the use of a diffusion process altogether, i.e., a regression model that generates samples via a single denoising step. Naturally, the regression models are trained separately, but we can still use an identical architecture, always feeding white noise for one frame, along with its corresponding noise level $\lambda _ { \operatorname* { m i n } }$ . Results are included in Table 3 and samples are included in Figure 7. + +Unsurprisingly, we find that both of these components are crucial to achieve good results. The use of many diffusion steps allows sampling sharp images that achieve much better (lower) FID scores than both prior work and the regression models, where in both cases, reconstructions appear blurry. Notably, the regression models achieve better PSNR and SSIM scores than 3DiM despite the severe blurriness, suggesting that these standardized metrics fail to meaningfully capture sample quality for geometry-free models, at least when comparing them to geometry-aware or non-stochastic reconstruction approaches. Similarly, FID also has a failure mode: na¨ıve image-to-image sampling improves (decreases) FID scores significantly, but severely worsens shape and texture inconsistencies between sampled frames. These findings suggest that none of these standardized metrics are sufficient to effectively evaluate geometry-free models for view synthesis; nevertheless, we observe that said metrics do correlate well with sample quality across 3DiMs and can still be useful indicators for hyperparameter tuning despite their individual failures. + +Table 5: Architecture comparison – study comparing the X-UNet and Concat-UNet neural architectures. Results are included for novel view synthesis from a single image on the SRN ShapeNet benchmark, with the re-split chairs as detailed in Section 3. $( ^ { * } )$ Re-split chairs dataset as detailed in Section 3 – these numbers are not comparable to those in Table 2. + +
PSNR (↑) SSIM (↑)carschairs*
FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Concat-UNet17.210.5221.5412.360.445.15
X-UNet21.010.578.9914.290.483.30
+ +# 3.2 UNET ARCHITECTURE COMPARISONS + +In order to demonstrate the benefit of our proposed modifications, we additionally compare our proposed X-UNet architecture from Section 2.3 to the simpler UNet architecture following Saharia et al. (2021b;a) (which we simply call “Concat-UNet”). The Concat-UNet architecture, unlike ours, does not share weights across frames; instead, it simply concatenates the conditioning image to the noisy input image along the channel axis. To do this comparison, we train 3DiMs on the ConcatUNet architecture with the same number of hidden channels, and keep all other hyperparameters identical. Because our architecture has the additional cross-attention layer at the coarse-resolution blocks, our architecture has a slightly larger number of parameters than the Concat-UNet ${ \sim } 4 7 1 \mathrm { M }$ v.s. ${ \sim } 4 2 1 \mathrm { M }$ ). Results are included in Table 5 and Figure 7. + +While the Concat-UNet architecture is able to sample frames that resemble the data distribution, we find that 3DiMs trained with our proposed X-UNet architecture suffer much less from 3D inconsistency and alignment to the conditioning frame. Moreover, while the metrics should be taken with a grain of salt as previously discussed, we find that all the metrics significantly worsen with the Concat-UNet. We hypothesize that our X-UNet architecture better exploits symmetries between frames and poses due to our proposed weight-sharing mechanism, and that the cross-attention helps significantly to align with the content of the conditioning view. + +# 4 EVALUATING 3D CONSISTENCY IN GEOMETRY-FREE VIEW SYNTHESIS + +As we demonstrate in Sections 3.1 and 3.2, the standardized metrics in the literature have failure modes when specifically applied to geometry-free novel view synthesis models, e.g., their inability to successfully measure 3D consistency, and the possibility of improving them with worse models. Leveraging the fact that volumetric rendering of colored density fields are 3D-consistent by design, we thus propose an additional evaluation scheme called “3D consistency scoring”. Our metrics should satisfy the following desiderata: + +1. The metric must penalize outputs that are not 3D consistent. +2. The metric must not penalize outputs that are 3D consistent but deviate from the ground truth. +3. The metric must penalize outputs that do not align with the conditioning view(s). + +In order to satisfy the second requirement, we cannot compare output renders to ground-truth views. Thus, one straightforward way to satisfy all desiderata is to sample many views from the geometryfree model given a single view, train a NeRF-like neural field (Mildenhall et al., 2020) on a fraction of these views, and compute a set of metrics that compare neural field renders on the remaining views. This way, if the geometry-free model outputs inconsistent views, the training of neural field will be hindered and classical image evaluation metrics should clearly reflect this. Additionally, to enforce the third requirement, we simply include the conditioning view(s) that were used to generate the rest as part of the training data. We report PSNR, SSIM and FID on the held-out views, although one could use other metrics under our proposed evaluation scheme. + +We evaluate 3DiMs on the SRN benchmark, and for comparison, we additionally include metrics for models trained on (1) the real test views and (2) on image-to-image samples from the 3DiMs, i.e., without our proposed sampler like we reported in Section 3. To maintain comparability, we sample the same number of views from the different 3DiMs we evaluate in this section, all at the same poses and conditioned on the same single views. We leave out $10 \%$ of the test views (25 out of 251) from neural field training, picking 25 random indices once and maintaining this choice of indices for all + +
Training view sourceSRN carsSRN chairs*
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Original data (3D consistent)28.210.9610.5724.870.9317.05
3DiM(~1.3B params)28.480.9629.5522.900.8658.61
3DiM(~471M params)28.530.9622.0918.840.7998.78
+ no stochastic conditioning25.780.9430.5117.610.75116.16
+ +Table 6: 3D consistency scores – For neural fields trained with different view sources, we compare renders to held-out views from the same sources. + +subsequent evaluations. We also train models with more parameters $( \sim 1 . 3 \mathrm { B } )$ in order to investigate whether increasing model capacity can further improve 3D consistency. See Supplementary Material (Sec.7) for more details on the neural fields we chose and their hyperparameters. + +We find that our proposed evaluation scheme clearly punishes 3D inconsistency as desired – the neural fields trained on image-to-image 3DiM samples have worse scores across all metrics compared to neural fields trained on 3DiM outputs sampled via our stochastic conditioning. This helps quantify the value of our proposed sampler, and also prevents the metrics from punishing reasonable outputs that are coherent with the input view(s) but do not match the target views – a desirable property due to the stochasticity of generative models. Moreover, we qualitatively find that the smaller model reported in the rest of the paper is comparable in quality to the ${ \sim } 1 . 3 \mathrm { B }$ parameter model on cars, though on chairs, we do observe a significant improvement on 3D consistency in our samples. Importantly, 3D consistency scoring agrees with our qualitative observations. + +# 5 CONCLUSION AND FUTURE WORK + +We propose 3DiM, a diffusion model for 3D novel view synthesis. Combining improvements in our X-UNet neural architecture (Section 2.3), with our novel stochastic conditioning sampling strategy that enables autoregressive generation over frames (Section 2.2), we show that from as few as a single image we can generate approximately 3D consistent views with very sharp sample quality. We additionally introduce 3D consistency scoring to evaluate the 3D consistency of geometry-free generative models by training neural fields on model output views, as their performance will be hindered increasingly with inconsistent training data (Section 4). We thus show, both quantitatively and visually, that 3DiMs with stochastic conditioning can achieve approximate 3D consistency and high sample quality simultaneously, and how classical metrics fail to capture both sharp modes and 3D inconsistency. + +A noteworthy limitation of 3DiM is that, due to the geometry-free setup, it can only handle distributions of poses that it is exposed to durining training. In particular, further study is required where more variations to the poses are introduced. For example, varying focal lengths, sensor widths, making the camera not look exactly at the objects, and varying the distances from the camera to the objects more aggressively. Our use of stochastic conditioning might also exacerbate the need for many denoising steps. While we believe that finding effective architectures that allow conditioning on sets of images remains an important open problem, stochastic conditioning unlocks the ability to train on examples that only have two views and helps overcome memory-intensive neural architectures. Similar approaches using randomized, sparse conditioning are already being explored in follow-up work on video generation (Davtyan et al., 2022) in order to enable efficient training when modeling the joint distribution is computationally prohibitive. + +We are most excited about the possibility of applying 3DiM, which can model entire datasets with a single model, to the largest 3D datasets from the real world – though more research is required to handle noisy poses (due to the need for pose estimation) and other challenges present in this context. Developing end-to-end approaches for high-quality generation that are 3D consistent by design like more recent work (Muller et al., 2022) and that also yield high-quality 3D meshes remain important ¨ research problems. Another potentially significant application of 3DiM is its use as a prior in order to achieve 3D consistent generation via approaches that operate exclusively at sampling time (Poole et al., 2022; Zhou & Tulsiani, 2022; Xu et al., 2022) in order to enable wide adoption of similar, disruptive technologies on the 3D design industry. + +# REFERENCES + +Jonathan T. Barron, Ben Mildenhall, Dor Verbin, Pratul P. Srinivasan, and Peter Hedman. Mip-nerf 360: Unbounded anti-aliased neural radiance fields. CVPR, 2022. + +James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. JAX: composable transformations of Python+NumPy programs, 2018. 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Sparsefusion: Distilling view-conditioned diffusion for 3d reconstruction. arXiv preprint arXiv:2212.00792, 2022. + +# Novel View Synthesis with Diffusion Models + +Supplementary Material + +# 6 ARCHITECTURE DETAILS + +In order to maximize the reproducibility of our results, we provide code in JAX (Bradbury et al., 2018) for our proposed X-UNet neural architecture from Section 2.3. Assuming we have an input batch with elements each containing + +{ "z", # noisy input image (HxWx3 tensor) "x", # conditioning view (HxWx3 tensor) "logsnr", # log signal-to-noise ratio of noisy image (scalar) "t", # camera positions (two 3d vectors) "R", # camera rotations (two 3x3 matrices) "K", # camera intrinsics (a.k.a. calibration matrix) (3x3 matrix) +} + +our neural network module is as follows: + +from typing import Optional import flax.linen as nn import jax.numpy as jnp import numpy as onp import visu3d as v3d + +nonlinearity $=$ nn.swish + +def out_init_scale(): """Zeros initializer.""" return nn.initializers.variance_scaling(0.0, 'fan_in', 'truncated_normal') + +def nearest_neighbor_upsample(h: jnp.ndarray): """Nearest neighbor upsampling for multiple frames.""" B, F, H, W, ${ \mathrm { ~ \small ~ \mathscr ~ { ~ C ~ } ~ } } = { \mathrm { ~ \small ~ h ~ } }$ .shape h $=$ h.reshape(B, F, H, 1, W, 1, C) h = jnp.broadcast_to(h, (B, F, H, 2, W, 2, C)) return h.reshape(B, F, H $\star$ 2, W $\star$ 2, C) + +def avgpool_downsample(h: jnp.ndarray, k: int $= 2$ ): """Average pooling downsampling for multiple frames.""" return nn.avg_pool(h, (1, k, k), (1, k, k)) + +def posenc_ddpm(timesteps: jnp.ndarray, emb_ch: int, max_time: int $=$ 1000.0): """Positional encodings for noise levels, following DDPM.""" # 1000 is the magic number from DDPM. With different timesteps, we # normalize by the number of steps but still multiply by 1000. timesteps \*= 1000.0 / max_time half_dim $=$ emb_ch // 2 # 10000 is the magic number from transformers. emb $=$ onp.log(10000) / (half_dim - 1) + +emb $=$ jnp.exp(jnp.arange(half_dim, dtype $=$ timesteps.dtype) $\star$ -emb) +emb $=$ emb.reshape( $\star$ ([1] $^ { \star }$ (timesteps.ndim - 1)), emb.shape[-1]) +emb $=$ timesteps[..., None] $\star$ emb +emb $=$ jnp.concatenate([jnp.sin(emb), jnp.cos(emb)], axis $= - 1$ ) +return emb + +def posenc_nerf(x: jnp.ndarray, min_deg: int $\qquad = \quad 0$ , max_deg: int $= ~ \perp 5$ ): """Concatenate x and its positional encodings, following NeRF.""" + +if min_deg $= =$ max_deg: return x +scales $=$ jnp.array([2\*\*i for i in range(min_deg, max_deg)]) +xb $=$ jnp.reshape( (x[..., None, :] $^ { \star }$ scales[:, None]), list(x.shape[:-1]) + [-1] +) +emb $=$ jnp.sin(jnp.concatenate([xb, xb $^ +$ onp.pi / 2.0], axis $: = - 1$ )) +return jnp.concatenate([x, emb], axis $= - 1$ ) + +class GroupNorm(nn.Module): """Group normalization, applied over frames.""" + +@nn.compact + +def __call__(self, h: jnp.ndarray):B, _, H, W, ${ \mathrm { ~ \small ~ \mathscr ~ { ~ C ~ } ~ } } = { \mathrm { ~ \small ~ h ~ } }$ .shapeh $=$ nn.GroupNorm(num_groups $= 3 2$ )(h.reshape(B $\star$ 2, 2, H, W, C))return h.reshape(B, 2, H, W, C) + +class FiLM(nn.Module): """Feature-wise linear modulation.""" + +features: int @nn.compact + +def __call__(self, h: jnp.ndarray, emb: jnp.ndarray): emb $=$ nn.Dense(2 \* self.features)(nonlinearity(emb)) scale, shift $=$ jnp.split(emb, $^ 2$ , axis $= - 1$ ) return h $^ { \star }$ (1.0 $^ +$ scale) $^ +$ shift + +class ResnetBlock(nn.Module): """BigGAN-style residual block, applied over frames.""" + +features: Optional[int] $=$ None dropout: float $= ~ 0 . 0$ resample: Optional[str] $=$ None + +# @nn.compact + +def __call__(self, h_in: jnp.ndarray, emb: jnp.ndarray, \*, train: bool): _, _, _, _, C $=$ h_in.shape features $=$ C if self.features is None else self.features + +$\mathrm { ~ \textbar ~ { ~ h ~ } ~ } =$ nonlinearity(GroupNorm()(h_in)) +if self.resample is not None: updown $= \quad \left\{ \begin{array} { r l r l } \end{array} \right.$ 'up': nearest_neighbor_upsample, 'down': avgpool_downsample, }[self.resample] + +$\mathrm { ~ \textit ~ { ~ h ~ } ~ } =$ updown(h) h_in $=$ updown(h_in) + +$\mathrm { ~ \textbar ~ { ~ h ~ } ~ } =$ nn.Conv(features, kernel_size $=$ (1, 3, 3), strides $=$ (1, 1, 1))(h) +$\mathrm { ~ \textbar ~ { ~ h ~ } ~ } =$ FiLM(features $=$ features)(GroupNorm()(h), emb) +h $=$ nonlinearity(h) +h $=$ nn.Dropout(rate $=$ self.dropout)(h, deterministic $: =$ not train) +$\mathrm { ~ \textbar ~ { ~ h ~ } ~ } =$ nn.Conv( features, kernel_size $=$ (1, 3, 3), strides $=$ (1, 1, 1), kernel_init $=$ out_init_scale(), + +)(h) + +if C $! =$ features: h_in $=$ nn.Dense(features)(h_in) return (h $^ +$ h_in) / onp.sqrt(2) + +class AttnLayer(nn.Module): + +"""Attention layer usable for self and cross attention.""" + +attn_heads: int $\qquad = \quad 4$ + +# @nn.compact + +def _call__(self, \*, q: jnp.ndarray, kv: jnp.ndarray): $\mathrm { ~ C ~ } = \mathrm { ~ q ~ }$ .shape[-1] head_dim $\begin{array} { r l } { \mathbf { \Sigma } } & { { } = \mathbf { \Sigma } \subset \mathbf { \Sigma } } \end{array}$ // self.attn_heads q $=$ nn.DenseGeneral((self.attn_heads, head_dim))(q) k $=$ nn.DenseGeneral((self.attn_heads, head_dim))(kv) $\begin{array} { r l } { \mathsf { v } } & { { } = } \end{array}$ nn.DenseGeneral((self.attn_heads, head_dim))(kv) return nn.dot_product_attention(q, k, v) + +class AttnBlock(nn.Module): """Attention block with skip connection.""" + +attn_type: str attn_heads: int $\qquad = \quad 4$ + +# @nn.compact + +def __call__(self, h_in: jnp.ndarray): B, _, H, W, C $=$ h_in.shape + +h $=$ GroupNorm()(h_in) h0 = h[:, 0].reshape(B, H \* W, C) h1 = h[:, 1].reshape(B, H $\star$ W, C) attn_layer $=$ AttnLayer(attn_heads $=$ self.attn_heads) + +if self.attn_type $= =$ 'self': $\mathrm { ~ \textit ~ { ~ h ~ O ~ } ~ } =$ attn_layer( $\mathtt { q } \mathrm { = h 0 }$ , $\mathtt { k v } { = } \mathtt { h } 0$ ) $\begin{array} { r l } { \operatorname { h } 1 } & { { } = } \end{array}$ attn_layer( $\mathtt { q } \mathrm { = h 1 }$ , $\mathrm { k v } { = } \mathrm { h } 1$ ) +elif self.attn_type $= =$ 'cross': $\mathrm { ~ \textit ~ { ~ h ~ O ~ } ~ } =$ attn_layer( $\mathtt { q } \mathrm { = h 0 }$ , $\mathrm { k v } { = } \mathrm { h } 1$ ) h1 $=$ attn_layer( $\mathtt { q } \mathrm { = h 1 }$ , $\mathtt { k v } { = } \mathtt { h } 0$ ) +else: raise NotImplementedError(self.attn_type) + +h = jnp.stack([h0, h1], axis $^ { = 1 }$ ) + +$\mathrm { ~ \textbar ~ { ~ h ~ } ~ } =$ h.reshape(B, 2, H, W, $^ { - 1 }$ ) +$\mathrm { ~ \textbar ~ { ~ h ~ } ~ } =$ nn.DenseGeneral(C, axis $ \mathfrak { s } = ( - 2$ , -1), kernel_init $=$ out_init_scale())(h) +return (h $^ +$ h_in) / onp.sqrt(2) + +class XUNetBlock(nn.Module): """X-UNet block.""" + +features: int use_attn: bool $=$ False attn_heads: int $\qquad = \quad 4$ dropout: float $= ~ 0 . 0$ + +# @nn.compact + +def call (self, x: jnp.ndarray, emb: jnp.ndarray, $\star$ , train: bool): h $=$ ResnetBlock(features $=$ self.features, dropout $=$ self.dropout)( x, emb, train $=$ train ) + +if self.use_attn: h $=$ AttnBlock(attn_type $=$ 'self', attn_heads $=$ self.attn_heads)(h) h $=$ AttnBlock(attn_type $= ^ { \parallel }$ cross', attn_heads $=$ self.attn_heads)(h) + +return h + +class ConditioningProcessor(nn.Module): """Process conditioning inputs into embeddings.""" + +emb_ch: int num_resolutions: int use_pos_emb: bool $=$ True use_ref_pose_emb: bool $=$ True + +@nn.compact +def __call__(self, batch: dict[str, jnp.ndarray], cond_mask: jnp.ndarray): B, H, W, _ $=$ batch['x'].shape # Log signal-to-noise-ratio embedding. +logsnr $=$ jnp.clip(batch['logsnr'], -20.0, 20.0) +logsnr $= \ 2 . 0 \times$ jnp.arctan(jnp.exp(-logsnr / 2.0)) / onp.pi +logsnr_emb $=$ posenc_ddpm(logsnr, emb_ch ${ \underline { { \mathbf { \Pi } } } } =$ self.emb_ch, max_time $^ { - 1 }$ .0) logsnr_emb $=$ nn.Dense(self.emb_ch)(logsnr_emb) +logsnr_emb $=$ nn.Dense(self.emb_ch)(nonlinearity(logsnr_emb)) + +# Pose embeddings. + +world_from_cam $=$ v3d.Transform( $\mathrm { R = }$ batch['R'], t $=$ batch['t']) cam_spec $=$ v3d.PinholeCamera(resolution ${ } = { }$ (H, W), $\mathrm { K } =$ batch['K']) rays $=$ v3d.Camera(spec $=$ cam_spec, world_from_cam $\cdot ^ { = }$ world_from_cam).rays() + +pose_emb_pos $=$ posenc_nerf(rays.pos, min_deg $= 0$ , max_deg $= \beth 5$ ) pose_emb_dir $=$ posenc_nerf(rays.dir, min_deg ${ \bf \bar { \theta } } = 0$ , max_deg ${ } = 8 { }$ ) pose_emb $=$ jnp.concatenate([pose_emb_pos, pose_emb_dir], axi $S { = } { - } 1$ ) + +# Enable classifier-free guidance over poses. + +D $=$ pose_emb.shape[-1] +assert cond_mask.shape $= =$ (B,) +cond_mask $=$ cond_mask[:, None, None, None, None] +pose_emb $=$ jnp.where(cond_mask, pose_emb, jnp.zeros_like(pose_emb)) + +# Learnable position embeddings over (H, W) of frames (optional). + +if self.use_pos_emb: pos_emb $=$ self.param( 'pos_emb', nn.initializers.normal(stddev $\ l = 1$ .0 / onp.sqrt(D)), (H, W, D), pose_emb.dtype, ) pose_emb $+ =$ pos_emb[None, None] + +# Binary embedding to let the model distinguish frames (optional) + +if self.use_ref_pose_emb: first_emb $=$ self.param( 'ref_pose_emb_first', nn.initializers.normal(stddev $^ { \cdot = 1 }$ .0 / onp.sqrt(D)), (D,), pose_emb.dtype, )[None, None, None, None] other_emb $=$ self.param( 'ref_pose_emb_other', nn.initializers.normal(stddev $\ l = 1$ .0 / onp.sqrt(D)), (D,), pose_emb.dtype, )[None, None, None, None] pose_emb $+ =$ jnp.concatenate([first_emb, other_emb], axis $^ { = 1 }$ + +# Downsample ray embeddings for each UNet resolution. +pose_embs $=$ [] +for i_level in range(self.num_resolutions): pose_embs.append( nn.Conv( features $=$ self.emb_ch, kernel_size $=$ (1, 3, 3), strides $=$ (1, $2 \star \star$ i_level, 2\*\*i_level), )(pose_emb) ) + +return logsnr_emb, pose_embs class XUNet(nn.Module): """Our proposed XUNet architecture.""" + +ch: int $= \ 2 5 6$ +ch_mult: tuple[int] $=$ (1, 2, 2, 4) +emb_ch: int $= ~ 1 0 2 4$ +num_res_blocks: int $= 3$ +attn_resolutions: tuple[int] $=$ (8, 16, 32) +attn_heads: int $\qquad = \quad 4$ +dropout: float $= ~ 0 . 1$ +use_pos_emb: bool $=$ True +use_ref_pose_emb: bool $=$ True + +@nn.compact def __call__( self, + +batch: dict[str, jnp.ndarray], \*, cond_mask: jnp.ndarray, train: bool, ): _, _, _, ${ \mathrm { ~ \small ~ \mathscr ~ { ~ C ~ } ~ } } =$ batch['x'].shape num_resolutions $=$ len(self.ch_mult) logsnr_emb, pose_embs $=$ ConditioningProcessor( emb_ch $=$ self.emb_ch, num_resolutions $=$ num_resolutions, use_pos_emb $=$ self.use_pos_emb, use_ref_pose_emb $=$ self.use_ref_pose_emb, )(batch, cond_mask) del cond_mask + +h $=$ jnp.stack([batch['x'], batch $[ { \mathrm { ~ ~ \cdot ~ } } _ { Z } { \mathrm { ~ ~ \cdot ~ } } ] ]$ , axis $: = 1$ ) h $=$ nn.Conv(self.ch, kernel_size $=$ (1, 3, 3), strides $=$ (1, 1, 1))(h) + +# Downsampling. + +hs $=$ [h] +for i_level in self.ch_mult: emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[i_level] + +for in range(self.num_res_blocks): use_attn $=$ h.shape[2] in self.attn_resolutions h $=$ XUNetBlock( features $=$ self.ch $\star$ self.ch_mult[i_level], dropout $=$ self.dropout, attn_heads $=$ self.attn_heads, use_attn ${ \bf \Phi } = { \bf \Phi }$ use_attn, )(h, emb, train $=$ train) hs.append(h) + +if i_level $\downarrow =$ num_resolutions - 1: emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[i_level + 1] h $=$ ResnetBlock(dropout $=$ self.dropout, resample $= 1$ down')( h, emb, train ${ \bf \Phi } = { \bf \Phi }$ train ) hs.append(h) + +# # Middle. + +emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[-1] +use_attn $=$ h.shape[2] in self.attn_resolutions +h $=$ XUNetBlock( features $=$ self.ch $\star$ self.ch_mult[i_level], dropout $=$ self.dropout, attn_heads $=$ self.attn_heads, use_attn $=$ use_attn, +)(h, emb, train $=$ train) + +$\#$ Upsampling. + +for i_level in reversed(range(num_resolutions)): emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[i_level] for _ in range(self.num_res_blocks + 1): use_attn $=$ hs[-1].shape[2] in self.attn_resolutions h $=$ jnp.concatenate([h, hs.pop()], axis $= - 1$ ) h $=$ XUNetBlock( + +features $=$ self.ch $\star$ self.ch_mult[i_level], dropout $=$ self.dropout, attn_heads $=$ self.attn_heads, use_attn ${ \bf \Phi } = { \bf \Phi }$ use_attn, )(h, emb, train $=$ train) + +if i_level $\ : \ 0$ : emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[i_level] $\mathrm { ~ \textbar ~ { ~ h ~ } ~ } =$ ResnetBlock(dropout $=$ self.dropout, resample $= ^ { \parallel }$ up')( h, emb, train ${ \bf \Phi } = { \bf \Phi }$ train ) + +# End. +assert not hs +$\mathrm { ~ \textbar ~ { ~ h ~ } ~ } =$ nonlinearity(GroupNorm()(h)) +return nn.Conv( C, kernel_size $=$ (1, 3, 3), strides $; =$ (1, 1, 1), kernel_init $=$ out_init_scale(), +)(h)[:, 1] + +# 7 HYPERPARAMETERS + +We now detail hyperparameter choices across our experiments. These include choices specific to the neural architecture, the training procedure, and also choices only relevant during inference. + +# 7.1 NEURAL ARCHITECTURE + +For our neural architecture, our main experiments use $_ { \mathrm { C h } = 2 5 6 }$ ( ${ \sim } 4 7 1 \mathrm { M }$ params), and we also experiment with $\cosh { = } 4 4 8$ $( \sim 1 . 3 \mathrm { B }$ params) in Section 4. One of our early findings that we kept throughout all experiments in the paper is that $\mathtt { c h \_ m u l t } = ( \mathtt { l } , \mathtt { \xi } _ { 2 } , \mathtt { \xi } _ { 2 } , \mathtt { \xi } _ { 4 } )$ (i.e., setting the lowest UNet resolution to 8x8) was sufficient to achieve good sample quality, wheras most prior work includes UNet resolutions up to $4 \mathbf { x } 4$ . We sweeped over the rest of the hyperparameters with the input and target views downsampled from their original $1 2 8 \mathrm { x } 1 2 8$ resolution to the $3 2 \mathrm { x } 3 2 $ , selecting values that led to the best qualitative improvements. These values are the default values present in the code we provide for the XUNet module in Section 6. We generally find that the best hyperparameter choices at low-resolution experiments transfer well when applied to the higher resolutions, and thus recommend this strategy for cheaper and more practical hyperparameter tuning. + +For the ${ \sim } 1 . 3 \mathrm { B }$ parameter models, we tried increasing the number of parameters of our proposed UNet architecture through several different ways: increasing the number of blocks per resolution, the number of attention heads, the number of cross-attention layers per block, and the base number of hidden channels. Among all of these, we only found the last to provide noticeably better sample quality. We thus run 3D consistency scoring for models scaled this way, with channel sizes per UNet resolution of $4 4 8 \times [ 1 , 2 , 2 , 4 ]$ instead of $2 5 6 \times [ 1 , 2 , 2 , 4 ]$ (we could not fit $_ { \mathrm { c h } = 5 1 2 }$ in TPUv4 memory without model parallelism). On cars, we find that the smaller model reported in the rest of the paper is comparable in quality to the ${ \sim } 1 . 3 \mathrm { B }$ parameter model, though on chairs, we do observe a significant improvement on 3D consistency, both visually and quantitatively (see Table 6). + +# 7.2 TRAINING + +Following Equation 3 in the paper, our neural network attempts to model the noise $\epsilon$ added to a real image in order to undo it given the noisy image. Other parameterizations are possible, e.g., predicting $_ { \textbf { \em x } }$ directly rather than predicting $\epsilon$ , though we did not sweep over these choices. For our noise schedule, we use a cosine-shaped log signal to noise ratio that monotonically decreases from 20 to -20. It can be implemented in JAX as follows: + +def logsnr_schedule_cosine(t, $\star$ , logsnr_min $= - 2 0$ ., logsnr_max $: = 2 0$ .): $\textrm { b } =$ onp.arctan(onp.exp(-.5 \* logsnr_max)) $\begin{array} { r l } { \exists } & { { } = } \end{array}$ onp.arctan(onp.exp(-.5 $\star$ logsnr_min)) - b return $^ { - 2 }$ . \* jnp.log(jnp.tan( $ { \sf a } \mathrm { ~ ~ \star ~ } \sf t + \mathrm { ~ ~ b ~ } )$ ) + +Additionally: + +• We use a learning rate with peak value 0.0001, using linear warmup for the first 10 million examples (where one batch has batch_size examples), following Karras et al. (2022). +• We use a global batch size of 128. +• We train each batch element as an unconditional example $10 \%$ of the time to enable classifier-free guidance. This is done by overriding the conditioning frame to be at the maximum noise level. We note other options are possible (e.g., zeroing-out the conditioning frame), but we chose the option which is most compatible with our neural architecture. +• We use the Adam optimizer (Kingma & Ba, 2014) with $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } = 0 . 9 9$ . +• We use EMA decay for the model parameters, with a half life of 500K examples (where one batch has batch_size examples) following Karras et al. (2022). + +# 7.3 SAMPLING + +While Ho et al. (2020) note that their ancestral sampler can be used with any variances between $\tilde { \beta } _ { t }$ (variances of $q ( \boldsymbol { z } _ { s } | \boldsymbol { z } _ { t } , \boldsymbol { x } ) )$ and $\beta _ { t }$ (variances of the underlying SDE), we simply use the former as those correspond to their ancestral sampler. We do not observe this choice to have a significant effect sample quality. All our samples are generated with 256 denoising steps. As is standard in the literature, we also clip each predicted $_ { \textbf { \em x } }$ at each denoising step to the normalized range of the images $[ - 1 , 1 ]$ . + +Classifier-free guidance. Our models use classifier-free guidance (Ho & Salimans, 2021), as we find that small guidance weights help encourage 3D consistency further. All our models were trained unconditionally with a probability of $10 \%$ for each minibatch element. For unconditional examples, we zero out the (positionally encoded) pose and replace the clean frame with standard Gaussian noise (leveraging our weight-sharing architecture, see Section 2.3). We swept over various guidance weights, and simply picked those where 3D inconsistency was qualitatively least apparent on sampled videos. For SRN cars, we use a guidance weight of 3.0, while for SRN chairs we use a weight of 2.0. + +# 7.4 3D CONSISTENCY SCORING + +Note that traditional NeRFs (Mildenhall et al., 2020) can be 3D inconsistent as the model allows for view-dependent radiance. We thus employ a simpler and faster to train version based on instantNGP (Muller et al., 2022) without view dependent components, and with additional distortion and ¨ orientation loss terms for improved convergence (Barron et al., 2022; Verbin et al., 2022). Note that the specific implementation details of the neural field method chosen will affect the metrics considerably, so it is of utmost importance to apply the same method and hyperparameters for the neural fields to make 3D consistency scores comparable across models. + +We design the neural fields as simple Multi-Layer Perceptrons (MLPs) of hidden size 64 and no skip connections. The density MLP has one hidden layer, while the MLP that predicts the color components uses 2 hidden layers. We use a learning rate of 0.01 linearly decayed to 0.001 for the first 100 steps. We use the Adam optimizer with weight decay set to 0.1 and clip gradients of norm exceeding 1.0. We only apply 1000 training steps for each scene and do not optimize camera poses. On each training step, we simply optimize over all available pixels, rather than sampling a random subset of pixels from all the training views. + +To render the neural fields after training, we set near and far bounds to $\begin{array} { r } { t _ { n } = \frac { 3 r _ { \mathrm { m i n } } } { 8 } , t _ { f } = \frac { 3 r _ { \mathrm { m a x } } } { 2 } . } \end{array}$ where $r _ { \operatorname* { m i n } } , r _ { \operatorname* { m a x } }$ are the minimum and maximum distances from the camera positions to the origin (center of each object) in the corresponding dataset. All renders post-training are also performed with differentiable volume rendering. \ No newline at end of file diff --git a/parse/dev/HtoA0oT30jC/HtoA0oT30jC_content_list.json b/parse/dev/HtoA0oT30jC/HtoA0oT30jC_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e4bea3d106c7c1185d09588fa348e02cc1fa9be5 --- /dev/null +++ b/parse/dev/HtoA0oT30jC/HtoA0oT30jC_content_list.json @@ -0,0 +1,2613 @@ +[ + { + "type": "text", + "text": "NOVEL VIEW SYNTHESIS WITH DIFFUSION MODELS ", + "text_level": 1, + "bbox": [ + 173, + 99, + 805, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Daniel Watson Google Research, Brain ", + "bbox": [ + 204, + 140, + 390, + 174 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "William Chan Google Research, Brain ", + "bbox": [ + 413, + 140, + 599, + 174 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ricardo Martin-Brualla Google Research ", + "bbox": [ + 620, + 141, + 821, + 174 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jonathan Ho Google Research, Brain ", + "bbox": [ + 204, + 190, + 390, + 223 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Andrea Tagliasacchi Google Research, Brain ", + "bbox": [ + 413, + 190, + 598, + 223 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mohammad Norouzi Google Research, Brain ", + "bbox": [ + 619, + 190, + 808, + 223 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 261, + 544, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present 3DiM, a diffusion model for 3D novel view synthesis, which is able to translate a single input view into consistent and sharp completions across many views. The core component of 3DiM is a pose-conditional image-to-image diffusion model, which is trained to take a source view and its pose as inputs, and generates a novel view for a target pose as output. 3DiM can then generate multiple views that are approximately 3D consistent using a novel technique called stochastic conditioning. At inference time, the output views are generated autoregressively. When generating each novel view, one selects a random conditioning view from the set of previously generated views at each denoising step. We demonstrate that stochastic conditioning significantly improves 3D consistency compared to a na¨ıve sampler for an image-to-image diffusion model, which involves conditioning on a single fixed view. We compare 3DiM to prior work on the SRN ShapeNet dataset, demonstrating that 3DiM’s generated completions from a single view achieve much higher fidelity, while being approximately 3D consistent. We also introduce a new evaluation methodology, 3D consistency scoring, to quantify the 3D consistency of a generated object by training a neural field on the model’s output views. 3DiM is geometry free, does not rely on hyper-networks or test-time optimization for novel view synthesis, and allows a single model to easily scale to a large number of scenes. ", + "bbox": [ + 232, + 289, + 766, + 553 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/476bebe18bd204ae2dce90f0a46a3eb7c0418fe2ef3bb88c505ec200a6c9160a.jpg", + "image_caption": [ + "Figure 1: Given a single input image on the left, 3DiM performs novel view synthesis and generates the four views on the right. We trained a single ${ \\sim } 4 7 1 \\mathrm { M }$ parameter 3DiM on all of ShapeNet (without classconditioning) and sample frames with 256 steps (512 score function evaluations with classifier-free guidance). See the Supplementary Website (https://3d-diffusion.github.io/) for video outputs. " + ], + "image_footnote": [], + "bbox": [ + 173, + 577, + 823, + 675 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 767, + 334, + 784 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Diffusion Probabilistic Models (DPMs) (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020), also known as simply diffusion models, have recently emerged as a powerful family of generative models, achieving state-of-the-art performance on audio and image synthesis (Chen et al., 2020; Dhariwal & Nichol, 2021), while admitting better training stability over adversarial approaches (Goodfellow et al., 2014), as well as likelihood computation, which enables further applications such as compression and density estimation (Song et al., 2021; Kingma et al., 2021). Diffusion models have achieved impressive empirical results in a variety of image-to-image translation tasks not limited to text-to-image, super-resolution, inpainting, colorization, uncropping, and artifact removal (Song et al., 2020; Saharia et al., 2021a; Ramesh et al., 2022; Saharia et al., 2022). ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One particular image-to-image translation problem where diffusion models have not been investigated is novel view synthesis, where, given a set of images of a given 3D scene, the task is to infer how the scene looks from novel viewpoints. Before the recent emergence of Scene Representation Networks (SRN) (Sitzmann et al., 2019) and Neural Radiance Fields (NeRF) (Mildenhall et al., 2020), state-of-the-art approaches to novel view synthesis were typically built on generative models (Sun et al., 2018) or more classical techniques on interpolation or disparity estimation (Park et al., 2017; Zhou et al., 2018). Today, these models have been outperformed by NeRF-class models (Yu et al., 2021; Niemeyer et al., 2021; Jang & Agapito, 2021), where 3D consistency is guaranteed by construction, as images are generated by volume rendering of a single underlying 3D representation (a.k.a. “geometry-aware” models). ", + "bbox": [ + 174, + 103, + 825, + 242 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Still, these approaches feature different limitations. Heavily regularized NeRFs for novel view synthesis with few images such as RegNeRF (Niemeyer et al., 2021) produce undesired artifacts when given very few images, and fail to leverage knowledge from multiple scenes (recall NeRFs are trained on a single scene, i.e., one model per scene), and given one or very few views of a novel scene, a reasonable model must extrapolate to complete the occluded parts of the scene. PixelNeRF (Yu et al., 2021) and VisionNeRF (Lin et al., 2022) address this by training NeRF-like models conditioned on feature maps that encode the novel input view(s). However, these approaches are regressive rather than generative, and as a result, they cannot yield different plausible modes and are prone to blurriness. This type of failure has also been previously observed in regression-based models (Saharia et al., 2021b). Other works such as CodeNeRF (Jang & Agapito, 2021) and LoLNeRF (Rebain et al., 2021) instead employ test-time optimization to handle novel scenes, but still have issues with sample quality. ", + "bbox": [ + 173, + 250, + 825, + 416 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In recent literature, geometry-free approaches (i.e., methods without explicit geometric inductive biases like those introduced by volume rendering) such as Light Field Networks (LFN) (Sitzmann et al., 2021) and Scene Representation Transformers (SRT) (Sajjadi et al., 2021) have achieved results competitive with 3D-aware methods in the “few-shot” setting, where the number of conditioning views is limited (i.e., 1-10 images vs. dozens of images as in the usual NeRF setting). Similarly to our approach, EG3D (Chan et al., 2022) provides approximate 3D consistency by leveraging generative models. EG3D employs a StyleGAN (Karras et al., 2019) with volumetric rendering, followed by generative super-resolution (the latter being responsible for the approximation). In comparison to this complex setup, we do not only provide a significantly simpler architecture, but also a simpler hyper-parameter tuning experience compared GANs, which are well-known to be notoriously difficult to tune (Mescheder et al., 2018). ", + "bbox": [ + 174, + 422, + 825, + 575 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Motivated by these observations and the success of diffusion models in image-to-image tasks, we introduce 3D Diffusion Models (3DiMs). 3DiMs are image-to-image diffusion models trained on pairs of images of the same scene, where we assume the poses of the two images are known. Drawing inspiration from Scene Representation Transformers (Sajjadi et al., 2021), 3DiMs are trained to build a conditional generative model of one view given another view and their poses. Our key discovery is that we can turn this image-to-image model into a model that can produce an entire set of 3D-consistent frames through autoregressive generation, which we enable with our novel stochastic conditioning sampling algorithm. We cover stochastic conditioning in more detail in Section 2.2 and provide an illustration in Figure 3. Compared to prior work, 3DiMs are generative (vs. regressive) geometry free models, they allow training to scale to a large number of scenes, and offer a simple end-to-end approach. ", + "bbox": [ + 174, + 583, + 825, + 736 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We now summarize our core contributions: ", + "bbox": [ + 176, + 742, + 455, + 757 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We introduce 3DiM, a geometry-free image-to-image diffusion model for novel view synthesis. \n2. We introduce the stochastic conditioning sampling algorithm, which encourages 3DiM to generate 3D-consistent outputs. \n3. We introduce $X$ -UNet, a new UNet architecture (Ronneberger et al., 2015) variant for 3D novel view synthesis, demonstrating that changes in architecture are critical for high fidelity results. \n4. We introduce an evaluation scheme for geometry-free view synthesis models, $3 D$ consistency scoring, that can numerically capture 3D consistency by training neural fields on model outputs. ", + "bbox": [ + 173, + 763, + 825, + 862 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/4f81f91c0027649513cafd37cab9de41c29642105305357d283cc0991c4575ab.jpg", + "image_caption": [ + "Figure 2: Pose-conditional image-to-image training – Example training inputs and outputs for pose-conditional image-to-image diffusion models, presented in Section 2.1. Given two frames from a common scene and their poses $( R , t )$ , the training task is to undo the noise added to one of the two frames. $( ^ { * } )$ In practice, our neural network is trained to predict the Gaussian noise $\\epsilon$ used to corrupt the original view – the predicted view is still just a linear combination of the noisy input and the predicted $\\epsilon$ . " + ], + "image_footnote": [], + "bbox": [ + 344, + 69, + 643, + 218 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 POSE-CONDITIONAL DIFFUSION MODELS ", + "text_level": 1, + "bbox": [ + 174, + 345, + 545, + 361 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To motivate 3DiMs, let us consider the problem of novel view synthesis given few images from a probabilistic perspective. Given a complete description of a 3D scene $s$ , for any pose $\\pmb { p }$ , the view $\\scriptstyle { \\pmb { x } } ^ { ( { \\pmb { p } } ) }$ at pose $\\pmb { p }$ is fully determined from $s$ , i.e., views are conditionally independent given $s$ . However, we are interested in modeling distributions of the form $q ( \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { m } | \\pmb { x } _ { m + 1 } , . . . , \\pmb { x } _ { n } )$ without $s$ , where views are no longer conditionally independent. A concrete example is the following: given the back of a person’s head, there are multiple plausible views for the front. An image-to-image model sampling front views given only the back should indeed yield different outputs for each front view – with no guarantees that they will be consistent with each other – especially if it learns the data distribution perfectly. Similarly, given a single view of an object that appears small, there is ambiguity on the pose itself: is it small and close, or simply far away? Thus, given the inherent ambiguity in the few-shot setting, we need a sampling scheme where generated views can depend on each other in order to achieve 3D consistency. This contrasts NeRF approaches, where query rays are conditionally independent given a 3D representation $s$ – an even stronger condition than imposing conditional independence among frames. Such approaches try to learn the richest possible representation for a single scene $s$ , while 3DiM avoids the difficulty of learning a generative model for $s$ altogether. ", + "bbox": [ + 173, + 377, + 825, + 601 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 IMAGE-TO-IMAGE DIFFUSION MODELS WITH POSE CONDITIONING ", + "text_level": 1, + "bbox": [ + 176, + 621, + 671, + 635 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given a data distribution $q ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } )$ of pairs of views from a common scene at poses $p _ { 1 } , p _ { 2 } \\in \\mathrm { S E } ( 3 )$ , we define an isotropic Gaussian process that adds increasing amounts of noise to data samples as the signal-to-noise-ratio $\\lambda$ decreases, following Salimans & Ho (2022): ", + "bbox": [ + 174, + 646, + 825, + 689 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/7306934ed6bb89ed18fa7ae3df5ed81b325d17bd060f8bf8b4d4d53472e3f064.jpg", + "text": "$$\nq ( z _ { k } ^ { ( \\lambda ) } | \\pmb { x } _ { k } ) : = \\mathcal { N } ( z _ { k } ^ { ( \\lambda ) } ; \\sigma ( \\lambda ) ^ { \\frac { 1 } { 2 } } \\pmb { x } _ { k } , \\sigma ( - \\lambda ) \\mathbf { I } )\n$$", + "text_format": "latex", + "bbox": [ + 354, + 696, + 642, + 719 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\sigma ( \\cdot )$ is the sigmoid function. We can apply the reparametrization trick (Kingma & Welling, 2013) and sample from these marginal distributions via ", + "bbox": [ + 171, + 727, + 821, + 756 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/bf27fb549e867a950564c2e1e0406dc4c6c7e7fdcec0e9b1f4f053610dc282cc.jpg", + "text": "$$\nz _ { k } ^ { ( \\lambda ) } = \\sigma ( \\lambda ) ^ { \\frac { 1 } { 2 } } x _ { k } + \\sigma ( - \\lambda ) ^ { \\frac { 1 } { 2 } } \\epsilon , \\quad \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )\n$$", + "text_format": "latex", + "bbox": [ + 346, + 763, + 651, + 785 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Then, given a pair of views, we learn to reverse this process in one of the two frames by minimizing the objective proposed by $\\mathrm { H o }$ et al. (2020), which has been shown to yield much better sample quality than maximizing the true evidence lower bound (ELBO): ", + "bbox": [ + 174, + 792, + 825, + 835 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0bff490b118041dae39794201061503813426cf1c1056d222495b14273a778fd.jpg", + "text": "$$\nL ( \\theta ) = \\mathbb { E } _ { q ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } ) } ~ \\mathbb { E } _ { \\lambda , \\epsilon } ~ \\| \\epsilon _ { \\theta } ( z _ { 2 } ^ { ( \\lambda ) } , \\pmb { x } _ { 1 } , \\lambda , \\pmb { p } _ { 1 } , \\pmb { p } _ { 2 } ) - \\epsilon \\| _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 316, + 844, + 679, + 866 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where \u000fθ is a neural network whose task is to denoise the frame z(λ)2 given a different (clean) frame $\\scriptstyle { \\mathbf { { \\mathscr { x } } } } _ { 1 }$ , and $\\lambda$ is the log signal-to-noise-ratio. To make our notation more legible, we slightly abuse notation and from now on we will simply write $\\epsilon _ { \\theta } ( z _ { 2 } ^ { ( \\lambda ) } , x _ { 1 } )$ . We illustrate training in Figure 2. ", + "bbox": [ + 174, + 876, + 825, + 925 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/385024aab77ac1d8f5ad70135ef69214f2f8ac11db03cb3e9024e3b64386e72b.jpg", + "image_caption": [ + "Figure 3: Stochastic conditioning sampler – We illustrate our proposed inference procedure for 3DiM, outlined in Section 2.2. There are two main components to our sampling procedure: (1) the autoregressive generation of multiple frames (illustrated vertically as “step $1 ^ { \\circ }$ , “step $2 ^ { \\circ }$ , etc.), and (2) the denoising process to generate each individual frame (illustrated horizontally). When generating a new frame, we select a previous frame as the conditioning frame randomly at each denoising step (illustrated with the dice). Note that this is not part of 3DiM training; also, we omit the pose inputs in the diagram to avoid overloading the figure. " + ], + "image_footnote": [], + "bbox": [ + 171, + 58, + 823, + 244 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 3D CONSISTENCY VIA STOCHASTIC CONDITIONING ", + "text_level": 1, + "bbox": [ + 174, + 401, + 571, + 415 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Motivation. We begin this section by motivating the need of our stochastic conditioning sampler. In the ideal situation, we would model our 3D scene frames using the chain rule decomposition: ", + "bbox": [ + 171, + 426, + 823, + 455 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d6aac7b809a902d38b5c9bc3e2ae9e783c10c340ddb7a3460959fa028f9edb16.jpg", + "text": "$$\np ( \\pmb { x } ) = \\prod _ { i } p ( \\pmb { x } _ { i } | \\pmb { x } _ { < i } )\n$$", + "text_format": "latex", + "bbox": [ + 424, + 457, + 573, + 489 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This factorization is ideal, as it models the distribution exactly without making any conditional independence assumptions. Each frame is generated autoregressively, conditioned on all the previous frames. However, we found this solution to perform poorly. Due to memory limitations, we can only condition on a limited number of frames in practice, (i.e., a $k$ -Markovian model). We also find that, as we increase the maximum number of input frames $k$ , the worse the sample quality becomes. In order to achieve the best possible sample quality, we thus opt for the bare minimum of $k = 2$ (i.e., an image-to-image model). Our key discovery is that, with $k = 2$ , we can still achieve approximate 3D consistency. Instead of using a sampler that is Markovian over frames, we leverage the iterative nature of diffusion sampling by varying the conditioning frame at each denoising step. ", + "bbox": [ + 173, + 491, + 825, + 617 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Stochastic Conditioning. We now detail our novel stochastic conditioning sampling procedure that allows us to generate 3D-consistent samples from a 3DiM. We start with a set of conditioning views $\\mathcal { X } = \\{ \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { k } \\}$ of a static scene, where typically $k = 1$ or is very small. We then generate a new frame by running a modified version of the standard denoising diffusion reverse process for steps $\\lambda _ { \\operatorname* { m i n } } = \\lambda _ { T } < \\lambda _ { T - 1 } < . . . < \\lambda _ { 0 } = \\lambda _ { \\operatorname* { m a x } }$ : ", + "bbox": [ + 174, + 623, + 823, + 694 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/54be09578c68e8ffee745836a6cd5d5ef08db9369edd69819a9dc58e1cf88046.jpg", + "text": "$$\n\\begin{array} { r c l } { { \\hat { \\pmb x } _ { k + 1 } } } & { { = } } & { { \\displaystyle \\frac { 1 } { \\sigma ( \\lambda _ { k } ) ^ { \\frac { 1 } { 2 } } } \\left( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } - \\sigma ( - \\lambda _ { t } ) ^ { \\frac { 1 } { 2 } } \\pmb \\epsilon _ { \\theta } ( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } , \\pmb x _ { i } ) \\right) } } \\\\ { { \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t - 1 } ) } } } & { { \\sim } } & { { q \\left( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t - 1 } ) } | \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } , \\hat { \\pmb x } _ { k + 1 } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 695, + 683, + 758 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where, crucially, $i \\sim \\operatorname { U n i f o r m } ( \\{ 1 , . . . , k \\} )$ is re-sampled at each denoising step. In other words, each individual denoising step is conditioned on a different random view from $\\mathcal { X }$ (the set that contains the input view(s) and the previously generated samples). Once we finish running this sampling chain and produce a final $\\mathbf { \\boldsymbol { x } } _ { k + 1 }$ , we simply add it to $\\mathcal { X }$ and repeat this procedure if we want to sample more frames. Given sufficient denoising steps, stochastic conditioning allows each generated frame to be guided by all previous frames. See Figure 3 for an illustration. In practice, we use 256 denoising steps, which we find to be sufficient to achieve both high sample quality and approximate 3D consistency. As usual in the literature, the first (noisiest sample) is just a Gaussian, i.e., $z _ { i } ^ { ( \\lambda _ { T } ) } \\sim$ $\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ , and at the last step $\\lambda _ { 0 }$ , we sample noiselessly. ", + "bbox": [ + 173, + 758, + 825, + 890 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can interpret stochastic conditioning as a na¨ıve approximation to true autoregressive sampling that works well in practice. True autoregressive sampling would require a score model of the form $\\nabla _ { z _ { k + 1 } ^ { ( \\lambda ) } } \\log q ( z _ { k + 1 } ^ { ( \\lambda ) } | \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { k } )$ , but this would strictly require multi-view training data, while we are ultimately interested in enabling novel view synthesis with as few as two training views per scene. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/ebe08a760122f6e8d1f7271054f456b935b0433fc3bd38c58fcdc2c918b6d1a0.jpg", + "image_caption": [ + "Figure 4: X-UNet Architecture – We modify the typical UNet architecture used by recent work on diffusion models to accomodate 3D novel view synthesis. We share the same UNet weights among the two input frames, the clean conditioning view and the denoising target view. We add cross attention layers to mix information between the input and output view, illustrated in yellow. " + ], + "image_footnote": [], + "bbox": [ + 209, + 66, + 799, + 286 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 405, + 823, + 441 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3 X-UNET ", + "text_level": 1, + "bbox": [ + 174, + 478, + 276, + 492 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The 3DiM model needs a neural network architecture that takes both the conditioning frame and the noisy frame as inputs. One natural way to do this is simply to concatenate the two images along the channel dimensions, and use the standard UNet architecture (Ronneberger et al., 2015; Ho et al., 2020). This “Concat-UNet” has found significant success in prior work of image-to-image diffusion models (Saharia et al., 2021b;a). However, in our early experiments, we found that the Concat-UNet yields very poor results – there were severe 3D inconsistencies and lack of alignment to the conditioning image. We hypothesize that, given limited model capacity and training data, it is difficult to learn complex, nonlinear image transformations that only rely on self-attention. We thus introduce our $X$ -UNet, whose core changes are (1) sharing parameters to process each of the two views, and (2) using cross attention between the two views. We find our X-UNet architecture to be very effective for 3D novel view synthesis. ", + "bbox": [ + 173, + 512, + 825, + 665 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now describe X-UNet in detail. We follow Ho et al. (2020); Song et al. (2020), and use the UNet (Ronneberger et al., 2015) with residual blocks and self-attention.We also take inspiration from Video Diffusion Models (Ho et al., 2022) by sharing weights over the two input frames for all the convolutional and self-attention layers, but with several key differences: ", + "bbox": [ + 174, + 672, + 825, + 728 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. We let each frame have its own noise level (recall that the inputs to a DDPM residual block are feature maps as well as a positional encoding for the noise level). We use a positional encoding of $\\lambda _ { \\mathrm { m a x } }$ for the clean frame. Ho et al. (2022) conversely denoise multiple frames simultaneously, each at the same noise level. 2. Alike Ho et al. (2020), we modulate each UNet block via FiLM (Dumoulin et al., 2018), but we use the sum of pose and noise-level positional encodings, as opposed to the noise-level embedding alone. Our pose encoding additionally differs in that they are of the same dimensionality as frames– they are camera rays, identical to those used by Sajjadi et al. (2021). 3. Instead of attending over “time” after each self-attention layer like Ho et al. (2022), which in our case would entail only two attention weights, we define a cross-attention layer and let each frame’s feature maps call this layer to query the other frame’s feature maps. ", + "bbox": [ + 173, + 736, + 825, + 888 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For more details on our proposed architecture, we refer the reader to the Supplementary Material (Sec.6). We also provide a comparison to the “Concat-UNet” architecture in Section 3.2. ", + "bbox": [ + 173, + 895, + 820, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/8d2bd63b699f0034354bdfb4e444b444cc03acff6739f74814c081e8aee2e157.jpg", + "image_caption": [ + "Figure 5: State-of-the-art comparisons – Example input and output views of a 3DiM trained on the SRN cars dataset at the 128x128 resolution, compared to existing geometry-aware methods. Results are best appreciated in the Supplementary Website (https://3d-diffusion.github.io/). " + ], + "image_footnote": [], + "bbox": [ + 191, + 97, + 812, + 227 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/38ff2d0dad50238e8aae50634ea92af7d4342d089dbbfcfbe060cc735b01f0a8.jpg", + "table_caption": [ + "Table 2: State-of-the-art comparisons – Results on the SRN ShapeNet benchmark comparing 3DiMs to prior work on novel view synthesis from a single image. $( ^ { * } )$ SSIM scores may not necessarily be computed with a Gaussian kernel following Wang et al. (2004) – in practice we observe this can lead to differences of up to 0.02. We report these marked numbers directly from prior work. " + ], + "table_footnote": [], + "table_body": "
SRN carsSRN chairs
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Geometry-aware
SRN22.250.8841.2122.890.8926.51
PixelNeRF23.170.8959.2423.720.9038.49
VisionNeRF22.880.9021.3124.480.9210.05
CodeNeRF23.80*0.9123.66*0.901
Geometry-free
LFN22.42*0.8922.26*0.90
ENR22.2622.831
3DiM (ours)21.010.578.9917.050.536.57
", + "bbox": [ + 178, + 299, + 821, + 484 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 580, + 326, + 595 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We benchmark 3DiMs on the SRN ShapeNet dataset (Sitzmann et al., 2019) to allow comparisons with prior work on novel view synthesis from a single image. This dataset consists of views and poses of car and chair ShapeNet (Chang et al., 2015) assets, rendered at the $1 2 8 \\mathrm { x } 1 2 8$ resolution. ", + "bbox": [ + 176, + 613, + 825, + 656 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We compare 3DiMs with Light Field Networks (LFN) (Sitzmann et al., 2021) and Equivariant Neural Rendering (ENR) (Dupont et al., 2020), two competitive geometry-free approaches, as well as geometry-aware approaches including Scene Representation Networks (SRN) (Sitzmann et al., 2019), PixelNeRF (Yu et al., 2021), CodeNeRF (Jang & Agapito, 2021) and the recent VisionNeRF (Lin et al., 2022). We include standard metrics in the literature: peak signal-to-noiseratio (PSNR), structural similarity (SSIM) (Wang et al., 2004), and the Frechet Inception Distance ´ (FID) (Heusel et al., 2017). To remain consistent with prior work, we evaluate over all scenes in the held-out dataset (each has 251 views), conditioning our models on a single view (view #64) to produce all other 250 views via a single call to our sampler outlined in Section 2.2, and then report the average PSNR and SSIM. Importantly, FID scores were computed with this same number of views, comparing the set of all generated views against the set of all the ground truth views. Because prior work did not include FID scores, we acquire the evaluation image outputs of SRN, PixelNeRF and VisionNeRF and compute the scores ourselves. We reproduce PSNR scores, and carefully note that some of the models do not follow Wang et al. (2004) on their SSIM computation (they use a uniform rather than Gaussian kernel), so we also recompute SSIM scores following (Wang et al., 2004). For more details, including hyperparameter choices, see Supplementary Material (Sec.7). ", + "bbox": [ + 174, + 662, + 825, + 886 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Out-of-distribution poses on SRN chairs. We note that the test split of the SRN ShapeNet chairs was unintentionally released with out-of-distribution poses compared to the train split: most test views are at a much larger distance to the object than those in the training dataset; we confirmed this via correspondence with the authors of Sitzmann et al. (2019). Because 3DiMs are geometry-free, we (unsurprisingly) observe that they do not perform well on this out-of-distribution evaluation task, as all the poses used at test-time are of scale never seen during training. However, simply merging, shuffling, and re-splitting the dataset completely fixes the issue. To maintain comparability with prior work, results on SRN chairs in Table 2 are those on the original dataset, and we denote the re-split dataset as “SRN chairs\\*” (note the \\*) in all subsequent tables to avoid confusion. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/7d1b0d0366e8db892efbd41eaf393921e833c5e8045fe86d2a6b9b3edbd37994.jpg", + "image_caption": [ + "Figure 6: 3DiM samples on OOD synthetic images – We qualitatively demonstrate the text-to-3D capabilities of 3DiM by synthethizing an input view with Imagen, a text-to-image model, and then create more views with 3DiM. " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 787, + 204 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 305, + 825, + 402 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "State-of-the-art comparisons. While 3DiM does not necessarily achieve superior reconstruction errors (PSNR and SSIM, see Table 2), qualitatively, we find that the fidelity of our generated videos can be strikingly better. We include an example in Figure 5. The use of diffusion models allows us to produce sharp samples, as opposed to regression models that are well-known to be prone to blurriness (Saharia et al., 2021b) in spite of high PSNR and SSIM scores. This is why we introduce and evaluate FID scores. In fact, we do not expect to achieve very low reconstruction errors due to the inherent ambiguity of novel view synthesis with a single image: constructing a consistent object with the given frame(s) is acceptable, but will still be punished by said reconstruction metrics if it differs from the ground truth views (even if achieving consistency). We additionally refer the reader to Section 3.1, where we include simple ablations demonstrating how worse models can improve the different standardized metrics in the literature. E.g., we show that with regression models, similarly to the baseline samples, PSNR and SSIM do not capture sharp modes well – they scores these models as much better despite the samples looking significantly more blurry (with PixelNeRF, which qualitatively seems the blurriest, achieving the best scores). ", + "bbox": [ + 173, + 412, + 825, + 607 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results on out-of-distribution images.. We additionally show qualitative results attempting to lift single images in the wild (without known poses) into 3D. To this end, we train a 471M parameter 3DiM on all of ShapeNet (except for 10 objects per class which we leave out for testing). This model has identical parameters to the 3DiM used for SRN cars. However, it differs in that we poses with relative camera extrinsics (i.e., the camera position and rotation corresponding to the noisy input frame is always the same) as we do not know the poses of images in the wild. Our qualitative results on single-image-to-3D generation include: ", + "bbox": [ + 173, + 617, + 825, + 715 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1. ShapeNet objects held out from the training data \n2. Images in the wild we directly took from the internet (with a white background, minimal shadow, and corresponding to any ShapeNet class) \n3. Images synthesized by Imagen (Saharia et al., 2022), a text-to-image diffusion model ", + "bbox": [ + 174, + 722, + 823, + 779 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In order to make Imagen synthesize objects with white backgrounds and minimal shadows at the $1 2 8 \\mathrm { x } 1 2 8$ resolution, in addition to natural language prompting, we inpaint a 5px white border at each denoising step when generating a $6 4 \\mathrm { x } 6 4$ image, and then upsample it using the second model on the Imagen cascade, i.e., a text-conditional super-resolution model. We find that this strategy is much more reliable than only relying on prompt engineering to create images with white backgrounds: this leads the text-to-image model to make the rest of background consistent with the fully white border, so the generated image results in an out-of-distribution object while maintaining an in-distribution background. We present example novel views given a ShapeNet test object in Figure 1, and given a synthetic image from Imagen in Figure 6. We include several video samples for all the cases described above in the Supplementary Website (https://3d-diffusion.github.io/). ", + "bbox": [ + 173, + 785, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/24fe033b344b3c9496f9ddbf4c0a3c7d978c06200af84173c3193235dc239aba.jpg", + "table_caption": [ + "SRN cars ", + "SRN chairs\\* " + ], + "table_footnote": [], + "table_body": "
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
3DiM21.010.578.9914.290.483.30
+ no stochastic conditioning23.820.591.8715.730.511.05
+regression22.550.8517.4516.850.7620.70
", + "bbox": [ + 181, + 116, + 820, + 178 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3: Ablation – study removing stochastic conditioning and diffusion altogether from 3DiM. \nResults are included for novel view synthesis from a single image on the SRN ShapeNet benchmark. \n$( ^ { * } )$ Re-split chairs dataset as detailed in Section 3 – not comparable to numbers in Table 2. ", + "bbox": [ + 174, + 188, + 823, + 231 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/d6dbef759ea682169ceebc121b6ec9861cd4dafcac248279f348c36b4c1adb3b.jpg", + "image_caption": [ + "Figure 7: Ablations – Example input and output views of 3DiM ablations on the SRN chairs dataset at the $1 2 8 \\mathrm { x } 1 2 8$ resolution. The second “Image-to-Image” column corresponds to removing our stochastic conditioning sampler from Section 2.2. The third “Concat-UNet” column corresponds to removing our proposed architecture in Section 2.3, resorting to a UNet that concatenates both images along the channel axis instead of weight sharing over frames. The fourth “Regression” column is one-step diffusion model trained from scratch. " + ], + "image_footnote": [], + "bbox": [ + 196, + 265, + 803, + 465 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We now present some ablation studies on 3DiM. First, we remove our proposed sampler and use a na¨ıve image-to-image model as discussed at the start of Section 2.2. We additionally remove the use of a diffusion process altogether, i.e., a regression model that generates samples via a single denoising step. Naturally, the regression models are trained separately, but we can still use an identical architecture, always feeding white noise for one frame, along with its corresponding noise level $\\lambda _ { \\operatorname* { m i n } }$ . Results are included in Table 3 and samples are included in Figure 7. ", + "bbox": [ + 174, + 666, + 825, + 750 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Unsurprisingly, we find that both of these components are crucial to achieve good results. The use of many diffusion steps allows sampling sharp images that achieve much better (lower) FID scores than both prior work and the regression models, where in both cases, reconstructions appear blurry. Notably, the regression models achieve better PSNR and SSIM scores than 3DiM despite the severe blurriness, suggesting that these standardized metrics fail to meaningfully capture sample quality for geometry-free models, at least when comparing them to geometry-aware or non-stochastic reconstruction approaches. Similarly, FID also has a failure mode: na¨ıve image-to-image sampling improves (decreases) FID scores significantly, but severely worsens shape and texture inconsistencies between sampled frames. These findings suggest that none of these standardized metrics are sufficient to effectively evaluate geometry-free models for view synthesis; nevertheless, we observe that said metrics do correlate well with sample quality across 3DiMs and can still be useful indicators for hyperparameter tuning despite their individual failures. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/faa7298931918687688ce393f7e14c449b7e2da082c8cabfb354a62bc379f0b0.jpg", + "table_caption": [ + "Table 5: Architecture comparison – study comparing the X-UNet and Concat-UNet neural architectures. Results are included for novel view synthesis from a single image on the SRN ShapeNet benchmark, with the re-split chairs as detailed in Section 3. $( ^ { * } )$ Re-split chairs dataset as detailed in Section 3 – these numbers are not comparable to those in Table 2. " + ], + "table_footnote": [], + "table_body": "
PSNR (↑) SSIM (↑)carschairs*
FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Concat-UNet17.210.5221.5412.360.445.15
X-UNet21.010.578.9914.290.483.30
", + "bbox": [ + 174, + 102, + 759, + 162 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.2 UNET ARCHITECTURE COMPARISONS ", + "text_level": 1, + "bbox": [ + 176, + 258, + 478, + 272 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In order to demonstrate the benefit of our proposed modifications, we additionally compare our proposed X-UNet architecture from Section 2.3 to the simpler UNet architecture following Saharia et al. (2021b;a) (which we simply call “Concat-UNet”). The Concat-UNet architecture, unlike ours, does not share weights across frames; instead, it simply concatenates the conditioning image to the noisy input image along the channel axis. To do this comparison, we train 3DiMs on the ConcatUNet architecture with the same number of hidden channels, and keep all other hyperparameters identical. Because our architecture has the additional cross-attention layer at the coarse-resolution blocks, our architecture has a slightly larger number of parameters than the Concat-UNet ${ \\sim } 4 7 1 \\mathrm { M }$ v.s. ${ \\sim } 4 2 1 \\mathrm { M }$ ). Results are included in Table 5 and Figure 7. ", + "bbox": [ + 174, + 284, + 825, + 410 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "While the Concat-UNet architecture is able to sample frames that resemble the data distribution, we find that 3DiMs trained with our proposed X-UNet architecture suffer much less from 3D inconsistency and alignment to the conditioning frame. Moreover, while the metrics should be taken with a grain of salt as previously discussed, we find that all the metrics significantly worsen with the Concat-UNet. We hypothesize that our X-UNet architecture better exploits symmetries between frames and poses due to our proposed weight-sharing mechanism, and that the cross-attention helps significantly to align with the content of the conditioning view. ", + "bbox": [ + 174, + 416, + 825, + 515 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 EVALUATING 3D CONSISTENCY IN GEOMETRY-FREE VIEW SYNTHESIS ", + "text_level": 1, + "bbox": [ + 173, + 537, + 785, + 551 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As we demonstrate in Sections 3.1 and 3.2, the standardized metrics in the literature have failure modes when specifically applied to geometry-free novel view synthesis models, e.g., their inability to successfully measure 3D consistency, and the possibility of improving them with worse models. Leveraging the fact that volumetric rendering of colored density fields are 3D-consistent by design, we thus propose an additional evaluation scheme called “3D consistency scoring”. Our metrics should satisfy the following desiderata: ", + "bbox": [ + 174, + 568, + 825, + 652 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "1. The metric must penalize outputs that are not 3D consistent. \n2. The metric must not penalize outputs that are 3D consistent but deviate from the ground truth. \n3. The metric must penalize outputs that do not align with the conditioning view(s). ", + "bbox": [ + 174, + 659, + 812, + 702 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In order to satisfy the second requirement, we cannot compare output renders to ground-truth views. Thus, one straightforward way to satisfy all desiderata is to sample many views from the geometryfree model given a single view, train a NeRF-like neural field (Mildenhall et al., 2020) on a fraction of these views, and compute a set of metrics that compare neural field renders on the remaining views. This way, if the geometry-free model outputs inconsistent views, the training of neural field will be hindered and classical image evaluation metrics should clearly reflect this. Additionally, to enforce the third requirement, we simply include the conditioning view(s) that were used to generate the rest as part of the training data. We report PSNR, SSIM and FID on the held-out views, although one could use other metrics under our proposed evaluation scheme. ", + "bbox": [ + 173, + 708, + 825, + 833 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We evaluate 3DiMs on the SRN benchmark, and for comparison, we additionally include metrics for models trained on (1) the real test views and (2) on image-to-image samples from the 3DiMs, i.e., without our proposed sampler like we reported in Section 3. To maintain comparability, we sample the same number of views from the different 3DiMs we evaluate in this section, all at the same poses and conditioned on the same single views. We leave out $10 \\%$ of the test views (25 out of 251) from neural field training, picking 25 random indices once and maintaining this choice of indices for all ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/06346efb1d8865985e9d156795692f5086aaf86f4d46d0c780996ae619b213bb.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Training view sourceSRN carsSRN chairs*
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Original data (3D consistent)28.210.9610.5724.870.9317.05
3DiM(~1.3B params)28.480.9629.5522.900.8658.61
3DiM(~471M params)28.530.9622.0918.840.7998.78
+ no stochastic conditioning25.780.9430.5117.610.75116.16
", + "bbox": [ + 176, + 101, + 821, + 184 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Table 6: 3D consistency scores – For neural fields trained with different view sources, we compare renders to held-out views from the same sources. ", + "bbox": [ + 173, + 193, + 823, + 222 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "subsequent evaluations. We also train models with more parameters $( \\sim 1 . 3 \\mathrm { B } )$ in order to investigate whether increasing model capacity can further improve 3D consistency. See Supplementary Material (Sec.7) for more details on the neural fields we chose and their hyperparameters. ", + "bbox": [ + 176, + 256, + 821, + 297 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We find that our proposed evaluation scheme clearly punishes 3D inconsistency as desired – the neural fields trained on image-to-image 3DiM samples have worse scores across all metrics compared to neural fields trained on 3DiM outputs sampled via our stochastic conditioning. This helps quantify the value of our proposed sampler, and also prevents the metrics from punishing reasonable outputs that are coherent with the input view(s) but do not match the target views – a desirable property due to the stochasticity of generative models. Moreover, we qualitatively find that the smaller model reported in the rest of the paper is comparable in quality to the ${ \\sim } 1 . 3 \\mathrm { B }$ parameter model on cars, though on chairs, we do observe a significant improvement on 3D consistency in our samples. Importantly, 3D consistency scoring agrees with our qualitative observations. ", + "bbox": [ + 174, + 304, + 825, + 429 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 CONCLUSION AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 176, + 458, + 488, + 473 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We propose 3DiM, a diffusion model for 3D novel view synthesis. Combining improvements in our X-UNet neural architecture (Section 2.3), with our novel stochastic conditioning sampling strategy that enables autoregressive generation over frames (Section 2.2), we show that from as few as a single image we can generate approximately 3D consistent views with very sharp sample quality. We additionally introduce 3D consistency scoring to evaluate the 3D consistency of geometry-free generative models by training neural fields on model output views, as their performance will be hindered increasingly with inconsistent training data (Section 4). We thus show, both quantitatively and visually, that 3DiMs with stochastic conditioning can achieve approximate 3D consistency and high sample quality simultaneously, and how classical metrics fail to capture both sharp modes and 3D inconsistency. ", + "bbox": [ + 174, + 492, + 825, + 632 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A noteworthy limitation of 3DiM is that, due to the geometry-free setup, it can only handle distributions of poses that it is exposed to durining training. In particular, further study is required where more variations to the poses are introduced. For example, varying focal lengths, sensor widths, making the camera not look exactly at the objects, and varying the distances from the camera to the objects more aggressively. Our use of stochastic conditioning might also exacerbate the need for many denoising steps. While we believe that finding effective architectures that allow conditioning on sets of images remains an important open problem, stochastic conditioning unlocks the ability to train on examples that only have two views and helps overcome memory-intensive neural architectures. Similar approaches using randomized, sparse conditioning are already being explored in follow-up work on video generation (Davtyan et al., 2022) in order to enable efficient training when modeling the joint distribution is computationally prohibitive. ", + "bbox": [ + 174, + 638, + 825, + 791 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We are most excited about the possibility of applying 3DiM, which can model entire datasets with a single model, to the largest 3D datasets from the real world – though more research is required to handle noisy poses (due to the need for pose estimation) and other challenges present in this context. Developing end-to-end approaches for high-quality generation that are 3D consistent by design like more recent work (Muller et al., 2022) and that also yield high-quality 3D meshes remain important ¨ research problems. Another potentially significant application of 3DiM is its use as a prior in order to achieve 3D consistent generation via approaches that operate exclusively at sampling time (Poole et al., 2022; Zhou & Tulsiani, 2022; Xu et al., 2022) in order to enable wide adoption of similar, disruptive technologies on the 3D design industry. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 103, + 287, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Jonathan T. Barron, Ben Mildenhall, Dor Verbin, Pratul P. Srinivasan, and Peter Hedman. Mip-nerf 360: Unbounded anti-aliased neural radiance fields. CVPR, 2022. 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", + "bbox": [ + 173, + 436, + 825, + 479 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Dor Verbin, Peter Hedman, Ben Mildenhall, Todd Zickler, Jonathan T. Barron, and Pratul P. Srinivasan. Ref-NeRF: Structured view-dependent appearance for neural radiance fields. CVPR, 2022. ", + "bbox": [ + 173, + 488, + 823, + 517 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 13(4):600– 612, 2004. ", + "bbox": [ + 173, + 525, + 823, + 569 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Dejia Xu, Yifan Jiang, Peihao Wang, Zhiwen Fan, Yi Wang, and Zhangyang Wang. Neurallift-360: Lifting an in-the-wild 2d photo to a 3d object with $3 6 0 ^ { \\circ }$ views. arXiv e-prints, pp. arXiv–2211, 2022. ", + "bbox": [ + 173, + 577, + 823, + 619 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Alex Yu, Vickie Ye, Matthew Tancik, and Angjoo Kanazawa. pixelnerf: Neural radiance fields from one or few images. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 4578–4587, 2021. ", + "bbox": [ + 174, + 628, + 823, + 672 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Tinghui Zhou, Richard Tucker, John Flynn, Graham Fyffe, and Noah Snavely. Stereo magnification: Learning view synthesis using multiplane images. arXiv preprint arXiv:1805.09817, 2018. ", + "bbox": [ + 171, + 680, + 823, + 710 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Zhizhuo Zhou and Shubham Tulsiani. Sparsefusion: Distilling view-conditioned diffusion for 3d reconstruction. arXiv preprint arXiv:2212.00792, 2022. ", + "bbox": [ + 169, + 718, + 823, + 747 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Novel View Synthesis with Diffusion Models ", + "text_level": 1, + "bbox": [ + 276, + 98, + 722, + 119 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Supplementary Material ", + "bbox": [ + 383, + 128, + 614, + 148 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "6 ARCHITECTURE DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 178, + 415, + 194 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In order to maximize the reproducibility of our results, we provide code in JAX (Bradbury et al., 2018) for our proposed X-UNet neural architecture from Section 2.3. Assuming we have an input batch with elements each containing ", + "bbox": [ + 174, + 208, + 825, + 252 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "{ \"z\", # noisy input image (HxWx3 tensor) \"x\", # conditioning view (HxWx3 tensor) \"logsnr\", # log signal-to-noise ratio of noisy image (scalar) \"t\", # camera positions (two 3d vectors) \"R\", # camera rotations (two 3x3 matrices) \"K\", # camera intrinsics (a.k.a. calibration matrix) (3x3 matrix) \n} ", + "bbox": [ + 173, + 262, + 859, + 375 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "our neural network module is as follows: ", + "bbox": [ + 174, + 386, + 442, + 400 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "from typing import Optional import flax.linen as nn import jax.numpy as jnp import numpy as onp import visu3d as v3d ", + "bbox": [ + 176, + 412, + 439, + 426 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 439, + 401, + 494 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "nonlinearity $=$ nn.swish ", + "bbox": [ + 174, + 523, + 401, + 536 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "def out_init_scale(): \"\"\"Zeros initializer.\"\"\" return nn.initializers.variance_scaling(0.0, 'fan_in', 'truncated_normal') ", + "bbox": [ + 173, + 564, + 918, + 606 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "def nearest_neighbor_upsample(h: jnp.ndarray): \"\"\"Nearest neighbor upsampling for multiple frames.\"\"\" B, F, H, W, ${ \\mathrm { ~ \\small ~ \\mathscr ~ { ~ C ~ } ~ } } = { \\mathrm { ~ \\small ~ h ~ } }$ .shape h $=$ h.reshape(B, F, H, 1, W, 1, C) h = jnp.broadcast_to(h, (B, F, H, 2, W, 2, C)) return h.reshape(B, F, H $\\star$ 2, W $\\star$ 2, C) ", + "bbox": [ + 173, + 633, + 725, + 717 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "def avgpool_downsample(h: jnp.ndarray, k: int $= 2$ ): \"\"\"Average pooling downsampling for multiple frames.\"\"\" return nn.avg_pool(h, (1, k, k), (1, k, k)) ", + "bbox": [ + 174, + 744, + 733, + 786 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "def posenc_ddpm(timesteps: jnp.ndarray, emb_ch: int, max_time: int $=$ 1000.0): \"\"\"Positional encodings for noise levels, following DDPM.\"\"\" # 1000 is the magic number from DDPM. With different timesteps, we # normalize by the number of steps but still multiply by 1000. timesteps \\*= 1000.0 / max_time half_dim $=$ emb_ch // 2 # 10000 is the magic number from transformers. emb $=$ onp.log(10000) / (half_dim - 1) ", + "bbox": [ + 173, + 813, + 926, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "emb $=$ jnp.exp(jnp.arange(half_dim, dtype $=$ timesteps.dtype) $\\star$ -emb) \nemb $=$ emb.reshape( $\\star$ ([1] $^ { \\star }$ (timesteps.ndim - 1)), emb.shape[-1]) \nemb $=$ timesteps[..., None] $\\star$ emb \nemb $=$ jnp.concatenate([jnp.sin(emb), jnp.cos(emb)], axis $= - 1$ ) \nreturn emb ", + "bbox": [ + 192, + 104, + 830, + 174 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "def posenc_nerf(x: jnp.ndarray, min_deg: int $\\qquad = \\quad 0$ , max_deg: int $= ~ \\perp 5$ ): \"\"\"Concatenate x and its positional encodings, following NeRF.\"\"\" ", + "bbox": [ + 173, + 202, + 844, + 229 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "if min_deg $= =$ max_deg: return x \nscales $=$ jnp.array([2\\*\\*i for i in range(min_deg, max_deg)]) \nxb $=$ jnp.reshape( (x[..., None, :] $^ { \\star }$ scales[:, None]), list(x.shape[:-1]) + [-1] \n) \nemb $=$ jnp.sin(jnp.concatenate([xb, xb $^ +$ onp.pi / 2.0], axis $: = - 1$ )) \nreturn jnp.concatenate([x, emb], axis $= - 1$ ) ", + "bbox": [ + 192, + 231, + 843, + 340 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "class GroupNorm(nn.Module): \"\"\"Group normalization, applied over frames.\"\"\" ", + "bbox": [ + 173, + 367, + 656, + 395 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "@nn.compact ", + "bbox": [ + 194, + 410, + 302, + 421 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "def __call__(self, h: jnp.ndarray):B, _, H, W, ${ \\mathrm { ~ \\small ~ \\mathscr ~ { ~ C ~ } ~ } } = { \\mathrm { ~ \\small ~ h ~ } }$ .shapeh $=$ nn.GroupNorm(num_groups $= 3 2$ )(h.reshape(B $\\star$ 2, 2, H, W, C))return h.reshape(B, 2, H, W, C)", + "bbox": [ + 192, + 424, + 810, + 478 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "class FiLM(nn.Module): \"\"\"Feature-wise linear modulation.\"\"\" ", + "bbox": [ + 173, + 506, + 558, + 534 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "features: int @nn.compact ", + "bbox": [ + 194, + 547, + 321, + 560 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 194, + 575, + 302, + 588 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "def __call__(self, h: jnp.ndarray, emb: jnp.ndarray): emb $=$ nn.Dense(2 \\* self.features)(nonlinearity(emb)) scale, shift $=$ jnp.split(emb, $^ 2$ , axis $= - 1$ ) return h $^ { \\star }$ (1.0 $^ +$ scale) $^ +$ shift ", + "bbox": [ + 199, + 590, + 722, + 643 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "class ResnetBlock(nn.Module): \"\"\"BigGAN-style residual block, applied over frames.\"\"\" ", + "bbox": [ + 173, + 672, + 735, + 699 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "features: Optional[int] $=$ None dropout: float $= ~ 0 . 0$ resample: Optional[str] $=$ None ", + "bbox": [ + 194, + 713, + 490, + 755 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "@nn.compact ", + "text_level": 1, + "bbox": [ + 194, + 770, + 302, + 781 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "def __call__(self, h_in: jnp.ndarray, emb: jnp.ndarray, \\*, train: bool): _, _, _, _, C $=$ h_in.shape features $=$ C if self.features is None else self.features ", + "bbox": [ + 194, + 784, + 898, + 824 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\mathrm { ~ \\textbar ~ { ~ h ~ } ~ } =$ nonlinearity(GroupNorm()(h_in)) \nif self.resample is not None: updown $= \\quad \\left\\{ \\begin{array} { r l r l } \\end{array} \\right.$ 'up': nearest_neighbor_upsample, 'down': avgpool_downsample, }[self.resample] ", + "bbox": [ + 210, + 838, + 584, + 921 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\mathrm { ~ \\textit ~ { ~ h ~ } ~ } =$ updown(h) h_in $=$ updown(h_in) ", + "bbox": [ + 230, + 104, + 418, + 132 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$\\mathrm { ~ \\textbar ~ { ~ h ~ } ~ } =$ nn.Conv(features, kernel_size $=$ (1, 3, 3), strides $=$ (1, 1, 1))(h) \n$\\mathrm { ~ \\textbar ~ { ~ h ~ } ~ } =$ FiLM(features $=$ features)(GroupNorm()(h), emb) \nh $=$ nonlinearity(h) \nh $=$ nn.Dropout(rate $=$ self.dropout)(h, deterministic $: =$ not train) \n$\\mathrm { ~ \\textbar ~ { ~ h ~ } ~ } =$ nn.Conv( features, kernel_size $=$ (1, 3, 3), strides $=$ (1, 1, 1), kernel_init $=$ out_init_scale(), ", + "bbox": [ + 210, + 145, + 856, + 273 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": ")(h) ", + "bbox": [ + 214, + 271, + 251, + 284 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "if C $! =$ features: h_in $=$ nn.Dense(features)(h_in) return (h $^ +$ h_in) / onp.sqrt(2) ", + "bbox": [ + 212, + 299, + 535, + 340 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "class AttnLayer(nn.Module): ", + "bbox": [ + 174, + 368, + 437, + 381 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "\"\"\"Attention layer usable for self and cross attention.\"\"\" ", + "bbox": [ + 187, + 382, + 764, + 395 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "attn_heads: int $\\qquad = \\quad 4$ ", + "bbox": [ + 194, + 410, + 380, + 422 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "@nn.compact ", + "text_level": 1, + "bbox": [ + 194, + 438, + 302, + 450 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "def _call__(self, \\*, q: jnp.ndarray, kv: jnp.ndarray): $\\mathrm { ~ C ~ } = \\mathrm { ~ q ~ }$ .shape[-1] head_dim $\\begin{array} { r l } { \\mathbf { \\Sigma } } & { { } = \\mathbf { \\Sigma } \\subset \\mathbf { \\Sigma } } \\end{array}$ // self.attn_heads q $=$ nn.DenseGeneral((self.attn_heads, head_dim))(q) k $=$ nn.DenseGeneral((self.attn_heads, head_dim))(kv) $\\begin{array} { r l } { \\mathsf { v } } & { { } = } \\end{array}$ nn.DenseGeneral((self.attn_heads, head_dim))(kv) return nn.dot_product_attention(q, k, v) ", + "bbox": [ + 192, + 452, + 728, + 547 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "class AttnBlock(nn.Module): \"\"\"Attention block with skip connection.\"\"\" ", + "bbox": [ + 176, + 575, + 617, + 603 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "attn_type: str attn_heads: int $\\qquad = \\quad 4$ ", + "bbox": [ + 194, + 617, + 380, + 643 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "@nn.compact ", + "text_level": 1, + "bbox": [ + 194, + 660, + 302, + 671 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "def __call__(self, h_in: jnp.ndarray): B, _, H, W, C $=$ h_in.shape ", + "bbox": [ + 196, + 672, + 565, + 700 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "h $=$ GroupNorm()(h_in) h0 = h[:, 0].reshape(B, H \\* W, C) h1 = h[:, 1].reshape(B, H $\\star$ W, C) attn_layer $=$ AttnLayer(attn_heads $=$ self.attn_heads) ", + "bbox": [ + 210, + 713, + 702, + 768 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "if self.attn_type $= =$ 'self': $\\mathrm { ~ \\textit ~ { ~ h ~ O ~ } ~ } =$ attn_layer( $\\mathtt { q } \\mathrm { = h 0 }$ , $\\mathtt { k v } { = } \\mathtt { h } 0$ ) $\\begin{array} { r l } { \\operatorname { h } 1 } & { { } = } \\end{array}$ attn_layer( $\\mathtt { q } \\mathrm { = h 1 }$ , $\\mathrm { k v } { = } \\mathrm { h } 1$ ) \nelif self.attn_type $= =$ 'cross': $\\mathrm { ~ \\textit ~ { ~ h ~ O ~ } ~ } =$ attn_layer( $\\mathtt { q } \\mathrm { = h 0 }$ , $\\mathrm { k v } { = } \\mathrm { h } 1$ ) h1 $=$ attn_layer( $\\mathtt { q } \\mathrm { = h 1 }$ , $\\mathtt { k v } { = } \\mathtt { h } 0$ ) \nelse: raise NotImplementedError(self.attn_type) ", + "bbox": [ + 210, + 784, + 645, + 895 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "h = jnp.stack([h0, h1], axis $^ { = 1 }$ ) ", + "bbox": [ + 212, + 907, + 514, + 921 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$\\mathrm { ~ \\textbar ~ { ~ h ~ } ~ } =$ h.reshape(B, 2, H, W, $^ { - 1 }$ ) \n$\\mathrm { ~ \\textbar ~ { ~ h ~ } ~ } =$ nn.DenseGeneral(C, axis $ \\mathfrak { s } = ( - 2$ , -1), kernel_init $=$ out_init_scale())(h) \nreturn (h $^ +$ h_in) / onp.sqrt(2) ", + "bbox": [ + 212, + 104, + 897, + 146 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "class XUNetBlock(nn.Module): \"\"\"X-UNet block.\"\"\" ", + "bbox": [ + 173, + 174, + 447, + 200 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "features: int use_attn: bool $=$ False attn_heads: int $\\qquad = \\quad 4$ dropout: float $= ~ 0 . 0$ ", + "bbox": [ + 192, + 215, + 411, + 270 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "@nn.compact ", + "text_level": 1, + "bbox": [ + 194, + 286, + 302, + 297 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "def call (self, x: jnp.ndarray, emb: jnp.ndarray, $\\star$ , train: bool): h $=$ ResnetBlock(features $=$ self.features, dropout $=$ self.dropout)( x, emb, train $=$ train ) ", + "bbox": [ + 194, + 299, + 867, + 356 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "if self.use_attn: h $=$ AttnBlock(attn_type $=$ 'self', attn_heads $=$ self.attn_heads)(h) h $=$ AttnBlock(attn_type $= ^ { \\parallel }$ cross', attn_heads $=$ self.attn_heads)(h) ", + "bbox": [ + 214, + 368, + 848, + 409 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "return h ", + "bbox": [ + 214, + 422, + 294, + 435 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "class ConditioningProcessor(nn.Module): \"\"\"Process conditioning inputs into embeddings.\"\"\" ", + "bbox": [ + 174, + 465, + 684, + 492 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "emb_ch: int num_resolutions: int use_pos_emb: bool $=$ True use_ref_pose_emb: bool $=$ True ", + "bbox": [ + 194, + 507, + 480, + 560 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "@nn.compact \ndef __call__(self, batch: dict[str, jnp.ndarray], cond_mask: jnp.ndarray): B, H, W, _ $=$ batch['x'].shape # Log signal-to-noise-ratio embedding. \nlogsnr $=$ jnp.clip(batch['logsnr'], -20.0, 20.0) \nlogsnr $= \\ 2 . 0 \\times$ jnp.arctan(jnp.exp(-logsnr / 2.0)) / onp.pi \nlogsnr_emb $=$ posenc_ddpm(logsnr, emb_ch ${ \\underline { { \\mathbf { \\Pi } } } } =$ self.emb_ch, max_time $^ { - 1 }$ .0) logsnr_emb $=$ nn.Dense(self.emb_ch)(logsnr_emb) \nlogsnr_emb $=$ nn.Dense(self.emb_ch)(nonlinearity(logsnr_emb)) ", + "bbox": [ + 194, + 575, + 918, + 617 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 210, + 631, + 857, + 713 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "# Pose embeddings. ", + "bbox": [ + 215, + 728, + 387, + 741 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "world_from_cam $=$ v3d.Transform( $\\mathrm { R = }$ batch['R'], t $=$ batch['t']) cam_spec $=$ v3d.PinholeCamera(resolution ${ } = { }$ (H, W), $\\mathrm { K } =$ batch['K']) rays $=$ v3d.Camera(spec $=$ cam_spec, world_from_cam $\\cdot ^ { = }$ world_from_cam).rays() ", + "bbox": [ + 214, + 742, + 897, + 782 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "pose_emb_pos $=$ posenc_nerf(rays.pos, min_deg $= 0$ , max_deg $= \\beth 5$ ) pose_emb_dir $=$ posenc_nerf(rays.dir, min_deg ${ \\bf \\bar { \\theta } } = 0$ , max_deg ${ } = 8 { }$ ) pose_emb $=$ jnp.concatenate([pose_emb_pos, pose_emb_dir], axi $S { = } { - } 1$ ) ", + "bbox": [ + 214, + 797, + 852, + 838 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "# Enable classifier-free guidance over poses. ", + "bbox": [ + 212, + 852, + 651, + 864 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "D $=$ pose_emb.shape[-1] \nassert cond_mask.shape $= =$ (B,) \ncond_mask $=$ cond_mask[:, None, None, None, None] \npose_emb $=$ jnp.where(cond_mask, pose_emb, jnp.zeros_like(pose_emb)) ", + "bbox": [ + 214, + 861, + 869, + 921 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "# Learnable position embeddings over (H, W) of frames (optional). ", + "bbox": [ + 207, + 118, + 846, + 132 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "if self.use_pos_emb: pos_emb $=$ self.param( 'pos_emb', nn.initializers.normal(stddev $\\ l = 1$ .0 / onp.sqrt(D)), (H, W, D), pose_emb.dtype, ) pose_emb $+ =$ pos_emb[None, None] ", + "bbox": [ + 215, + 133, + 751, + 243 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "# Binary embedding to let the model distinguish frames (optional) ", + "bbox": [ + 210, + 257, + 852, + 270 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "if self.use_ref_pose_emb: first_emb $=$ self.param( 'ref_pose_emb_first', nn.initializers.normal(stddev $^ { \\cdot = 1 }$ .0 / onp.sqrt(D)), (D,), pose_emb.dtype, )[None, None, None, None] other_emb $=$ self.param( 'ref_pose_emb_other', nn.initializers.normal(stddev $\\ l = 1$ .0 / onp.sqrt(D)), (D,), pose_emb.dtype, )[None, None, None, None] pose_emb $+ =$ jnp.concatenate([first_emb, other_emb], axis $^ { = 1 }$ ", + "bbox": [ + 220, + 272, + 800, + 492 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "# Downsample ray embeddings for each UNet resolution. \npose_embs $=$ [] \nfor i_level in range(self.num_resolutions): pose_embs.append( nn.Conv( features $=$ self.emb_ch, kernel_size $=$ (1, 3, 3), strides $=$ (1, $2 \\star \\star$ i_level, 2\\*\\*i_level), )(pose_emb) ) ", + "bbox": [ + 212, + 506, + 732, + 645 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "return logsnr_emb, pose_embs class XUNet(nn.Module): \"\"\"Our proposed XUNet architecture.\"\"\" ", + "bbox": [ + 215, + 659, + 488, + 671 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 700, + 568, + 727 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ch: int $= \\ 2 5 6$ \nch_mult: tuple[int] $=$ (1, 2, 2, 4) \nemb_ch: int $= ~ 1 0 2 4$ \nnum_res_blocks: int $= 3$ \nattn_resolutions: tuple[int] $=$ (8, 16, 32) \nattn_heads: int $\\qquad = \\quad 4$ \ndropout: float $= ~ 0 . 1$ \nuse_pos_emb: bool $=$ True \nuse_ref_pose_emb: bool $=$ True ", + "bbox": [ + 194, + 741, + 604, + 866 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "@nn.compact def __call__( self, ", + "bbox": [ + 192, + 880, + 325, + 920 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "batch: dict[str, jnp.ndarray], \\*, cond_mask: jnp.ndarray, train: bool, ): _, _, _, ${ \\mathrm { ~ \\small ~ \\mathscr ~ { ~ C ~ } ~ } } =$ batch['x'].shape num_resolutions $=$ len(self.ch_mult) logsnr_emb, pose_embs $=$ ConditioningProcessor( emb_ch $=$ self.emb_ch, num_resolutions $=$ num_resolutions, use_pos_emb $=$ self.use_pos_emb, use_ref_pose_emb $=$ self.use_ref_pose_emb, )(batch, cond_mask) del cond_mask ", + "bbox": [ + 194, + 104, + 669, + 311 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "h $=$ jnp.stack([batch['x'], batch $[ { \\mathrm { ~ ~ \\cdot ~ } } _ { Z } { \\mathrm { ~ ~ \\cdot ~ } } ] ]$ , axis $: = 1$ ) h $=$ nn.Conv(self.ch, kernel_size $=$ (1, 3, 3), strides $=$ (1, 1, 1))(h) ", + "bbox": [ + 207, + 327, + 846, + 354 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "# Downsampling. ", + "bbox": [ + 214, + 367, + 357, + 381 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "hs $=$ [h] \nfor i_level in self.ch_mult: emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[i_level] ", + "bbox": [ + 214, + 382, + 787, + 422 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "for in range(self.num_res_blocks): use_attn $=$ h.shape[2] in self.attn_resolutions h $=$ XUNetBlock( features $=$ self.ch $\\star$ self.ch_mult[i_level], dropout $=$ self.dropout, attn_heads $=$ self.attn_heads, use_attn ${ \\bf \\Phi } = { \\bf \\Phi }$ use_attn, )(h, emb, train $=$ train) hs.append(h) ", + "bbox": [ + 232, + 438, + 702, + 561 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "if i_level $\\downarrow =$ num_resolutions - 1: emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[i_level + 1] h $=$ ResnetBlock(dropout $=$ self.dropout, resample $= 1$ down')( h, emb, train ${ \\bf \\Phi } = { \\bf \\Phi }$ train ) hs.append(h) ", + "bbox": [ + 232, + 575, + 849, + 659 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "# Middle. ", + "text_level": 1, + "bbox": [ + 215, + 671, + 299, + 684 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[-1] \nuse_attn $=$ h.shape[2] in self.attn_resolutions \nh $=$ XUNetBlock( features $=$ self.ch $\\star$ self.ch_mult[i_level], dropout $=$ self.dropout, attn_heads $=$ self.attn_heads, use_attn $=$ use_attn, \n)(h, emb, train $=$ train) ", + "bbox": [ + 212, + 686, + 722, + 796 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$\\#$ Upsampling. ", + "bbox": [ + 214, + 810, + 339, + 824 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "for i_level in reversed(range(num_resolutions)): emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[i_level] for _ in range(self.num_res_blocks + 1): use_attn $=$ hs[-1].shape[2] in self.attn_resolutions h $=$ jnp.concatenate([h, hs.pop()], axis $= - 1$ ) h $=$ XUNetBlock( ", + "bbox": [ + 215, + 825, + 789, + 852 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 232, + 866, + 753, + 921 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "features $=$ self.ch $\\star$ self.ch_mult[i_level], dropout $=$ self.dropout, attn_heads $=$ self.attn_heads, use_attn ${ \\bf \\Phi } = { \\bf \\Phi }$ use_attn, )(h, emb, train $=$ train) ", + "bbox": [ + 251, + 104, + 692, + 174 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "if i_level $\\ : \\ 0$ : emb $=$ logsnr_emb[..., None, None, :] $^ +$ pose_embs[i_level] $\\mathrm { ~ \\textbar ~ { ~ h ~ } ~ } =$ ResnetBlock(dropout $=$ self.dropout, resample $= ^ { \\parallel }$ up')( h, emb, train ${ \\bf \\Phi } = { \\bf \\Phi }$ train ) ", + "bbox": [ + 232, + 188, + 810, + 257 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "# End. \nassert not hs \n$\\mathrm { ~ \\textbar ~ { ~ h ~ } ~ } =$ nonlinearity(GroupNorm()(h)) \nreturn nn.Conv( C, kernel_size $=$ (1, 3, 3), strides $; =$ (1, 1, 1), kernel_init $=$ out_init_scale(), \n)(h)[:, 1] ", + "bbox": [ + 212, + 271, + 535, + 396 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "7 HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 377, + 117 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We now detail hyperparameter choices across our experiments. These include choices specific to the neural architecture, the training procedure, and also choices only relevant during inference. ", + "bbox": [ + 173, + 135, + 823, + 162 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "7.1 NEURAL ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 181, + 390, + 195 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For our neural architecture, our main experiments use $_ { \\mathrm { C h } = 2 5 6 }$ ( ${ \\sim } 4 7 1 \\mathrm { M }$ params), and we also experiment with $\\cosh { = } 4 4 8$ $( \\sim 1 . 3 \\mathrm { B }$ params) in Section 4. One of our early findings that we kept throughout all experiments in the paper is that $\\mathtt { c h \\_ m u l t } = ( \\mathtt { l } , \\mathtt { \\xi } _ { 2 } , \\mathtt { \\xi } _ { 2 } , \\mathtt { \\xi } _ { 4 } )$ (i.e., setting the lowest UNet resolution to 8x8) was sufficient to achieve good sample quality, wheras most prior work includes UNet resolutions up to $4 \\mathbf { x } 4$ . We sweeped over the rest of the hyperparameters with the input and target views downsampled from their original $1 2 8 \\mathrm { x } 1 2 8$ resolution to the $3 2 \\mathrm { x } 3 2 $ , selecting values that led to the best qualitative improvements. These values are the default values present in the code we provide for the XUNet module in Section 6. We generally find that the best hyperparameter choices at low-resolution experiments transfer well when applied to the higher resolutions, and thus recommend this strategy for cheaper and more practical hyperparameter tuning. ", + "bbox": [ + 174, + 208, + 825, + 347 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For the ${ \\sim } 1 . 3 \\mathrm { B }$ parameter models, we tried increasing the number of parameters of our proposed UNet architecture through several different ways: increasing the number of blocks per resolution, the number of attention heads, the number of cross-attention layers per block, and the base number of hidden channels. Among all of these, we only found the last to provide noticeably better sample quality. We thus run 3D consistency scoring for models scaled this way, with channel sizes per UNet resolution of $4 4 8 \\times [ 1 , 2 , 2 , 4 ]$ instead of $2 5 6 \\times [ 1 , 2 , 2 , 4 ]$ (we could not fit $_ { \\mathrm { c h } = 5 1 2 }$ in TPUv4 memory without model parallelism). On cars, we find that the smaller model reported in the rest of the paper is comparable in quality to the ${ \\sim } 1 . 3 \\mathrm { B }$ parameter model, though on chairs, we do observe a significant improvement on 3D consistency, both visually and quantitatively (see Table 6). ", + "bbox": [ + 174, + 353, + 825, + 478 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "7.2 TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 497, + 285, + 511 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Following Equation 3 in the paper, our neural network attempts to model the noise $\\epsilon$ added to a real image in order to undo it given the noisy image. Other parameterizations are possible, e.g., predicting $_ { \\textbf { \\em x } }$ directly rather than predicting $\\epsilon$ , though we did not sweep over these choices. For our noise schedule, we use a cosine-shaped log signal to noise ratio that monotonically decreases from 20 to -20. It can be implemented in JAX as follows: ", + "bbox": [ + 174, + 523, + 825, + 593 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "def logsnr_schedule_cosine(t, $\\star$ , logsnr_min $= - 2 0$ ., logsnr_max $: = 2 0$ .): $\\textrm { b } =$ onp.arctan(onp.exp(-.5 \\* logsnr_max)) $\\begin{array} { r l } { \\exists } & { { } = } \\end{array}$ onp.arctan(onp.exp(-.5 $\\star$ logsnr_min)) - b return $^ { - 2 }$ . \\* jnp.log(jnp.tan( $ { \\sf a } \\mathrm { ~ ~ \\star ~ } \\sf t + \\mathrm { ~ ~ b ~ } )$ ) ", + "bbox": [ + 176, + 606, + 820, + 662 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Additionally: ", + "bbox": [ + 174, + 674, + 261, + 689 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• We use a learning rate with peak value 0.0001, using linear warmup for the first 10 million examples (where one batch has batch_size examples), following Karras et al. (2022). \n• We use a global batch size of 128. \n• We train each batch element as an unconditional example $10 \\%$ of the time to enable classifier-free guidance. This is done by overriding the conditioning frame to be at the maximum noise level. We note other options are possible (e.g., zeroing-out the conditioning frame), but we chose the option which is most compatible with our neural architecture. \n• We use the Adam optimizer (Kingma & Ba, 2014) with $\\beta _ { 1 } = 0 . 9$ and $\\beta _ { 2 } = 0 . 9 9$ . \n• We use EMA decay for the model parameters, with a half life of 500K examples (where one batch has batch_size examples) following Karras et al. (2022). ", + "bbox": [ + 173, + 695, + 826, + 835 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "7.3 SAMPLING ", + "text_level": 1, + "bbox": [ + 174, + 853, + 290, + 867 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "While Ho et al. (2020) note that their ancestral sampler can be used with any variances between $\\tilde { \\beta } _ { t }$ (variances of $q ( \\boldsymbol { z } _ { s } | \\boldsymbol { z } _ { t } , \\boldsymbol { x } ) )$ and $\\beta _ { t }$ (variances of the underlying SDE), we simply use the former as those correspond to their ancestral sampler. We do not observe this choice to have a significant effect sample quality. All our samples are generated with 256 denoising steps. As is standard in the literature, we also clip each predicted $_ { \\textbf { \\em x } }$ at each denoising step to the normalized range of the images $[ - 1 , 1 ]$ . ", + "bbox": [ + 176, + 880, + 823, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Classifier-free guidance. Our models use classifier-free guidance (Ho & Salimans, 2021), as we find that small guidance weights help encourage 3D consistency further. All our models were trained unconditionally with a probability of $10 \\%$ for each minibatch element. For unconditional examples, we zero out the (positionally encoded) pose and replace the clean frame with standard Gaussian noise (leveraging our weight-sharing architecture, see Section 2.3). We swept over various guidance weights, and simply picked those where 3D inconsistency was qualitatively least apparent on sampled videos. For SRN cars, we use a guidance weight of 3.0, while for SRN chairs we use a weight of 2.0. ", + "bbox": [ + 174, + 156, + 825, + 267 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "7.4 3D CONSISTENCY SCORING ", + "text_level": 1, + "bbox": [ + 176, + 284, + 405, + 297 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Note that traditional NeRFs (Mildenhall et al., 2020) can be 3D inconsistent as the model allows for view-dependent radiance. We thus employ a simpler and faster to train version based on instantNGP (Muller et al., 2022) without view dependent components, and with additional distortion and ¨ orientation loss terms for improved convergence (Barron et al., 2022; Verbin et al., 2022). Note that the specific implementation details of the neural field method chosen will affect the metrics considerably, so it is of utmost importance to apply the same method and hyperparameters for the neural fields to make 3D consistency scores comparable across models. ", + "bbox": [ + 174, + 309, + 825, + 407 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We design the neural fields as simple Multi-Layer Perceptrons (MLPs) of hidden size 64 and no skip connections. The density MLP has one hidden layer, while the MLP that predicts the color components uses 2 hidden layers. We use a learning rate of 0.01 linearly decayed to 0.001 for the first 100 steps. We use the Adam optimizer with weight decay set to 0.1 and clip gradients of norm exceeding 1.0. We only apply 1000 training steps for each scene and do not optimize camera poses. On each training step, we simply optimize over all available pixels, rather than sampling a random subset of pixels from all the training views. ", + "bbox": [ + 174, + 415, + 825, + 512 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To render the neural fields after training, we set near and far bounds to $\\begin{array} { r } { t _ { n } = \\frac { 3 r _ { \\mathrm { m i n } } } { 8 } , t _ { f } = \\frac { 3 r _ { \\mathrm { m a x } } } { 2 } . } \\end{array}$ where $r _ { \\operatorname* { m i n } } , r _ { \\operatorname* { m a x } }$ are the minimum and maximum distances from the camera positions to the origin (center of each object) in the corresponding dataset. All renders post-training are also performed with differentiable volume rendering. 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The core component of 3DiM is a pose-conditional image-to-image dif-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 263, + 469, + 276 + ], + "spans": [ + { + "bbox": [ + 141, + 263, + 469, + 276 + ], + "score": 1.0, + "content": "fusion model, which is trained to take a source view and its pose as inputs, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 275, + 470, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 470, + 286 + ], + "score": 1.0, + "content": "generates a novel view for a target pose as output. 3DiM can then generate mul-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 285, + 470, + 297 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 470, + 297 + ], + "score": 1.0, + "content": "tiple views that are approximately 3D consistent using a novel technique called", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 297, + 469, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 297, + 469, + 308 + ], + "score": 1.0, + "content": "stochastic conditioning. At inference time, the output views are generated autore-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 308, + 469, + 319 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 469, + 319 + ], + "score": 1.0, + "content": "gressively. When generating each novel view, one selects a random condition-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 318, + 469, + 330 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 469, + 330 + ], + "score": 1.0, + "content": "ing view from the set of previously generated views at each denoising step. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 329, + 469, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 329, + 469, + 342 + ], + "score": 1.0, + "content": "demonstrate that stochastic conditioning significantly improves 3D consistency", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 340, + 469, + 353 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 469, + 353 + ], + "score": 1.0, + "content": "compared to a na¨ıve sampler for an image-to-image diffusion model, which in-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 350, + 469, + 363 + ], + "spans": [ + { + "bbox": [ + 142, + 350, + 469, + 363 + ], + "score": 1.0, + "content": "volves conditioning on a single fixed view. We compare 3DiM to prior work on the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 361, + 470, + 374 + ], + "spans": [ + { + "bbox": [ + 141, + 361, + 470, + 374 + ], + "score": 1.0, + "content": "SRN ShapeNet dataset, demonstrating that 3DiM’s generated completions from a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 373, + 470, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 373, + 470, + 386 + ], + "score": 1.0, + "content": "single view achieve much higher fidelity, while being approximately 3D consis-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 384, + 469, + 396 + ], + "spans": [ + { + "bbox": [ + 141, + 384, + 469, + 396 + ], + "score": 1.0, + "content": "tent. We also introduce a new evaluation methodology, 3D consistency scoring, to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 395, + 470, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 395, + 470, + 408 + ], + "score": 1.0, + "content": "quantify the 3D consistency of a generated object by training a neural field on the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 407, + 469, + 419 + ], + "spans": [ + { + "bbox": [ + 141, + 407, + 469, + 419 + ], + "score": 1.0, + "content": "model’s output views. 3DiM is geometry free, does not rely on hyper-networks", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 417, + 469, + 429 + ], + "spans": [ + { + "bbox": [ + 141, + 417, + 469, + 429 + ], + "score": 1.0, + "content": "or test-time optimization for novel view synthesis, and allows a single model to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 428, + 304, + 440 + ], + "spans": [ + { + "bbox": [ + 142, + 428, + 304, + 440 + ], + "score": 1.0, + "content": "easily scale to a large number of scenes.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 23, + "bbox_fs": [ + 141, + 230, + 470, + 440 + ] + }, + { + "type": "image", + "bbox": [ + 106, + 457, + 504, + 535 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 457, + 504, + 535 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 457, + 504, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 504, + 535 + ], + "score": 0.965, + "type": "image", + "image_path": "476bebe18bd204ae2dce90f0a46a3eb7c0418fe2ef3bb88c505ec200a6c9160a.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 106, + 457, + 504, + 483.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 106, + 483.0, + 504, + 509.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 106, + 509.0, + 504, + 535.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 546, + 505, + 587 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "Figure 1: Given a single input image on the left, 3DiM performs novel view synthesis and generates the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 555, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 277, + 567 + ], + "score": 1.0, + "content": "four views on the right. We trained a single", + "type": "text" + }, + { + "bbox": [ + 277, + 557, + 308, + 566 + ], + "score": 0.67, + "content": "{ \\sim } 4 7 1 \\mathrm { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 555, + 504, + 567 + ], + "score": 1.0, + "content": "parameter 3DiM on all of ShapeNet (without class-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "conditioning) and sample frames with 256 steps (512 score function evaluations with classifier-free guidance).", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 575, + 402, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 402, + 588 + ], + "score": 1.0, + "content": "See the Supplementary Website (https://3d-diffusion.github.io/) for video outputs.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + } + ], + "index": 35.75 + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 205, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 208, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 208, + 623 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Diffusion Probabilistic Models (DPMs) (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "et al., 2020), also known as simply diffusion models, have recently emerged as a powerful family", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "of generative models, achieving state-of-the-art performance on audio and image synthesis (Chen", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "et al., 2020; Dhariwal & Nichol, 2021), while admitting better training stability over adversarial", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 504, + 690 + ], + "score": 1.0, + "content": "approaches (Goodfellow et al., 2014), as well as likelihood computation, which enables further ap-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 504, + 700 + ], + "score": 1.0, + "content": "plications such as compression and density estimation (Song et al., 2021; Kingma et al., 2021).", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Diffusion models have achieved impressive empirical results in a variety of image-to-image trans-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "lation tasks not limited to text-to-image, super-resolution, inpainting, colorization, uncropping, and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 721, + 502, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 502, + 732 + ], + "score": 1.0, + "content": "artifact removal (Song et al., 2020; Saharia et al., 2021a; Ramesh et al., 2022; Saharia et al., 2022).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 633, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "One particular image-to-image translation problem where diffusion models have not been investi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "gated is novel view synthesis, where, given a set of images of a given 3D scene, the task is to infer", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "how the scene looks from novel viewpoints. Before the recent emergence of Scene Representation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Networks (SRN) (Sitzmann et al., 2019) and Neural Radiance Fields (NeRF) (Mildenhall et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "2020), state-of-the-art approaches to novel view synthesis were typically built on generative mod-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "els (Sun et al., 2018) or more classical techniques on interpolation or disparity estimation (Park et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "2017; Zhou et al., 2018). Today, these models have been outperformed by NeRF-class models (Yu", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "et al., 2021; Niemeyer et al., 2021; Jang & Agapito, 2021), where 3D consistency is guaranteed by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "construction, as images are generated by volume rendering of a single underlying 3D representation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 247, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 247, + 194 + ], + "score": 1.0, + "content": "(a.k.a. “geometry-aware” models).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Still, these approaches feature different limitations. Heavily regularized NeRFs for novel view syn-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 221 + ], + "score": 1.0, + "content": "thesis with few images such as RegNeRF (Niemeyer et al., 2021) produce undesired artifacts when", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "given very few images, and fail to leverage knowledge from multiple scenes (recall NeRFs are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "trained on a single scene, i.e., one model per scene), and given one or very few views of a novel", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "scene, a reasonable model must extrapolate to complete the occluded parts of the scene. Pixel-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "NeRF (Yu et al., 2021) and VisionNeRF (Lin et al., 2022) address this by training NeRF-like models", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "conditioned on feature maps that encode the novel input view(s). However, these approaches are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "regressive rather than generative, and as a result, they cannot yield different plausible modes and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "are prone to blurriness. This type of failure has also been previously observed in regression-based", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "models (Saharia et al., 2021b). Other works such as CodeNeRF (Jang & Agapito, 2021) and LoL-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "NeRF (Rebain et al., 2021) instead employ test-time optimization to handle novel scenes, but still", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 236, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 236, + 332 + ], + "score": 1.0, + "content": "have issues with sample quality.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 335, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "In recent literature, geometry-free approaches (i.e., methods without explicit geometric inductive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 359 + ], + "score": 1.0, + "content": "biases like those introduced by volume rendering) such as Light Field Networks (LFN) (Sitzmann", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 504, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 504, + 370 + ], + "score": 1.0, + "content": "et al., 2021) and Scene Representation Transformers (SRT) (Sajjadi et al., 2021) have achieved re-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "sults competitive with 3D-aware methods in the “few-shot” setting, where the number of condition-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "ing views is limited (i.e., 1-10 images vs. dozens of images as in the usual NeRF setting). Similarly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "score": 1.0, + "content": "to our approach, EG3D (Chan et al., 2022) provides approximate 3D consistency by leveraging", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "generative models. EG3D employs a StyleGAN (Karras et al., 2019) with volumetric rendering,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "followed by generative super-resolution (the latter being responsible for the approximation). In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "comparison to this complex setup, we do not only provide a significantly simpler architecture, but", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "also a simpler hyper-parameter tuning experience compared GANs, which are well-known to be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 446, + 320, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 320, + 457 + ], + "score": 1.0, + "content": "notoriously difficult to tune (Mescheder et al., 2018).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Motivated by these observations and the success of diffusion models in image-to-image tasks, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "introduce 3D Diffusion Models (3DiMs). 3DiMs are image-to-image diffusion models trained on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "pairs of images of the same scene, where we assume the poses of the two images are known. 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We cover stochastic conditioning in more detail in Section 2.2 and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "provide an illustration in Figure 3. 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Today, these models have been outperformed by NeRF-class models (Yu", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "et al., 2021; Niemeyer et al., 2021; Jang & Agapito, 2021), where 3D consistency is guaranteed by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "construction, as images are generated by volume rendering of a single underlying 3D representation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 247, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 247, + 194 + ], + "score": 1.0, + "content": "(a.k.a. “geometry-aware” models).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 82, + 506, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Still, these approaches feature different limitations. Heavily regularized NeRFs for novel view syn-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 221 + ], + "score": 1.0, + "content": "thesis with few images such as RegNeRF (Niemeyer et al., 2021) produce undesired artifacts when", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "given very few images, and fail to leverage knowledge from multiple scenes (recall NeRFs are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "trained on a single scene, i.e., one model per scene), and given one or very few views of a novel", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "scene, a reasonable model must extrapolate to complete the occluded parts of the scene. Pixel-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "NeRF (Yu et al., 2021) and VisionNeRF (Lin et al., 2022) address this by training NeRF-like models", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "conditioned on feature maps that encode the novel input view(s). However, these approaches are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "regressive rather than generative, and as a result, they cannot yield different plausible modes and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "are prone to blurriness. This type of failure has also been previously observed in regression-based", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "models (Saharia et al., 2021b). Other works such as CodeNeRF (Jang & Agapito, 2021) and LoL-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "NeRF (Rebain et al., 2021) instead employ test-time optimization to handle novel scenes, but still", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 236, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 236, + 332 + ], + "score": 1.0, + "content": "have issues with sample quality.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 198, + 505, + 332 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 335, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "In recent literature, geometry-free approaches (i.e., methods without explicit geometric inductive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 359 + ], + "score": 1.0, + "content": "biases like those introduced by volume rendering) such as Light Field Networks (LFN) (Sitzmann", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 504, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 504, + 370 + ], + "score": 1.0, + "content": "et al., 2021) and Scene Representation Transformers (SRT) (Sajjadi et al., 2021) have achieved re-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "sults competitive with 3D-aware methods in the “few-shot” setting, where the number of condition-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "ing views is limited (i.e., 1-10 images vs. dozens of images as in the usual NeRF setting). Similarly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "score": 1.0, + "content": "to our approach, EG3D (Chan et al., 2022) provides approximate 3D consistency by leveraging", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "generative models. EG3D employs a StyleGAN (Karras et al., 2019) with volumetric rendering,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "followed by generative super-resolution (the latter being responsible for the approximation). In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "comparison to this complex setup, we do not only provide a significantly simpler architecture, but", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "also a simpler hyper-parameter tuning experience compared GANs, which are well-known to be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 446, + 320, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 320, + 457 + ], + "score": 1.0, + "content": "notoriously difficult to tune (Mescheder et al., 2018).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 336, + 506, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Motivated by these observations and the success of diffusion models in image-to-image tasks, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "introduce 3D Diffusion Models (3DiMs). 3DiMs are image-to-image diffusion models trained on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "pairs of images of the same scene, where we assume the poses of the two images are known. Draw-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "ing inspiration from Scene Representation Transformers (Sajjadi et al., 2021), 3DiMs are trained to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "build a conditional generative model of one view given another view and their poses. Our key dis-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "score": 1.0, + "content": "covery is that we can turn this image-to-image model into a model that can produce an entire set of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 528, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 540 + ], + "score": 1.0, + "content": "3D-consistent frames through autoregressive generation, which we enable with our novel stochastic", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 540, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 550 + ], + "score": 1.0, + "content": "conditioning sampling algorithm. 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How-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 333, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 347, + 348 + ], + "score": 1.0, + "content": "ever, we are interested in modeling distributions of the form", + "type": "text" + }, + { + "bbox": [ + 348, + 334, + 459, + 345 + ], + "score": 0.92, + "content": "q ( \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { m } | \\pmb { x } _ { m + 1 } , . . . , \\pmb { x } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 333, + 493, + 348 + ], + "score": 1.0, + "content": "without", + "type": "text" + }, + { + "bbox": [ + 493, + 334, + 501, + 344 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 333, + 506, + 348 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 358 + ], + "score": 1.0, + "content": "where views are no longer conditionally independent. 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We can apply the reparametrization trick (Kingma & Welling,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 586, + 330, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 330, + 600 + ], + "score": 1.0, + "content": "2013) and sample from these marginal distributions via", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 605, + 399, + 622 + ], + "lines": [ + { + "bbox": [ + 212, + 605, + 399, + 622 + ], + "spans": [ + { + "bbox": [ + 212, + 605, + 399, + 622 + ], + "score": 0.92, + "content": "z _ { k } ^ { ( \\lambda ) } = \\sigma ( \\lambda ) ^ { \\frac { 1 } { 2 } } x _ { k } + \\sigma ( - \\lambda ) ^ { \\frac { 1 } { 2 } } \\epsilon , \\quad \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )", + "type": "interline_equation", + "image_path": "bf27fb549e867a950564c2e1e0406dc4c6c7e7fdcec0e9b1f4f053610dc282cc.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 212, + 605, + 399, + 622 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "Then, given a pair of views, we learn to reverse this process in one of the two frames by minimizing", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 216, + 653 + ], + "score": 1.0, + "content": "the objective proposed by", + "type": "text" + }, + { + "bbox": [ + 216, + 640, + 230, + 650 + ], + "score": 0.31, + "content": "\\mathrm { H o }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "et al. (2020), which has been shown to yield much better sample", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 651, + 367, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 367, + 663 + ], + "score": 1.0, + "content": "quality than maximizing the true evidence lower bound (ELBO):", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 669, + 416, + 686 + ], + "lines": [ + { + "bbox": [ + 194, + 669, + 416, + 686 + ], + "spans": [ + { + "bbox": [ + 194, + 669, + 416, + 686 + ], + "score": 0.9, + "content": "L ( \\theta ) = \\mathbb { E } _ { q ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } ) } ~ \\mathbb { E } _ { \\lambda , \\epsilon } ~ \\| \\epsilon _ { \\theta } ( z _ { 2 } ^ { ( \\lambda ) } , \\pmb { x } _ { 1 } , \\lambda , \\pmb { p } _ { 1 } , \\pmb { p } _ { 2 } ) - \\epsilon \\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "0bff490b118041dae39794201061503813426cf1c1056d222495b14273a778fd.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 194, + 669, + 416, + 686 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 694, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 691, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 691, + 378, + 712 + ], + "score": 1.0, + "content": "where \u000fθ is a neural network whose task is to denoise the frame z(λ)2", + "type": "text" + }, + { + "bbox": [ + 381, + 693, + 506, + 711 + ], + "score": 1.0, + "content": "given a different (clean) frame", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 118, + 718 + ], + "score": 0.83, + "content": "\\scriptstyle { \\mathbf { { \\mathscr { x } } } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 707, + 141, + 720 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 142, + 708, + 149, + 717 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 707, + 505, + 720 + ], + "score": 1.0, + "content": "is the log signal-to-noise-ratio. To make our notation more legible, we slightly abuse", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 103, + 716, + 490, + 736 + ], + "spans": [ + { + "bbox": [ + 103, + 716, + 297, + 736 + ], + "score": 1.0, + "content": "notation and from now on we will simply write", + "type": "text" + }, + { + "bbox": [ + 297, + 718, + 348, + 733 + ], + "score": 0.94, + "content": "\\epsilon _ { \\theta } ( z _ { 2 } ^ { ( \\lambda ) } , x _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 716, + 490, + 736 + ], + "score": 1.0, + "content": ". 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How-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 333, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 347, + 348 + ], + "score": 1.0, + "content": "ever, we are interested in modeling distributions of the form", + "type": "text" + }, + { + "bbox": [ + 348, + 334, + 459, + 345 + ], + "score": 0.92, + "content": "q ( \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { m } | \\pmb { x } _ { m + 1 } , . . . , \\pmb { x } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 333, + 493, + 348 + ], + "score": 1.0, + "content": "without", + "type": "text" + }, + { + "bbox": [ + 493, + 334, + 501, + 344 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 333, + 506, + 348 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 358 + ], + "score": 1.0, + "content": "where views are no longer conditionally independent. A concrete example is the following: given", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 369 + ], + "score": 1.0, + "content": "the back of a person’s head, there are multiple plausible views for the front. An image-to-image", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "model sampling front views given only the back should indeed yield different outputs for each front", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 378, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 389 + ], + "score": 1.0, + "content": "view – with no guarantees that they will be consistent with each other – especially if it learns the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "data distribution perfectly. Similarly, given a single view of an object that appears small, there is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "ambiguity on the pose itself: is it small and close, or simply far away? Thus, given the inherent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "ambiguity in the few-shot setting, we need a sampling scheme where generated views can depend", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 420, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 104, + 420, + 505, + 435 + ], + "score": 1.0, + "content": "on each other in order to achieve 3D consistency. This contrasts NeRF approaches, where query", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 356, + 444 + ], + "score": 1.0, + "content": "rays are conditionally independent given a 3D representation", + "type": "text" + }, + { + "bbox": [ + 357, + 433, + 365, + 442 + ], + "score": 0.58, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 433, + 505, + 444 + ], + "score": 1.0, + "content": "– an even stronger condition than", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "imposing conditional independence among frames. Such approaches try to learn the richest possible", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 454, + 237, + 467 + ], + "score": 1.0, + "content": "representation for a single scene", + "type": "text" + }, + { + "bbox": [ + 237, + 455, + 245, + 464 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 454, + 506, + 467 + ], + "score": 1.0, + "content": ", while 3DiM avoids the difficulty of learning a generative model", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 174, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 120, + 477 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 465, + 128, + 475 + ], + "score": 0.82, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 465, + 174, + 477 + ], + "score": 1.0, + "content": "altogether.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 299, + 506, + 477 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 411, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 412, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 412, + 505 + ], + "score": 1.0, + "content": "2.1 IMAGE-TO-IMAGE DIFFUSION MODELS WITH POSE CONDITIONING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 206, + 525 + ], + "score": 1.0, + "content": "Given a data distribution", + "type": "text" + }, + { + "bbox": [ + 206, + 513, + 246, + 525 + ], + "score": 0.93, + "content": "q ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 511, + 437, + 525 + ], + "score": 1.0, + "content": "of pairs of views from a common scene at poses", + "type": "text" + }, + { + "bbox": [ + 438, + 513, + 501, + 525 + ], + "score": 0.9, + "content": "p _ { 1 } , p _ { 2 } \\in \\mathrm { S E } ( 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 511, + 505, + 525 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "we define an isotropic Gaussian process that adds increasing amounts of noise to data samples as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 535, + 393, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 204, + 547 + ], + "score": 1.0, + "content": "the signal-to-noise-ratio", + "type": "text" + }, + { + "bbox": [ + 205, + 536, + 212, + 545 + ], + "score": 0.83, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 535, + 393, + 547 + ], + "score": 1.0, + "content": "decreases, following Salimans & Ho (2022):", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 511, + 505, + 547 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 552, + 393, + 570 + ], + "lines": [ + { + "bbox": [ + 217, + 552, + 393, + 570 + ], + "spans": [ + { + "bbox": [ + 217, + 552, + 393, + 570 + ], + "score": 0.93, + "content": "q ( z _ { k } ^ { ( \\lambda ) } | \\pmb { x } _ { k } ) : = \\mathcal { N } ( z _ { k } ^ { ( \\lambda ) } ; \\sigma ( \\lambda ) ^ { \\frac { 1 } { 2 } } \\pmb { x } _ { k } , \\sigma ( - \\lambda ) \\mathbf { I } )", + "type": "interline_equation", + "image_path": "7306934ed6bb89ed18fa7ae3df5ed81b325d17bd060f8bf8b4d4d53472e3f064.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 217, + 552, + 393, + 570 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 576, + 503, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 133, + 590 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 576, + 151, + 588 + ], + "score": 0.91, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 574, + 505, + 590 + ], + "score": 1.0, + "content": "is the sigmoid function. We can apply the reparametrization trick (Kingma & Welling,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 586, + 330, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 330, + 600 + ], + "score": 1.0, + "content": "2013) and sample from these marginal distributions via", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 574, + 505, + 600 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 605, + 399, + 622 + ], + "lines": [ + { + "bbox": [ + 212, + 605, + 399, + 622 + ], + "spans": [ + { + "bbox": [ + 212, + 605, + 399, + 622 + ], + "score": 0.92, + "content": "z _ { k } ^ { ( \\lambda ) } = \\sigma ( \\lambda ) ^ { \\frac { 1 } { 2 } } x _ { k } + \\sigma ( - \\lambda ) ^ { \\frac { 1 } { 2 } } \\epsilon , \\quad \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )", + "type": "interline_equation", + "image_path": "bf27fb549e867a950564c2e1e0406dc4c6c7e7fdcec0e9b1f4f053610dc282cc.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 212, + 605, + 399, + 622 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "Then, given a pair of views, we learn to reverse this process in one of the two frames by minimizing", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 216, + 653 + ], + "score": 1.0, + "content": "the objective proposed by", + "type": "text" + }, + { + "bbox": [ + 216, + 640, + 230, + 650 + ], + "score": 0.31, + "content": "\\mathrm { H o }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "et al. (2020), which has been shown to yield much better sample", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 651, + 367, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 367, + 663 + ], + "score": 1.0, + "content": "quality than maximizing the true evidence lower bound (ELBO):", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 628, + 505, + 663 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 669, + 416, + 686 + ], + "lines": [ + { + "bbox": [ + 194, + 669, + 416, + 686 + ], + "spans": [ + { + "bbox": [ + 194, + 669, + 416, + 686 + ], + "score": 0.9, + "content": "L ( \\theta ) = \\mathbb { E } _ { q ( \\pmb { x } _ { 1 } , \\pmb { x } _ { 2 } ) } ~ \\mathbb { E } _ { \\lambda , \\epsilon } ~ \\| \\epsilon _ { \\theta } ( z _ { 2 } ^ { ( \\lambda ) } , \\pmb { x } _ { 1 } , \\lambda , \\pmb { p } _ { 1 } , \\pmb { p } _ { 2 } ) - \\epsilon \\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "0bff490b118041dae39794201061503813426cf1c1056d222495b14273a778fd.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 194, + 669, + 416, + 686 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 694, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 691, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 691, + 378, + 712 + ], + "score": 1.0, + "content": "where \u000fθ is a neural network whose task is to denoise the frame z(λ)2", + "type": "text" + }, + { + "bbox": [ + 381, + 693, + 506, + 711 + ], + "score": 1.0, + "content": "given a different (clean) frame", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 118, + 718 + ], + "score": 0.83, + "content": "\\scriptstyle { \\mathbf { { \\mathscr { x } } } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 707, + 141, + 720 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 142, + 708, + 149, + 717 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 707, + 505, + 720 + ], + "score": 1.0, + "content": "is the log signal-to-noise-ratio. To make our notation more legible, we slightly abuse", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 103, + 716, + 490, + 736 + ], + "spans": [ + { + "bbox": [ + 103, + 716, + 297, + 736 + ], + "score": 1.0, + "content": "notation and from now on we will simply write", + "type": "text" + }, + { + "bbox": [ + 297, + 718, + 348, + 733 + ], + "score": 0.94, + "content": "\\epsilon _ { \\theta } ( z _ { 2 } ^ { ( \\lambda ) } , x _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 716, + 490, + 736 + ], + "score": 1.0, + "content": ". We illustrate training in Figure 2.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 103, + 691, + 506, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 46, + 504, + 194 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 46, + 504, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 46, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 46, + 504, + 194 + ], + "score": 0.96, + "type": "image", + "image_path": "385024aab77ac1d8f5ad70135ef69214f2f8ac11db03cb3e9024e3b64386e72b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 46, + 504, + 95.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 95.33333333333334, + 504, + 144.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 144.66666666666669, + 504, + 194.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 221, + 505, + 299 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "Figure 3: Stochastic conditioning sampler – We illustrate our proposed inference procedure for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "3DiM, outlined in Section 2.2. There are two main components to our sampling procedure: (1) the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 402, + 257 + ], + "score": 1.0, + "content": "autoregressive generation of multiple frames (illustrated vertically as “step", + "type": "text" + }, + { + "bbox": [ + 402, + 244, + 412, + 254 + ], + "score": 0.46, + "content": "1 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 243, + 438, + 257 + ], + "score": 1.0, + "content": ", “step", + "type": "text" + }, + { + "bbox": [ + 438, + 244, + 449, + 254 + ], + "score": 0.57, + "content": "2 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 243, + 505, + 257 + ], + "score": 1.0, + "content": ", etc.), and (2)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 253, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 269 + ], + "score": 1.0, + "content": "the denoising process to generate each individual frame (illustrated horizontally). When generating", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 264, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 264, + 506, + 280 + ], + "score": 1.0, + "content": "a new frame, we select a previous frame as the conditioning frame randomly at each denoising step", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 275, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 290 + ], + "score": 1.0, + "content": "(illustrated with the dice). Note that this is not part of 3DiM training; also, we omit the pose inputs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 288, + 294, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 294, + 300 + ], + "score": 1.0, + "content": "in the diagram to avoid overloading the figure.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 318, + 350, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 350, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 350, + 330 + ], + "score": 1.0, + "content": "2.2 3D CONSISTENCY VIA STOCHASTIC CONDITIONING", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 105, + 338, + 504, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 504, + 351 + ], + "score": 1.0, + "content": "Motivation. We begin this section by motivating the need of our stochastic conditioning sampler.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 348, + 491, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 491, + 362 + ], + "score": 1.0, + "content": "In the ideal situation, we would model our 3D scene frames using the chain rule decomposition:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 362, + 351, + 388 + ], + "lines": [ + { + "bbox": [ + 260, + 362, + 351, + 388 + ], + "spans": [ + { + "bbox": [ + 260, + 362, + 351, + 388 + ], + "score": 0.96, + "content": "p ( \\pmb { x } ) = \\prod _ { i } p ( \\pmb { x } _ { i } | \\pmb { x } _ { < i } )", + "type": "interline_equation", + "image_path": "d6aac7b809a902d38b5c9bc3e2ae9e783c10c340ddb7a3460959fa028f9edb16.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 260, + 362, + 351, + 388 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 504, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 504, + 402 + ], + "score": 1.0, + "content": "This factorization is ideal, as it models the distribution exactly without making any conditional in-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "dependence assumptions. Each frame is generated autoregressively, conditioned on all the previous", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "frames. However, we found this solution to perform poorly. Due to memory limitations, we can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 364, + 435 + ], + "score": 1.0, + "content": "only condition on a limited number of frames in practice, (i.e., a", + "type": "text" + }, + { + "bbox": [ + 365, + 423, + 371, + 433 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "-Markovian model). We also find", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 339, + 447 + ], + "score": 1.0, + "content": "that, as we increase the maximum number of input frames", + "type": "text" + }, + { + "bbox": [ + 339, + 434, + 346, + 444 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 433, + 506, + 447 + ], + "score": 1.0, + "content": ", the worse the sample quality becomes.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 459, + 458 + ], + "score": 1.0, + "content": "In order to achieve the best possible sample quality, we thus opt for the bare minimum of", + "type": "text" + }, + { + "bbox": [ + 459, + 445, + 484, + 455 + ], + "score": 0.89, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 343, + 469 + ], + "score": 1.0, + "content": "an image-to-image model). Our key discovery is that, with", + "type": "text" + }, + { + "bbox": [ + 343, + 456, + 368, + 466 + ], + "score": 0.9, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 455, + 505, + 469 + ], + "score": 1.0, + "content": ", we can still achieve approximate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "3D consistency. Instead of using a sampler that is Markovian over frames, we leverage the iterative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 476, + 453, + 492 + ], + "spans": [ + { + "bbox": [ + 104, + 476, + 453, + 492 + ], + "score": 1.0, + "content": "nature of diffusion sampling by varying the conditioning frame at each denoising step.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 504, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "Stochastic Conditioning. We now detail our novel stochastic conditioning sampling procedure that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "allows us to generate 3D-consistent samples from a 3DiM. We start with a set of conditioning views", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 515, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 107, + 516, + 179, + 528 + ], + "score": 0.93, + "content": "\\mathcal { X } = \\{ \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 515, + 313, + 531 + ], + "score": 1.0, + "content": "of a static scene, where typically", + "type": "text" + }, + { + "bbox": [ + 314, + 516, + 339, + 527 + ], + "score": 0.89, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 515, + 506, + 531 + ], + "score": 1.0, + "content": "or is very small. We then generate a new", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 541 + ], + "score": 1.0, + "content": "frame by running a modified version of the standard denoising diffusion reverse process for steps", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 538, + 267, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 263, + 550 + ], + "score": 0.9, + "content": "\\lambda _ { \\operatorname* { m i n } } = \\lambda _ { T } < \\lambda _ { T - 1 } < . . . < \\lambda _ { 0 } = \\lambda _ { \\operatorname* { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 538, + 267, + 552 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 551, + 418, + 601 + ], + "lines": [ + { + "bbox": [ + 190, + 551, + 418, + 601 + ], + "spans": [ + { + "bbox": [ + 190, + 551, + 418, + 601 + ], + "score": 0.92, + "content": "\\begin{array} { r c l } { { \\hat { \\pmb x } _ { k + 1 } } } & { { = } } & { { \\displaystyle \\frac { 1 } { \\sigma ( \\lambda _ { k } ) ^ { \\frac { 1 } { 2 } } } \\left( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } - \\sigma ( - \\lambda _ { t } ) ^ { \\frac { 1 } { 2 } } \\pmb \\epsilon _ { \\theta } ( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } , \\pmb x _ { i } ) \\right) } } \\\\ { { \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t - 1 } ) } } } & { { \\sim } } & { { q \\left( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t - 1 } ) } | \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } , \\hat { \\pmb x } _ { k + 1 } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "54be09578c68e8ffee745836a6cd5d5ef08db9369edd69819a9dc58e1cf88046.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 190, + 551, + 418, + 567.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 190, + 567.6666666666666, + 418, + 584.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 190, + 584.3333333333333, + 418, + 600.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 174, + 615 + ], + "score": 1.0, + "content": "where, crucially,", + "type": "text" + }, + { + "bbox": [ + 175, + 602, + 272, + 614 + ], + "score": 0.83, + "content": "i \\sim \\operatorname { U n i f o r m } ( \\{ 1 , . . . , k \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "is re-sampled at each denoising step. In other words, each", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 396, + 626 + ], + "score": 1.0, + "content": "individual denoising step is conditioned on a different random view from", + "type": "text" + }, + { + "bbox": [ + 397, + 614, + 406, + 623 + ], + "score": 0.81, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "(the set that contains the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 625, + 504, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 504, + 636 + ], + "score": 1.0, + "content": "input view(s) and the previously generated samples). Once we finish running this sampling chain", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 188, + 648 + ], + "score": 1.0, + "content": "and produce a final", + "type": "text" + }, + { + "bbox": [ + 188, + 637, + 211, + 647 + ], + "score": 0.89, + "content": "\\mathbf { \\boldsymbol { x } } _ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 635, + 298, + 648 + ], + "score": 1.0, + "content": ", we simply add it to", + "type": "text" + }, + { + "bbox": [ + 299, + 636, + 308, + 645 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "and repeat this procedure if we want to sample", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 660 + ], + "score": 1.0, + "content": "more frames. Given sufficient denoising steps, stochastic conditioning allows each generated frame", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "to be guided by all previous frames. See Figure 3 for an illustration. In practice, we use 256", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "denoising steps, which we find to be sufficient to achieve both high sample quality and approximate", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 678, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 469, + 697 + ], + "score": 1.0, + "content": "3D consistency. As usual in the literature, the first (noisiest sample) is just a Gaussian, i.e.,", + "type": "text" + }, + { + "bbox": [ + 469, + 679, + 505, + 694 + ], + "score": 0.91, + "content": "z _ { i } ^ { ( \\lambda _ { T } ) } \\sim", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 692, + 327, + 705 + ], + "spans": [ + { + "bbox": [ + 107, + 693, + 139, + 705 + ], + "score": 0.92, + "content": "\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 692, + 219, + 705 + ], + "score": 1.0, + "content": ", and at the last step", + "type": "text" + }, + { + "bbox": [ + 219, + 693, + 231, + 704 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 692, + 327, + 705 + ], + "score": 1.0, + "content": ", we sample noiselessly.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We can interpret stochastic conditioning as a na¨ıve approximation to true autoregressive sampling", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "that works well in practice. True autoregressive sampling would require a score model of the form", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 46, + 504, + 194 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 46, + 504, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 46, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 46, + 504, + 194 + ], + "score": 0.96, + "type": "image", + "image_path": "385024aab77ac1d8f5ad70135ef69214f2f8ac11db03cb3e9024e3b64386e72b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 46, + 504, + 95.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 95.33333333333334, + 504, + 144.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 144.66666666666669, + 504, + 194.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 221, + 505, + 299 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "Figure 3: Stochastic conditioning sampler – We illustrate our proposed inference procedure for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "score": 1.0, + "content": "3DiM, outlined in Section 2.2. There are two main components to our sampling procedure: (1) the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 402, + 257 + ], + "score": 1.0, + "content": "autoregressive generation of multiple frames (illustrated vertically as “step", + "type": "text" + }, + { + "bbox": [ + 402, + 244, + 412, + 254 + ], + "score": 0.46, + "content": "1 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 243, + 438, + 257 + ], + "score": 1.0, + "content": ", “step", + "type": "text" + }, + { + "bbox": [ + 438, + 244, + 449, + 254 + ], + "score": 0.57, + "content": "2 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 243, + 505, + 257 + ], + "score": 1.0, + "content": ", etc.), and (2)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 253, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 269 + ], + "score": 1.0, + "content": "the denoising process to generate each individual frame (illustrated horizontally). When generating", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 264, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 264, + 506, + 280 + ], + "score": 1.0, + "content": "a new frame, we select a previous frame as the conditioning frame randomly at each denoising step", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 275, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 290 + ], + "score": 1.0, + "content": "(illustrated with the dice). Note that this is not part of 3DiM training; also, we omit the pose inputs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 288, + 294, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 294, + 300 + ], + "score": 1.0, + "content": "in the diagram to avoid overloading the figure.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 318, + 350, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 350, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 350, + 330 + ], + "score": 1.0, + "content": "2.2 3D CONSISTENCY VIA STOCHASTIC CONDITIONING", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 105, + 338, + 504, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 504, + 351 + ], + "score": 1.0, + "content": "Motivation. We begin this section by motivating the need of our stochastic conditioning sampler.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 348, + 491, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 491, + 362 + ], + "score": 1.0, + "content": "In the ideal situation, we would model our 3D scene frames using the chain rule decomposition:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 337, + 504, + 362 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 362, + 351, + 388 + ], + "lines": [ + { + "bbox": [ + 260, + 362, + 351, + 388 + ], + "spans": [ + { + "bbox": [ + 260, + 362, + 351, + 388 + ], + "score": 0.96, + "content": "p ( \\pmb { x } ) = \\prod _ { i } p ( \\pmb { x } _ { i } | \\pmb { x } _ { < i } )", + "type": "interline_equation", + "image_path": "d6aac7b809a902d38b5c9bc3e2ae9e783c10c340ddb7a3460959fa028f9edb16.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 260, + 362, + 351, + 388 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 504, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 504, + 402 + ], + "score": 1.0, + "content": "This factorization is ideal, as it models the distribution exactly without making any conditional in-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "dependence assumptions. Each frame is generated autoregressively, conditioned on all the previous", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "frames. However, we found this solution to perform poorly. Due to memory limitations, we can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 364, + 435 + ], + "score": 1.0, + "content": "only condition on a limited number of frames in practice, (i.e., a", + "type": "text" + }, + { + "bbox": [ + 365, + 423, + 371, + 433 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "-Markovian model). We also find", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 339, + 447 + ], + "score": 1.0, + "content": "that, as we increase the maximum number of input frames", + "type": "text" + }, + { + "bbox": [ + 339, + 434, + 346, + 444 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 433, + 506, + 447 + ], + "score": 1.0, + "content": ", the worse the sample quality becomes.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 444, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 459, + 458 + ], + "score": 1.0, + "content": "In order to achieve the best possible sample quality, we thus opt for the bare minimum of", + "type": "text" + }, + { + "bbox": [ + 459, + 445, + 484, + 455 + ], + "score": 0.89, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 444, + 505, + 458 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 343, + 469 + ], + "score": 1.0, + "content": "an image-to-image model). Our key discovery is that, with", + "type": "text" + }, + { + "bbox": [ + 343, + 456, + 368, + 466 + ], + "score": 0.9, + "content": "k = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 455, + 505, + 469 + ], + "score": 1.0, + "content": ", we can still achieve approximate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "3D consistency. Instead of using a sampler that is Markovian over frames, we leverage the iterative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 476, + 453, + 492 + ], + "spans": [ + { + "bbox": [ + 104, + 476, + 453, + 492 + ], + "score": 1.0, + "content": "nature of diffusion sampling by varying the conditioning frame at each denoising step.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 390, + 506, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 504, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "Stochastic Conditioning. We now detail our novel stochastic conditioning sampling procedure that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "allows us to generate 3D-consistent samples from a 3DiM. We start with a set of conditioning views", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 515, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 107, + 516, + 179, + 528 + ], + "score": 0.93, + "content": "\\mathcal { X } = \\{ \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 515, + 313, + 531 + ], + "score": 1.0, + "content": "of a static scene, where typically", + "type": "text" + }, + { + "bbox": [ + 314, + 516, + 339, + 527 + ], + "score": 0.89, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 515, + 506, + 531 + ], + "score": 1.0, + "content": "or is very small. We then generate a new", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 526, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 541 + ], + "score": 1.0, + "content": "frame by running a modified version of the standard denoising diffusion reverse process for steps", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 538, + 267, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 263, + 550 + ], + "score": 0.9, + "content": "\\lambda _ { \\operatorname* { m i n } } = \\lambda _ { T } < \\lambda _ { T - 1 } < . . . < \\lambda _ { 0 } = \\lambda _ { \\operatorname* { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 538, + 267, + 552 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 493, + 506, + 552 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 551, + 418, + 601 + ], + "lines": [ + { + "bbox": [ + 190, + 551, + 418, + 601 + ], + "spans": [ + { + "bbox": [ + 190, + 551, + 418, + 601 + ], + "score": 0.92, + "content": "\\begin{array} { r c l } { { \\hat { \\pmb x } _ { k + 1 } } } & { { = } } & { { \\displaystyle \\frac { 1 } { \\sigma ( \\lambda _ { k } ) ^ { \\frac { 1 } { 2 } } } \\left( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } - \\sigma ( - \\lambda _ { t } ) ^ { \\frac { 1 } { 2 } } \\pmb \\epsilon _ { \\theta } ( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } , \\pmb x _ { i } ) \\right) } } \\\\ { { \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t - 1 } ) } } } & { { \\sim } } & { { q \\left( \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t - 1 } ) } | \\pmb z _ { k + 1 } ^ { ( \\lambda _ { t } ) } , \\hat { \\pmb x } _ { k + 1 } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "54be09578c68e8ffee745836a6cd5d5ef08db9369edd69819a9dc58e1cf88046.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 190, + 551, + 418, + 567.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 190, + 567.6666666666666, + 418, + 584.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 190, + 584.3333333333333, + 418, + 600.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 174, + 615 + ], + "score": 1.0, + "content": "where, crucially,", + "type": "text" + }, + { + "bbox": [ + 175, + 602, + 272, + 614 + ], + "score": 0.83, + "content": "i \\sim \\operatorname { U n i f o r m } ( \\{ 1 , . . . , k \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "is re-sampled at each denoising step. In other words, each", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 396, + 626 + ], + "score": 1.0, + "content": "individual denoising step is conditioned on a different random view from", + "type": "text" + }, + { + "bbox": [ + 397, + 614, + 406, + 623 + ], + "score": 0.81, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "(the set that contains the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 625, + 504, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 504, + 636 + ], + "score": 1.0, + "content": "input view(s) and the previously generated samples). Once we finish running this sampling chain", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 188, + 648 + ], + "score": 1.0, + "content": "and produce a final", + "type": "text" + }, + { + "bbox": [ + 188, + 637, + 211, + 647 + ], + "score": 0.89, + "content": "\\mathbf { \\boldsymbol { x } } _ { k + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 635, + 298, + 648 + ], + "score": 1.0, + "content": ", we simply add it to", + "type": "text" + }, + { + "bbox": [ + 299, + 636, + 308, + 645 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "and repeat this procedure if we want to sample", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 660 + ], + "score": 1.0, + "content": "more frames. Given sufficient denoising steps, stochastic conditioning allows each generated frame", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "to be guided by all previous frames. See Figure 3 for an illustration. In practice, we use 256", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "denoising steps, which we find to be sufficient to achieve both high sample quality and approximate", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 678, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 469, + 697 + ], + "score": 1.0, + "content": "3D consistency. As usual in the literature, the first (noisiest sample) is just a Gaussian, i.e.,", + "type": "text" + }, + { + "bbox": [ + 469, + 679, + 505, + 694 + ], + "score": 0.91, + "content": "z _ { i } ^ { ( \\lambda _ { T } ) } \\sim", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 692, + 327, + 705 + ], + "spans": [ + { + "bbox": [ + 107, + 693, + 139, + 705 + ], + "score": 0.92, + "content": "\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 692, + 219, + 705 + ], + "score": 1.0, + "content": ", and at the last step", + "type": "text" + }, + { + "bbox": [ + 219, + 693, + 231, + 704 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 692, + 327, + 705 + ], + "score": 1.0, + "content": ", we sample noiselessly.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 601, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We can interpret stochastic conditioning as a na¨ıve approximation to true autoregressive sampling", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "that works well in practice. True autoregressive sampling would require a score model of the form", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 315, + 509, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 225, + 340 + ], + "score": 0.92, + "content": "\\nabla _ { z _ { k + 1 } ^ { ( \\lambda ) } } \\log q ( z _ { k + 1 } ^ { ( \\lambda ) } | \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { k } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 226, + 315, + 509, + 343 + ], + "score": 1.0, + "content": ", but this would strictly require multi-view training data, while we are", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 336, + 500, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 500, + 352 + ], + "score": 1.0, + "content": "ultimately interested in enabling novel view synthesis with as few as two training views per scene.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 40.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 128, + 53, + 489, + 227 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 53, + 489, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 53, + 489, + 227 + ], + "spans": [ + { + "bbox": [ + 128, + 53, + 489, + 227 + ], + "score": 0.972, + "type": "image", + "image_path": "ebe08a760122f6e8d1f7271054f456b935b0433fc3bd38c58fcdc2c918b6d1a0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 128, + 53, + 489, + 111.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 128, + 111.0, + 489, + 169.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 128, + 169.0, + 489, + 227.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 241, + 505, + 287 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Figure 4: X-UNet Architecture – We modify the typical UNet architecture used by recent work on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 250, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 267 + ], + "score": 1.0, + "content": "diffusion models to accomodate 3D novel view synthesis. We share the same UNet weights among", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "the two input frames, the clean conditioning view and the denoising target view. We add cross", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 276, + 473, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 473, + 288 + ], + "score": 1.0, + "content": "attention layers to mix information between the input and output view, illustrated in yellow.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 108, + 321, + 504, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 509, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 225, + 340 + ], + "score": 0.92, + "content": "\\nabla _ { z _ { k + 1 } ^ { ( \\lambda ) } } \\log q ( z _ { k + 1 } ^ { ( \\lambda ) } | \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 315, + 509, + 343 + ], + "score": 1.0, + "content": ", but this would strictly require multi-view training data, while we are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 336, + 500, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 500, + 352 + ], + "score": 1.0, + "content": "ultimately interested in enabling novel view synthesis with as few as two training views per scene.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 379, + 169, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 171, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 171, + 392 + ], + "score": 1.0, + "content": "2.3 X-UNET", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "The 3DiM model needs a neural network architecture that takes both the conditioning frame and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "the noisy frame as inputs. One natural way to do this is simply to concatenate the two images", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "along the channel dimensions, and use the standard UNet architecture (Ronneberger et al., 2015; Ho", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "score": 1.0, + "content": "et al., 2020). This “Concat-UNet” has found significant success in prior work of image-to-image", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "diffusion models (Saharia et al., 2021b;a). However, in our early experiments, we found that the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "Concat-UNet yields very poor results – there were severe 3D inconsistencies and lack of alignment", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "score": 1.0, + "content": "to the conditioning image. We hypothesize that, given limited model capacity and training data, it is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "difficult to learn complex, nonlinear image transformations that only rely on self-attention. We thus", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 163, + 507 + ], + "score": 1.0, + "content": "introduce our", + "type": "text" + }, + { + "bbox": [ + 164, + 495, + 172, + 504 + ], + "score": 0.33, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "-UNet, whose core changes are (1) sharing parameters to process each of the two", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "views, and (2) using cross attention between the two views. We find our X-UNet architecture to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 516, + 279, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 279, + 529 + ], + "score": 1.0, + "content": "very effective for 3D novel view synthesis.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "We now describe X-UNet in detail. We follow Ho et al. (2020); Song et al. (2020), and use the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "UNet (Ronneberger et al., 2015) with residual blocks and self-attention.We also take inspiration", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "from Video Diffusion Models (Ho et al., 2022) by sharing weights over the two input frames for all", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 565, + 410, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 410, + 578 + ], + "score": 1.0, + "content": "the convolutional and self-attention layers, but with several key differences:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "1. We let each frame have its own noise level (recall that the inputs to a DDPM residual block are", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 117, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 117, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "feature maps as well as a positional encoding for the noise level). We use a positional encoding", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 118, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 118, + 604, + 129, + 618 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 130, + 605, + 149, + 616 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "for the clean frame. Ho et al. (2022) conversely denoise multiple frames simultaneously,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 118, + 617, + 234, + 628 + ], + "spans": [ + { + "bbox": [ + 118, + 617, + 234, + 628 + ], + "score": 1.0, + "content": "each at the same noise level.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "2. Alike Ho et al. (2020), we modulate each UNet block via FiLM (Dumoulin et al., 2018), but we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 117, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 117, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "use the sum of pose and noise-level positional encodings, as opposed to the noise-level embed-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 118, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 118, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "ding alone. Our pose encoding additionally differs in that they are of the same dimensionality as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 118, + 660, + 429, + 673 + ], + "spans": [ + { + "bbox": [ + 118, + 660, + 429, + 673 + ], + "score": 1.0, + "content": "frames– they are camera rays, identical to those used by Sajjadi et al. (2021).", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "3. Instead of attending over “time” after each self-attention layer like Ho et al. (2022), which in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 118, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 118, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "our case would entail only two attention weights, we define a cross-attention layer and let each", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 118, + 693, + 422, + 705 + ], + "spans": [ + { + "bbox": [ + 118, + 693, + 422, + 705 + ], + "score": 1.0, + "content": "frame’s feature maps call this layer to query the other frame’s feature maps.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "For more details on our proposed architecture, we refer the reader to the Supplementary Mate-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 720, + 479, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 479, + 732 + ], + "score": 1.0, + "content": "rial (Sec.6). We also provide a comparison to the “Concat-UNet” architecture in Section 3.2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 128, + 53, + 489, + 227 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 53, + 489, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 53, + 489, + 227 + ], + "spans": [ + { + "bbox": [ + 128, + 53, + 489, + 227 + ], + "score": 0.972, + "type": "image", + "image_path": "ebe08a760122f6e8d1f7271054f456b935b0433fc3bd38c58fcdc2c918b6d1a0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 128, + 53, + 489, + 111.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 128, + 111.0, + 489, + 169.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 128, + 169.0, + 489, + 227.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 241, + 505, + 287 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Figure 4: X-UNet Architecture – We modify the typical UNet architecture used by recent work on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 250, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 267 + ], + "score": 1.0, + "content": "diffusion models to accomodate 3D novel view synthesis. We share the same UNet weights among", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "the two input frames, the clean conditioning view and the denoising target view. We add cross", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 276, + 473, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 473, + 288 + ], + "score": 1.0, + "content": "attention layers to mix information between the input and output view, illustrated in yellow.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 108, + 321, + 504, + 350 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 315, + 509, + 352 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 379, + 169, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 171, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 171, + 392 + ], + "score": 1.0, + "content": "2.3 X-UNET", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "The 3DiM model needs a neural network architecture that takes both the conditioning frame and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "the noisy frame as inputs. One natural way to do this is simply to concatenate the two images", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "along the channel dimensions, and use the standard UNet architecture (Ronneberger et al., 2015; Ho", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "score": 1.0, + "content": "et al., 2020). This “Concat-UNet” has found significant success in prior work of image-to-image", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "diffusion models (Saharia et al., 2021b;a). However, in our early experiments, we found that the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "Concat-UNet yields very poor results – there were severe 3D inconsistencies and lack of alignment", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "score": 1.0, + "content": "to the conditioning image. We hypothesize that, given limited model capacity and training data, it is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "difficult to learn complex, nonlinear image transformations that only rely on self-attention. We thus", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 163, + 507 + ], + "score": 1.0, + "content": "introduce our", + "type": "text" + }, + { + "bbox": [ + 164, + 495, + 172, + 504 + ], + "score": 0.33, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "-UNet, whose core changes are (1) sharing parameters to process each of the two", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "views, and (2) using cross attention between the two views. We find our X-UNet architecture to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 516, + 279, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 279, + 529 + ], + "score": 1.0, + "content": "very effective for 3D novel view synthesis.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 407, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "We now describe X-UNet in detail. We follow Ho et al. (2020); Song et al. (2020), and use the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "UNet (Ronneberger et al., 2015) with residual blocks and self-attention.We also take inspiration", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "from Video Diffusion Models (Ho et al., 2022) by sharing weights over the two input frames for all", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 565, + 410, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 410, + 578 + ], + "score": 1.0, + "content": "the convolutional and self-attention layers, but with several key differences:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 532, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "1. We let each frame have its own noise level (recall that the inputs to a DDPM residual block are", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 117, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 117, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "feature maps as well as a positional encoding for the noise level). We use a positional encoding", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 118, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 118, + 604, + 129, + 618 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 130, + 605, + 149, + 616 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "for the clean frame. Ho et al. (2022) conversely denoise multiple frames simultaneously,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 118, + 617, + 234, + 628 + ], + "spans": [ + { + "bbox": [ + 118, + 617, + 234, + 628 + ], + "score": 1.0, + "content": "each at the same noise level.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "2. Alike Ho et al. (2020), we modulate each UNet block via FiLM (Dumoulin et al., 2018), but we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 117, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 117, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "use the sum of pose and noise-level positional encodings, as opposed to the noise-level embed-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 118, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 118, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "ding alone. Our pose encoding additionally differs in that they are of the same dimensionality as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 118, + 660, + 429, + 673 + ], + "spans": [ + { + "bbox": [ + 118, + 660, + 429, + 673 + ], + "score": 1.0, + "content": "frames– they are camera rays, identical to those used by Sajjadi et al. 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Results", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 213, + 438, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 438, + 225 + ], + "score": 1.0, + "content": "are best appreciated in the Supplementary Website (https://3d-diffusion.github.io/).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 109, + 237, + 503, + 384 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 237, + 503, + 384 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 237, + 503, + 384 + ], + "spans": [ + { + "bbox": [ + 109, + 237, + 503, + 384 + ], + "score": 0.979, + "html": "
SRN carsSRN chairs
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Geometry-aware
SRN22.250.8841.2122.890.8926.51
PixelNeRF23.170.8959.2423.720.9038.49
VisionNeRF22.880.9021.3124.480.9210.05
CodeNeRF23.80*0.9123.66*0.901
Geometry-free
LFN22.42*0.8922.26*0.90
ENR22.2622.831
3DiM (ours)21.010.578.9917.050.536.57
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(2004) – in practice we observe", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "this can lead to differences of up to 0.02. We report these marked numbers directly from prior work.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 200, + 472 + ], + "lines": [ + { + "bbox": [ + 104, + 458, + 202, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 458, + 202, + 475 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "We benchmark 3DiMs on the SRN ShapeNet dataset (Sitzmann et al., 2019) to allow comparisons", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "with prior work on novel view synthesis from a single image. This dataset consists of views and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 509, + 491, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 409, + 521 + ], + "score": 1.0, + "content": "poses of car and chair ShapeNet (Chang et al., 2015) assets, rendered at the", + "type": "text" + }, + { + "bbox": [ + 409, + 509, + 446, + 519 + ], + "score": 0.63, + "content": "1 2 8 \\mathrm { x } 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 509, + 491, + 521 + ], + "score": 1.0, + "content": "resolution.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "We compare 3DiMs with Light Field Networks (LFN) (Sitzmann et al., 2021) and Equivariant", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "Neural Rendering (ENR) (Dupont et al., 2020), two competitive geometry-free approaches, as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 548, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 504, + 560 + ], + "score": 1.0, + "content": "well as geometry-aware approaches including Scene Representation Networks (SRN) (Sitzmann", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "et al., 2019), PixelNeRF (Yu et al., 2021), CodeNeRF (Jang & Agapito, 2021) and the recent Vi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "sionNeRF (Lin et al., 2022). 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Results", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 213, + 438, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 438, + 225 + ], + "score": 1.0, + "content": "are best appreciated in the Supplementary Website (https://3d-diffusion.github.io/).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 109, + 237, + 503, + 384 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 237, + 503, + 384 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 237, + 503, + 384 + ], + "spans": [ + { + "bbox": [ + 109, + 237, + 503, + 384 + ], + "score": 0.979, + "html": "
SRN carsSRN chairs
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Geometry-aware
SRN22.250.8841.2122.890.8926.51
PixelNeRF23.170.8959.2423.720.9038.49
VisionNeRF22.880.9021.3124.480.9210.05
CodeNeRF23.80*0.9123.66*0.901
Geometry-free
LFN22.42*0.8922.26*0.90
ENR22.2622.831
3DiM (ours)21.010.578.9917.050.536.57
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We report these marked numbers directly from prior work.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 200, + 472 + ], + "lines": [ + { + "bbox": [ + 104, + 458, + 202, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 458, + 202, + 475 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 486, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "We benchmark 3DiMs on the SRN ShapeNet dataset (Sitzmann et al., 2019) to allow comparisons", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "with prior work on novel view synthesis from a single image. This dataset consists of views and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 509, + 491, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 409, + 521 + ], + "score": 1.0, + "content": "poses of car and chair ShapeNet (Chang et al., 2015) assets, rendered at the", + "type": "text" + }, + { + "bbox": [ + 409, + 509, + 446, + 519 + ], + "score": 0.63, + "content": "1 2 8 \\mathrm { x } 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 509, + 491, + 521 + ], + "score": 1.0, + "content": "resolution.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 486, + 505, + 521 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "We compare 3DiMs with Light Field Networks (LFN) (Sitzmann et al., 2021) and Equivariant", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "Neural Rendering (ENR) (Dupont et al., 2020), two competitive geometry-free approaches, as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 548, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 504, + 560 + ], + "score": 1.0, + "content": "well as geometry-aware approaches including Scene Representation Networks (SRN) (Sitzmann", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "et al., 2019), PixelNeRF (Yu et al., 2021), CodeNeRF (Jang & Agapito, 2021) and the recent Vi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "sionNeRF (Lin et al., 2022). We include standard metrics in the literature: peak signal-to-noise-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "ratio (PSNR), structural similarity (SSIM) (Wang et al., 2004), and the Frechet Inception Distance ´", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "(FID) (Heusel et al., 2017). To remain consistent with prior work, we evaluate over all scenes in the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "held-out dataset (each has 251 views), conditioning our models on a single view (view #64) to pro-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "duce all other 250 views via a single call to our sampler outlined in Section 2.2, and then report the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "average PSNR and SSIM. 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We reproduce PSNR scores, and carefully note that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "some of the models do not follow Wang et al. (2004) on their SSIM computation (they use a uniform", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "rather than Gaussian kernel), so we also recompute SSIM scores following (Wang et al., 2004). For", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 690, + 447, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 447, + 703 + ], + "score": 1.0, + "content": "more details, including hyperparameter choices, see Supplementary Material (Sec.7).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 525, + 506, + 703 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Out-of-distribution poses on SRN chairs. We note that the test split of the SRN ShapeNet chairs", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "was unintentionally released with out-of-distribution poses compared to the train split: most test", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "views are at a much larger distance to the object than those in the training dataset; we confirmed this", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "via correspondence with the authors of Sitzmann et al. (2019). Because 3DiMs are geometry-free,", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "score": 1.0, + "content": "we (unsurprisingly) observe that they do not perform well on this out-of-distribution evaluation task,", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 274, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 274, + 506, + 289 + ], + "score": 1.0, + "content": "as all the poses used at test-time are of scale never seen during training. However, simply merging,", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "shuffling, and re-splitting the dataset completely fixes the issue. To maintain comparability with", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "prior work, results on SRN chairs in Table 2 are those on the original dataset, and we denote the", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 308, + 462, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 462, + 321 + ], + "score": 1.0, + "content": "re-split dataset as “SRN chairs*” (note the *) in all subsequent tables to avoid confusion.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 79, + 482, + 162 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 79, + 482, + 162 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 79, + 482, + 162 + ], + "spans": [ + { + "bbox": [ + 108, + 79, + 482, + 162 + ], + "score": 0.967, + "type": "image", + "image_path": "7d1b0d0366e8db892efbd41eaf393921e833c5e8045fe86d2a6b9b3edbd37994.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 79, + 482, + 106.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 106.66666666666667, + 482, + 134.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 134.33333333333334, + 482, + 162.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 172, + 505, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "Figure 6: 3DiM samples on OOD synthetic images – We qualitatively demonstrate the text-to-3D", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 195 + ], + "score": 1.0, + "content": "capabilities of 3DiM by synthethizing an input view with Imagen, a text-to-image model, and then", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 195, + 230, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 230, + 205 + ], + "score": 1.0, + "content": "create more views with 3DiM.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "views are at a much larger distance to the object than those in the training dataset; we confirmed this", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "via correspondence with the authors of Sitzmann et al. (2019). Because 3DiMs are geometry-free,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "score": 1.0, + "content": "we (unsurprisingly) observe that they do not perform well on this out-of-distribution evaluation task,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 274, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 274, + 506, + 289 + ], + "score": 1.0, + "content": "as all the poses used at test-time are of scale never seen during training. However, simply merging,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "shuffling, and re-splitting the dataset completely fixes the issue. To maintain comparability with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "prior work, results on SRN chairs in Table 2 are those on the original dataset, and we denote the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 308, + 462, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 462, + 321 + ], + "score": 1.0, + "content": "re-split dataset as “SRN chairs*” (note the *) in all subsequent tables to avoid confusion.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "State-of-the-art comparisons. While 3DiM does not necessarily achieve superior reconstruction", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "errors (PSNR and SSIM, see Table 2), qualitatively, we find that the fidelity of our generated videos", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "can be strikingly better. We include an example in Figure 5. The use of diffusion models allows", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 373 + ], + "score": 1.0, + "content": "us to produce sharp samples, as opposed to regression models that are well-known to be prone to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "blurriness (Saharia et al., 2021b) in spite of high PSNR and SSIM scores. This is why we introduce", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "and evaluate FID scores. In fact, we do not expect to achieve very low reconstruction errors due to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "score": 1.0, + "content": "the inherent ambiguity of novel view synthesis with a single image: constructing a consistent object", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "with the given frame(s) is acceptable, but will still be punished by said reconstruction metrics if it", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "differs from the ground truth views (even if achieving consistency). We additionally refer the reader", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "to Section 3.1, where we include simple ablations demonstrating how worse models can improve the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 435, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 451 + ], + "score": 1.0, + "content": "different standardized metrics in the literature. E.g., we show that with regression models, similarly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "to the baseline samples, PSNR and SSIM do not capture sharp modes well – they scores these", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "models as much better despite the samples looking significantly more blurry (with PixelNeRF, which", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 470, + 345, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 345, + 482 + ], + "score": 1.0, + "content": "qualitatively seems the blurriest, achieving the best scores).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Results on out-of-distribution images.. We additionally show qualitative results attempting to lift", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "single images in the wild (without known poses) into 3D. To this end, we train a 471M parameter", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "3DiM on all of ShapeNet (except for 10 objects per class which we leave out for testing). This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "model has identical parameters to the 3DiM used for SRN cars. However, it differs in that we poses", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "score": 1.0, + "content": "with relative camera extrinsics (i.e., the camera position and rotation corresponding to the noisy", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "input frame is always the same) as we do not know the poses of images in the wild. Our qualitative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 556, + 307, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 307, + 567 + ], + "score": 1.0, + "content": "results on single-image-to-3D generation include:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 504, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 315, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 315, + 584 + ], + "score": 1.0, + "content": "1. ShapeNet objects held out from the training data", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "2. Images in the wild we directly took from the internet (with a white background, minimal shadow,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 118, + 594, + 289, + 606 + ], + "spans": [ + { + "bbox": [ + 118, + 594, + 289, + 606 + ], + "score": 1.0, + "content": "and corresponding to any ShapeNet class)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 461, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 461, + 617 + ], + "score": 1.0, + "content": "3. Images synthesized by Imagen (Saharia et al., 2022), a text-to-image diffusion model", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "In order to make Imagen synthesize objects with white backgrounds and minimal shadows at the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 633, + 143, + 644 + ], + "score": 0.52, + "content": "1 2 8 \\mathrm { x } 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "resolution, in addition to natural language prompting, we inpaint a 5px white border at each", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 241, + 656 + ], + "score": 1.0, + "content": "denoising step when generating a", + "type": "text" + }, + { + "bbox": [ + 241, + 644, + 267, + 654 + ], + "score": 0.75, + "content": "6 4 \\mathrm { x } 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "image, and then upsample it using the second model on the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Imagen cascade, i.e., a text-conditional super-resolution model. We find that this strategy is much", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "more reliable than only relying on prompt engineering to create images with white backgrounds: this", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "leads the text-to-image model to make the rest of background consistent with the fully white border,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "so the generated image results in an out-of-distribution object while maintaining an in-distribution", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "background. We present example novel views given a ShapeNet test object in Figure 1, and given", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "a synthetic image from Imagen in Figure 6. We include several video samples for all the cases", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 423, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 423, + 732 + ], + "score": 1.0, + "content": "described above in the Supplementary Website (https://3d-diffusion.github.io/).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 79, + 482, + 162 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 79, + 482, + 162 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 79, + 482, + 162 + ], + "spans": [ + { + "bbox": [ + 108, + 79, + 482, + 162 + ], + "score": 0.967, + "type": "image", + "image_path": "7d1b0d0366e8db892efbd41eaf393921e833c5e8045fe86d2a6b9b3edbd37994.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 79, + 482, + 106.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 106.66666666666667, + 482, + 134.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 134.33333333333334, + 482, + 162.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 172, + 505, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "Figure 6: 3DiM samples on OOD synthetic images – We qualitatively demonstrate the text-to-3D", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 195 + ], + "score": 1.0, + "content": "capabilities of 3DiM by synthethizing an input view with Imagen, a text-to-image model, and then", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 195, + 230, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 230, + 205 + ], + "score": 1.0, + "content": "create more views with 3DiM.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 319 + ], + "lines": [], + "index": 9, + "bbox_fs": [ + 104, + 243, + 506, + 321 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "State-of-the-art comparisons. While 3DiM does not necessarily achieve superior reconstruction", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "errors (PSNR and SSIM, see Table 2), qualitatively, we find that the fidelity of our generated videos", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "can be strikingly better. We include an example in Figure 5. The use of diffusion models allows", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 373 + ], + "score": 1.0, + "content": "us to produce sharp samples, as opposed to regression models that are well-known to be prone to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "blurriness (Saharia et al., 2021b) in spite of high PSNR and SSIM scores. This is why we introduce", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "and evaluate FID scores. In fact, we do not expect to achieve very low reconstruction errors due to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "score": 1.0, + "content": "the inherent ambiguity of novel view synthesis with a single image: constructing a consistent object", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "with the given frame(s) is acceptable, but will still be punished by said reconstruction metrics if it", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "differs from the ground truth views (even if achieving consistency). We additionally refer the reader", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "to Section 3.1, where we include simple ablations demonstrating how worse models can improve the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 435, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 451 + ], + "score": 1.0, + "content": "different standardized metrics in the literature. E.g., we show that with regression models, similarly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "to the baseline samples, PSNR and SSIM do not capture sharp modes well – they scores these", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "models as much better despite the samples looking significantly more blurry (with PixelNeRF, which", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 470, + 345, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 345, + 482 + ], + "score": 1.0, + "content": "qualitatively seems the blurriest, achieving the best scores).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 327, + 506, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Results on out-of-distribution images.. We additionally show qualitative results attempting to lift", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "single images in the wild (without known poses) into 3D. To this end, we train a 471M parameter", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "3DiM on all of ShapeNet (except for 10 objects per class which we leave out for testing). This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "model has identical parameters to the 3DiM used for SRN cars. However, it differs in that we poses", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "score": 1.0, + "content": "with relative camera extrinsics (i.e., the camera position and rotation corresponding to the noisy", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "input frame is always the same) as we do not know the poses of images in the wild. Our qualitative", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 556, + 307, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 307, + 567 + ], + "score": 1.0, + "content": "results on single-image-to-3D generation include:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 489, + 505, + 567 + ] + }, + { + "type": "list", + "bbox": [ + 107, + 572, + 504, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 315, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 315, + 584 + ], + "score": 1.0, + "content": "1. ShapeNet objects held out from the training data", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "2. Images in the wild we directly took from the internet (with a white background, minimal shadow,", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 594, + 289, + 606 + ], + "spans": [ + { + "bbox": [ + 118, + 594, + 289, + 606 + ], + "score": 1.0, + "content": "and corresponding to any ShapeNet class)", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 604, + 461, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 461, + 617 + ], + "score": 1.0, + "content": "3. Images synthesized by Imagen (Saharia et al., 2022), a text-to-image diffusion model", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 572, + 506, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "In order to make Imagen synthesize objects with white backgrounds and minimal shadows at the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 633, + 143, + 644 + ], + "score": 0.52, + "content": "1 2 8 \\mathrm { x } 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "resolution, in addition to natural language prompting, we inpaint a 5px white border at each", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 241, + 656 + ], + "score": 1.0, + "content": "denoising step when generating a", + "type": "text" + }, + { + "bbox": [ + 241, + 644, + 267, + 654 + ], + "score": 0.75, + "content": "6 4 \\mathrm { x } 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "image, and then upsample it using the second model on the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Imagen cascade, i.e., a text-conditional super-resolution model. We find that this strategy is much", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "more reliable than only relying on prompt engineering to create images with white backgrounds: this", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "leads the text-to-image model to make the rest of background consistent with the fully white border,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "so the generated image results in an out-of-distribution object while maintaining an in-distribution", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "background. We present example novel views given a ShapeNet test object in Figure 1, and given", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "a synthetic image from Imagen in Figure 6. We include several video samples for all the cases", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 423, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 423, + 732 + ], + "score": 1.0, + "content": "described above in the Supplementary Website (https://3d-diffusion.github.io/).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 621, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 111, + 92, + 502, + 141 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 282, + 81, + 318, + 90 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 281, + 79, + 320, + 92 + ], + "spans": [ + { + "bbox": [ + 281, + 79, + 320, + 92 + ], + "score": 1.0, + "content": "SRN cars", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_caption", + "bbox": [ + 410, + 81, + 458, + 90 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 410, + 80, + 459, + 91 + ], + "spans": [ + { + "bbox": [ + 410, + 80, + 459, + 91 + ], + "score": 1.0, + "content": "SRN chairs*", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 111, + 92, + 502, + 141 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 92, + 502, + 141 + ], + "spans": [ + { + "bbox": [ + 111, + 92, + 502, + 141 + ], + "score": 0.942, + "html": "
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
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", + "type": "table", + "image_path": "24fe033b344b3c9496f9ddbf4c0a3c7d978c06200af84173c3193235dc239aba.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 111, + 92, + 502, + 108.33333333333333 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 108.33333333333333, + 502, + 124.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 111, + 124.66666666666666, + 502, + 141.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 149, + 504, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 148, + 504, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 504, + 162 + ], + "score": 1.0, + "content": "Table 3: Ablation – study removing stochastic conditioning and diffusion altogether from 3DiM.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 504, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 504, + 173 + ], + "score": 1.0, + "content": "Results are included for novel view synthesis from a single image on the SRN ShapeNet benchmark.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 172, + 469, + 183 + ], + "spans": [ + { + "bbox": [ + 107, + 172, + 119, + 183 + ], + "score": 0.58, + "content": "( ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 172, + 469, + 183 + ], + "score": 1.0, + "content": "Re-split chairs dataset as detailed in Section 3 – not comparable to numbers in Table 2.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "image", + "bbox": [ + 120, + 210, + 492, + 369 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 120, + 210, + 492, + 369 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 210, + 492, + 369 + ], + "spans": [ + { + "bbox": [ + 120, + 210, + 492, + 369 + ], + "score": 0.964, + "type": "image", + "image_path": "d6dbef759ea682169ceebc121b6ec9861cd4dafcac248279f348c36b4c1adb3b.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 120, + 210, + 492, + 263.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 120, + 263.0, + 492, + 316.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 120, + 316.0, + 492, + 369.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 378, + 505, + 445 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "Figure 7: Ablations – Example input and output views of 3DiM ablations on the SRN chairs dataset", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 133, + 403 + ], + "score": 1.0, + "content": "at the", + "type": "text" + }, + { + "bbox": [ + 134, + 390, + 171, + 400 + ], + "score": 0.5, + "content": "1 2 8 \\mathrm { x } 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "resolution. The second “Image-to-Image” column corresponds to removing our", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "stochastic conditioning sampler from Section 2.2. The third “Concat-UNet” column corresponds to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "removing our proposed architecture in Section 2.3, resorting to a UNet that concatenates both images", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "along the channel axis instead of weight sharing over frames. The fourth “Regression” column is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 434, + 293, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 293, + 445 + ], + "score": 1.0, + "content": "one-step diffusion model trained from scratch.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + } + ], + "index": 11.25 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "We now present some ablation studies on 3DiM. First, we remove our proposed sampler and use", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "a na¨ıve image-to-image model as discussed at the start of Section 2.2. We additionally remove the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "use of a diffusion process altogether, i.e., a regression model that generates samples via a single", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "denoising step. Naturally, the regression models are trained separately, but we can still use an", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "identical architecture, always feeding white noise for one frame, along with its corresponding noise", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 583, + 429, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 127, + 595 + ], + "score": 1.0, + "content": "level", + "type": "text" + }, + { + "bbox": [ + 128, + 583, + 146, + 594 + ], + "score": 0.89, + "content": "\\lambda _ { \\operatorname* { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 583, + 429, + 595 + ], + "score": 1.0, + "content": ". Results are included in Table 3 and samples are included in Figure 7.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "Unsurprisingly, we find that both of these components are crucial to achieve good results. The use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "of many diffusion steps allows sampling sharp images that achieve much better (lower) FID scores", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "than both prior work and the regression models, where in both cases, reconstructions appear blurry.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Notably, the regression models achieve better PSNR and SSIM scores than 3DiM despite the severe", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 643, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 658 + ], + "score": 1.0, + "content": "blurriness, suggesting that these standardized metrics fail to meaningfully capture sample quality", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "for geometry-free models, at least when comparing them to geometry-aware or non-stochastic re-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "construction approaches. Similarly, FID also has a failure mode: na¨ıve image-to-image sampling", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "improves (decreases) FID scores significantly, but severely worsens shape and texture inconsisten-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "cies between sampled frames. 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PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
3DiM21.010.578.9914.290.483.30
+ no stochastic conditioning23.820.591.8715.730.511.05
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Results are included in Table 3 and samples are included in Figure 7.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 527, + 505, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "Unsurprisingly, we find that both of these components are crucial to achieve good results. 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Similarly, FID also has a failure mode: na¨ıve image-to-image sampling", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "improves (decreases) FID scores significantly, but severely worsens shape and texture inconsisten-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "cies between sampled frames. 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PSNR (↑) SSIM (↑)carschairs*
FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Concat-UNet17.210.5221.5412.360.445.15
X-UNet21.010.578.9914.290.483.30
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(2021b;a) (which we simply call “Concat-UNet”). The Concat-UNet architecture, unlike ours,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "does not share weights across frames; instead, it simply concatenates the conditioning image to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "noisy input image along the channel axis. 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Results are included in Table 5 and Figure 7.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "While the Concat-UNet architecture is able to sample frames that resemble the data distribution,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "we find that 3DiMs trained with our proposed X-UNet architecture suffer much less from 3D in-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "consistency and alignment to the conditioning frame. 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We hypothesize that our X-UNet architecture better exploits symmetries between", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "frames and poses due to our proposed weight-sharing mechanism, and that the cross-attention helps", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 397, + 359, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 359, + 409 + ], + "score": 1.0, + "content": "significantly to align with the content of the conditioning view.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 106, + 426, + 481, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 482, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 482, + 438 + ], + "score": 1.0, + "content": "4 EVALUATING 3D CONSISTENCY IN GEOMETRY-FREE VIEW SYNTHESIS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "As we demonstrate in Sections 3.1 and 3.2, the standardized metrics in the literature have failure", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 462, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 504, + 473 + ], + "score": 1.0, + "content": "modes when specifically applied to geometry-free novel view synthesis models, e.g., their inability", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 504, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 504, + 485 + ], + "score": 1.0, + "content": "to successfully measure 3D consistency, and the possibility of improving them with worse models.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 482, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 505, + 497 + ], + "score": 1.0, + "content": "Leveraging the fact that volumetric rendering of colored density fields are 3D-consistent by design,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "we thus propose an additional evaluation scheme called “3D consistency scoring”. Our metrics", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 505, + 266, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 266, + 518 + ], + "score": 1.0, + "content": "should satisfy the following desiderata:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 497, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 360, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 360, + 534 + ], + "score": 1.0, + "content": "1. The metric must penalize outputs that are not 3D consistent.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 532, + 496, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 496, + 546 + ], + "score": 1.0, + "content": "2. 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The metric must penalize outputs that do not align with the conditioning view(s).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "In order to satisfy the second requirement, we cannot compare output renders to ground-truth views.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 586 + ], + "score": 1.0, + "content": "Thus, one straightforward way to satisfy all desiderata is to sample many views from the geometry-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "free model given a single view, train a NeRF-like neural field (Mildenhall et al., 2020) on a fraction", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 592, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 506, + 608 + ], + "score": 1.0, + "content": "of these views, and compute a set of metrics that compare neural field renders on the remaining", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "views. This way, if the geometry-free model outputs inconsistent views, the training of neural field", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "will be hindered and classical image evaluation metrics should clearly reflect this. Additionally, to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "enforce the third requirement, we simply include the conditioning view(s) that were used to generate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "the rest as part of the training data. 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To maintain comparability, we sample", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "the same number of views from the different 3DiMs we evaluate in this section, all at the same poses", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 332, + 721 + ], + "score": 1.0, + "content": "and conditioned on the same single views. 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PSNR (↑) SSIM (↑)carschairs*
FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Concat-UNet17.210.5221.5412.360.445.15
X-UNet21.010.578.9914.290.483.30
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Results are included for novel view synthesis from a single image on the SRN ShapeNet", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 345, + 172 + ], + "score": 1.0, + "content": "benchmark, with the re-split chairs as detailed in Section 3.", + "type": "text" + }, + { + "bbox": [ + 345, + 160, + 358, + 171 + ], + "score": 0.35, + "content": "( ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "Re-split chairs dataset as detailed in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 371, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 371, + 182 + ], + "score": 1.0, + "content": "Section 3 – these numbers are not comparable to those in Table 2.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 205, + 293, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 295, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 295, + 217 + ], + "score": 1.0, + "content": "3.2 UNET ARCHITECTURE COMPARISONS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "In order to demonstrate the benefit of our proposed modifications, we additionally compare our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "proposed X-UNet architecture from Section 2.3 to the simpler UNet architecture following Saharia", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "et al. (2021b;a) (which we simply call “Concat-UNet”). The Concat-UNet architecture, unlike ours,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "does not share weights across frames; instead, it simply concatenates the conditioning image to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "noisy input image along the channel axis. To do this comparison, we train 3DiMs on the Concat-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "UNet architecture with the same number of hidden channels, and keep all other hyperparameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "identical. Because our architecture has the additional cross-attention layer at the coarse-resolution", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 470, + 315 + ], + "score": 1.0, + "content": "blocks, our architecture has a slightly larger number of parameters than the Concat-UNet", + "type": "text" + }, + { + "bbox": [ + 471, + 303, + 504, + 313 + ], + "score": 0.62, + "content": "{ \\sim } 4 7 1 \\mathrm { M }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 313, + 343, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 122, + 325 + ], + "score": 1.0, + "content": "v.s.", + "type": "text" + }, + { + "bbox": [ + 123, + 313, + 156, + 324 + ], + "score": 0.6, + "content": "{ \\sim } 4 2 1 \\mathrm { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 313, + 343, + 325 + ], + "score": 1.0, + "content": "). Results are included in Table 5 and Figure 7.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 225, + 506, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "While the Concat-UNet architecture is able to sample frames that resemble the data distribution,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "we find that 3DiMs trained with our proposed X-UNet architecture suffer much less from 3D in-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "consistency and alignment to the conditioning frame. Moreover, while the metrics should be taken", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 364, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 504, + 375 + ], + "score": 1.0, + "content": "with a grain of salt as previously discussed, we find that all the metrics significantly worsen with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "the Concat-UNet. We hypothesize that our X-UNet architecture better exploits symmetries between", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "frames and poses due to our proposed weight-sharing mechanism, and that the cross-attention helps", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 397, + 359, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 359, + 409 + ], + "score": 1.0, + "content": "significantly to align with the content of the conditioning view.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 329, + 505, + 409 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 426, + 481, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 482, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 482, + 438 + ], + "score": 1.0, + "content": "4 EVALUATING 3D CONSISTENCY IN GEOMETRY-FREE VIEW SYNTHESIS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "As we demonstrate in Sections 3.1 and 3.2, the standardized metrics in the literature have failure", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 462, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 504, + 473 + ], + "score": 1.0, + "content": "modes when specifically applied to geometry-free novel view synthesis models, e.g., their inability", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 504, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 504, + 485 + ], + "score": 1.0, + "content": "to successfully measure 3D consistency, and the possibility of improving them with worse models.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 482, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 505, + 497 + ], + "score": 1.0, + "content": "Leveraging the fact that volumetric rendering of colored density fields are 3D-consistent by design,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "we thus propose an additional evaluation scheme called “3D consistency scoring”. Our metrics", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 505, + 266, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 266, + 518 + ], + "score": 1.0, + "content": "should satisfy the following desiderata:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 450, + 505, + 518 + ] + }, + { + "type": "index", + "bbox": [ + 107, + 522, + 497, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 360, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 360, + 534 + ], + "score": 1.0, + "content": "1. The metric must penalize outputs that are not 3D consistent.", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 532, + 496, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 496, + 546 + ], + "score": 1.0, + "content": "2. The metric must not penalize outputs that are 3D consistent but deviate from the ground truth.", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 104, + 543, + 444, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 444, + 557 + ], + "score": 1.0, + "content": "3. The metric must penalize outputs that do not align with the conditioning view(s).", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + } + ], + "index": 32, + "bbox_fs": [ + 104, + 523, + 496, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "In order to satisfy the second requirement, we cannot compare output renders to ground-truth views.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 586 + ], + "score": 1.0, + "content": "Thus, one straightforward way to satisfy all desiderata is to sample many views from the geometry-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "free model given a single view, train a NeRF-like neural field (Mildenhall et al., 2020) on a fraction", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 592, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 506, + 608 + ], + "score": 1.0, + "content": "of these views, and compute a set of metrics that compare neural field renders on the remaining", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "views. This way, if the geometry-free model outputs inconsistent views, the training of neural field", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "will be hindered and classical image evaluation metrics should clearly reflect this. Additionally, to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "enforce the third requirement, we simply include the conditioning view(s) that were used to generate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "the rest as part of the training data. We report PSNR, SSIM and FID on the held-out views, although", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 648, + 377, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 377, + 662 + ], + "score": 1.0, + "content": "one could use other metrics under our proposed evaluation scheme.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 561, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We evaluate 3DiMs on the SRN benchmark, and for comparison, we additionally include metrics for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "models trained on (1) the real test views and (2) on image-to-image samples from the 3DiMs, i.e.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "without our proposed sampler like we reported in Section 3. To maintain comparability, we sample", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "the same number of views from the different 3DiMs we evaluate in this section, all at the same poses", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 332, + 721 + ], + "score": 1.0, + "content": "and conditioned on the same single views. We leave out", + "type": "text" + }, + { + "bbox": [ + 333, + 710, + 352, + 720 + ], + "score": 0.88, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "of the test views (25 out of 251) from", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "neural field training, picking 25 random indices once and maintaining this choice of indices for all", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 665, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 80, + 503, + 146 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 80, + 503, + 146 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 503, + 146 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 503, + 146 + ], + "score": 0.974, + "html": "
Training view sourceSRN carsSRN chairs*
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Original data (3D consistent)28.210.9610.5724.870.9317.05
3DiM(~1.3B params)28.480.9629.5522.900.8658.61
3DiM(~471M params)28.530.9622.0918.840.7998.78
+ no stochastic conditioning25.780.9430.5117.610.75116.16
", + "type": "table", + "image_path": "06346efb1d8865985e9d156795692f5086aaf86f4d46d0c780996ae619b213bb.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 80, + 503, + 102.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 102.0, + 503, + 124.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 124.0, + 503, + 146.0 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 153, + 504, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 505, + 166 + ], + "score": 1.0, + "content": "Table 6: 3D consistency scores – For neural fields trained with different view sources, we compare", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 164, + 304, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 304, + 177 + ], + "score": 1.0, + "content": "renders to held-out views from the same sources.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 108, + 203, + 503, + 236 + ], + "lines": [ + { + "bbox": [ + 107, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 107, + 203, + 389, + 215 + ], + "score": 1.0, + "content": "subsequent evaluations. We also train models with more parameters", + "type": "text" + }, + { + "bbox": [ + 390, + 203, + 424, + 214 + ], + "score": 0.83, + "content": "( \\sim 1 . 3 \\mathrm { B } )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "in order to investi-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "gate whether increasing model capacity can further improve 3D consistency. See Supplementary", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 224, + 466, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 466, + 237 + ], + "score": 1.0, + "content": "Material (Sec.7) for more details on the neural fields we chose and their hyperparameters.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "We find that our proposed evaluation scheme clearly punishes 3D inconsistency as desired – the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "neural fields trained on image-to-image 3DiM samples have worse scores across all metrics com-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "pared to neural fields trained on 3DiM outputs sampled via our stochastic conditioning. This helps", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "quantify the value of our proposed sampler, and also prevents the metrics from punishing reasonable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "outputs that are coherent with the input view(s) but do not match the target views – a desirable prop-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "erty due to the stochasticity of generative models. Moreover, we qualitatively find that the smaller", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 392, + 320 + ], + "score": 1.0, + "content": "model reported in the rest of the paper is comparable in quality to the", + "type": "text" + }, + { + "bbox": [ + 392, + 307, + 420, + 318 + ], + "score": 0.85, + "content": "{ \\sim } 1 . 3 \\mathrm { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "parameter model on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "cars, though on chairs, we do observe a significant improvement on 3D consistency in our samples.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 329, + 416, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 416, + 343 + ], + "score": 1.0, + "content": "Importantly, 3D consistency scoring agrees with our qualitative observations.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 363, + 299, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 301, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 301, + 377 + ], + "score": 1.0, + "content": "5 CONCLUSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 404 + ], + "score": 1.0, + "content": "We propose 3DiM, a diffusion model for 3D novel view synthesis. Combining improvements in our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "score": 1.0, + "content": "X-UNet neural architecture (Section 2.3), with our novel stochastic conditioning sampling strategy", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "that enables autoregressive generation over frames (Section 2.2), we show that from as few as a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "score": 1.0, + "content": "single image we can generate approximately 3D consistent views with very sharp sample quality.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "We additionally introduce 3D consistency scoring to evaluate the 3D consistency of geometry-free", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "generative models by training neural fields on model output views, as their performance will be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "hindered increasingly with inconsistent training data (Section 4). We thus show, both quantitatively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "and visually, that 3DiMs with stochastic conditioning can achieve approximate 3D consistency and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "high sample quality simultaneously, and how classical metrics fail to capture both sharp modes and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 180, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 180, + 503 + ], + "score": 1.0, + "content": "3D inconsistency.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "A noteworthy limitation of 3DiM is that, due to the geometry-free setup, it can only handle distri-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "butions of poses that it is exposed to durining training. In particular, further study is required where", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "more variations to the poses are introduced. For example, varying focal lengths, sensor widths,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "making the camera not look exactly at the objects, and varying the distances from the camera to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "objects more aggressively. Our use of stochastic conditioning might also exacerbate the need for", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "many denoising steps. While we believe that finding effective architectures that allow conditioning", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "on sets of images remains an important open problem, stochastic conditioning unlocks the ability", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "to train on examples that only have two views and helps overcome memory-intensive neural archi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "tectures. 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Another potentially significant application of 3DiM is its use as a prior in order", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "to achieve 3D consistent generation via approaches that operate exclusively at sampling time (Poole", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "et al., 2022; Zhou & Tulsiani, 2022; Xu et al., 2022) in order to enable wide adoption of similar,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 308, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 308, + 734 + ], + "score": 1.0, + "content": "disruptive technologies on the 3D design industry.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 80, + 503, + 146 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 80, + 503, + 146 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 503, + 146 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 503, + 146 + ], + "score": 0.974, + "html": "
Training view sourceSRN carsSRN chairs*
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Original data (3D consistent)28.210.9610.5724.870.9317.05
3DiM(~1.3B params)28.480.9629.5522.900.8658.61
3DiM(~471M params)28.530.9622.0918.840.7998.78
+ no stochastic conditioning25.780.9430.5117.610.75116.16
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We also train models with more parameters", + "type": "text" + }, + { + "bbox": [ + 390, + 203, + 424, + 214 + ], + "score": 0.83, + "content": "( \\sim 1 . 3 \\mathrm { B } )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "in order to investi-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "gate whether increasing model capacity can further improve 3D consistency. See Supplementary", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 224, + 466, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 466, + 237 + ], + "score": 1.0, + "content": "Material (Sec.7) for more details on the neural fields we chose and their hyperparameters.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 203, + 505, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "We find that our proposed evaluation scheme clearly punishes 3D inconsistency as desired – the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "neural fields trained on image-to-image 3DiM samples have worse scores across all metrics com-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "pared to neural fields trained on 3DiM outputs sampled via our stochastic conditioning. This helps", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "quantify the value of our proposed sampler, and also prevents the metrics from punishing reasonable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "outputs that are coherent with the input view(s) but do not match the target views – a desirable prop-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "erty due to the stochasticity of generative models. Moreover, we qualitatively find that the smaller", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 392, + 320 + ], + "score": 1.0, + "content": "model reported in the rest of the paper is comparable in quality to the", + "type": "text" + }, + { + "bbox": [ + 392, + 307, + 420, + 318 + ], + "score": 0.85, + "content": "{ \\sim } 1 . 3 \\mathrm { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "parameter model on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "cars, though on chairs, we do observe a significant improvement on 3D consistency in our samples.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 329, + 416, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 416, + 343 + ], + "score": 1.0, + "content": "Importantly, 3D consistency scoring agrees with our qualitative observations.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 241, + 506, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 363, + 299, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 301, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 301, + 377 + ], + "score": 1.0, + "content": "5 CONCLUSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 404 + ], + "score": 1.0, + "content": "We propose 3DiM, a diffusion model for 3D novel view synthesis. Combining improvements in our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "score": 1.0, + "content": "X-UNet neural architecture (Section 2.3), with our novel stochastic conditioning sampling strategy", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "that enables autoregressive generation over frames (Section 2.2), we show that from as few as a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "score": 1.0, + "content": "single image we can generate approximately 3D consistent views with very sharp sample quality.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "We additionally introduce 3D consistency scoring to evaluate the 3D consistency of geometry-free", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "generative models by training neural fields on model output views, as their performance will be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "hindered increasingly with inconsistent training data (Section 4). We thus show, both quantitatively", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "and visually, that 3DiMs with stochastic conditioning can achieve approximate 3D consistency and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "high sample quality simultaneously, and how classical metrics fail to capture both sharp modes and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 180, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 180, + 503 + ], + "score": 1.0, + "content": "3D inconsistency.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 389, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "A noteworthy limitation of 3DiM is that, due to the geometry-free setup, it can only handle distri-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "butions of poses that it is exposed to durining training. 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Assuming we have an input", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 187, + 254, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 254, + 201 + ], + "score": 1.0, + "content": "batch with elements each containing", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 166, + 505, + 201 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 208, + 526, + 297 + ], + "lines": [ + { + "bbox": [ + 107, + 210, + 115, + 219 + ], + "spans": [ + { + "bbox": [ + 107, + 210, + 115, + 219 + ], + "score": 1.0, + "content": "{", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 216, + 371, + 234 + ], + "spans": [ + { + "bbox": [ + 128, + 216, + 157, + 234 + ], + "score": 1.0, + "content": "\"z\",", + "type": "text" + }, + { + "bbox": [ + 165, + 218, + 371, + 232 + ], + "score": 1.0, + "content": "# noisy input image (HxWx3 tensor)", + "type": "text" + } + ], + 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(2020) note that their ancestral sampler can be used with any variances between", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 117, + 721 + ], + "score": 0.87, + "content": "\\tilde { \\beta } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 708, + 174, + 723 + ], + "score": 1.0, + "content": "(variances of", + "type": "text" + }, + { + "bbox": [ + 174, + 709, + 222, + 722 + ], + "score": 0.91, + "content": "q ( \\boldsymbol { z } _ { s } | \\boldsymbol { z } _ { t } , \\boldsymbol { x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 708, + 242, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 243, + 710, + 253, + 721 + ], + "score": 0.87, + "content": "\\beta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "(variances of the underlying SDE), we simply use the former", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "as those correspond to their ancestral sampler. 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We generally find that the best hyperparameter", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "choices at low-resolution experiments transfer well when applied to the higher resolutions, and thus", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 263, + 426, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 426, + 277 + ], + "score": 1.0, + "content": "recommend this strategy for cheaper and more practical hyperparameter tuning.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 164, + 506, + 277 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 138, + 293 + ], + "score": 1.0, + "content": "For the", + "type": "text" + }, + { + "bbox": [ + 139, + 281, + 168, + 291 + ], + "score": 0.84, + "content": "{ \\sim } 1 . 3 \\mathrm { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "parameter models, we tried increasing the number of parameters of our proposed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "UNet architecture through several different ways: increasing the number of blocks per resolution,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "the number of attention heads, the number of cross-attention layers per block, and the base number", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "of hidden channels. 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We do not observe this choice to have a significant", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "effect sample quality. All our samples are generated with 256 denoising steps. 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All our samples are generated with 256 denoising steps. As is standard in the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 256, + 107 + ], + "score": 1.0, + "content": "literature, we also clip each predicted", + "type": "text" + }, + { + "bbox": [ + 256, + 95, + 264, + 104 + ], + "score": 0.74, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "at each denoising step to the normalized range of the images", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 104, + 140, + 118 + ], + "spans": [ + { + "bbox": [ + 107, + 104, + 135, + 117 + ], + "score": 0.83, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 104, + 140, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 124, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 123, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 506, + 137 + ], + "score": 1.0, + "content": "Classifier-free guidance. Our models use classifier-free guidance (Ho & Salimans, 2021), as we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "find that small guidance weights help encourage 3D consistency further. All our models were trained", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 255, + 159 + ], + "score": 1.0, + "content": "unconditionally with a probability of", + "type": "text" + }, + { + "bbox": [ + 255, + 146, + 275, + 156 + ], + "score": 0.88, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 146, + 505, + 159 + ], + "score": 1.0, + "content": "for each minibatch element. For unconditional examples,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 169 + ], + "score": 1.0, + "content": "we zero out the (positionally encoded) pose and replace the clean frame with standard Gaussian", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "noise (leveraging our weight-sharing architecture, see Section 2.3). We swept over various guidance", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 192 + ], + "score": 1.0, + "content": "weights, and simply picked those where 3D inconsistency was qualitatively least apparent on sam-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "score": 1.0, + "content": "pled videos. For SRN cars, we use a guidance weight of 3.0, while for SRN chairs we use a weight", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 200, + 135, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 135, + 212 + ], + "score": 1.0, + "content": "of 2.0.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 108, + 225, + 248, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 250, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 250, + 238 + ], + "score": 1.0, + "content": "7.4 3D CONSISTENCY SCORING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "Note that traditional NeRFs (Mildenhall et al., 2020) can be 3D inconsistent as the model allows", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "for view-dependent radiance. We thus employ a simpler and faster to train version based on instant-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "NGP (Muller et al., 2022) without view dependent components, and with additional distortion and ¨", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "orientation loss terms for improved convergence (Barron et al., 2022; Verbin et al., 2022). Note", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "that the specific implementation details of the neural field method chosen will affect the metrics", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "considerably, so it is of utmost importance to apply the same method and hyperparameters for the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 393, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 393, + 324 + ], + "score": 1.0, + "content": "neural fields to make 3D consistency scores comparable across models.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 342 + ], + "score": 1.0, + "content": "We design the neural fields as simple Multi-Layer Perceptrons (MLPs) of hidden size 64 and no", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "skip connections. 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Note", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "that the specific implementation details of the neural field method chosen will affect the metrics", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "considerably, so it is of utmost importance to apply the same method and hyperparameters for the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 393, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 393, + 324 + ], + "score": 1.0, + "content": "neural fields to make 3D consistency scores comparable across models.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 246, + 505, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 342 + ], + "score": 1.0, + "content": "We design the neural fields as simple Multi-Layer Perceptrons (MLPs) of hidden size 64 and no", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "skip connections. 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SRN carsSRN chairs
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Geometry-aware
SRN22.250.8841.2122.890.8926.51
PixelNeRF23.170.8959.2423.720.9038.49
VisionNeRF22.880.9021.3124.480.9210.05
CodeNeRF23.80*0.9123.66*0.901
Geometry-free
LFN22.42*0.8922.26*0.90
ENR22.2622.831
3DiM (ours)21.010.578.9917.050.536.57
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PSNR (↑) SSIM (↑)carschairs*
FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Concat-UNet17.210.5221.5412.360.445.15
X-UNet21.010.578.9914.290.483.30
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Training view sourceSRN carsSRN chairs*
PSNR (↑)SSIM (↑)FID (↓)PSNR (↑)SSIM (↑)FID (↓)
Original data (3D consistent)28.210.9610.5724.870.9317.05
3DiM(~1.3B params)28.480.9629.5522.900.8658.61
3DiM(~471M params)28.530.9622.0918.840.7998.78
+ no stochastic conditioning25.780.9430.5117.610.75116.16
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], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 224, + 544, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neural data compression based on nonlinear transform coding has made great progress over the last few years, mainly due to improvements in prior models, quantization methods and nonlinear transforms. A general trend in many recent works pushing the limit of rate-distortion performance is to use ever more expensive prior models that can lead to prohibitively slow decoding. Instead, we focus on more expressive transforms that result in a better rate-distortioncomputation trade-off. Specifically, we show that nonlinear transforms built on Swin-transformers can achieve better compression efficiency than transforms built on convolutional neural networks (ConvNets), while requiring fewer parameters and shorter decoding time. Paired with a compute-efficient Channel-wise AutoRegressive Model prior, our SwinT-ChARM model outperforms VTM-12.1 by $3 . 6 8 \\%$ in BD-rate on Kodak with comparable decoding speed. In P-frame video compression setting, we are able to outperform the popular ConvNet-based scalespace-flow model by $1 2 . 3 5 \\%$ in BD-rate on UVG. We provide model scaling studies to verify the computational efficiency of the proposed solutions and conduct several analyses to reveal the source of coding gain of transformers over ConvNets, including better spatial decorrelation, flexible effective receptive field, and more localized response of latent pixels during progressive decoding. ", + "bbox": [ + 233, + 257, + 764, + 505 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 532, + 336, + 547 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Transform coding (Goyal, 2001) is the dominant paradigm for compression of multi-media signals, and serves as the technical foundation for many successful coding standards such as JPEG, AAC, and HEVC/VVC. Codecs based on transform coding divide the task of lossy compression into three modularized components: transform, quantization, and entropy coding. All three components can be enhanced by deep neural networks: autoencoder networks are adopted as flexible nonlinear transforms, deep generative models are used as powerful learnable entropy models, and various differentiable quantization schemes are proposed to aid end-to-end training. Thanks to these advancements, we have seen rapid progress in the domain of image and video compression. Particularly, the hyperprior line of work (Balle et al., 2018; Minnen et al., 2018; Lee et al., 2019; Agustsson et al., 2020; ´ Minnen & Singh, 2020) has led to steady progress of neural compression performance over the past two years, reaching or even surpassing state-of-the-art traditional codecs. For example, in image compression, BPG444 was surpassed by a neural codec in 2018 (Minnen et al., 2018), and (Cheng et al., 2020; Xie et al., 2021; Ma et al., 2021; Guo et al., 2021; Wu et al., 2020) have claimed on-par or better performance than VTM (a test model of the state-of-the-art non-learned VVC standard). ", + "bbox": [ + 174, + 564, + 825, + 757 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One general trend in the advancement of neural image compression schemes is to develop ever more expressive yet expensive prior models based on spatial context. However, the rate-distortion improvement from context based prior modeling often comes with a hefty price $\\mathrm { { t a g } ^ { 1 } }$ in terms of decoding complexity. Noteably, all existing works that claimed on-par or better performance than VTM (Cheng et al., 2020; Xie et al., 2021; Ma et al., 2021; Guo et al., 2021; Wu et al., 2020) rely on slow and expensive spatial context based prior models. ", + "bbox": [ + 174, + 765, + 825, + 848 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The development of nonlinear transforms, on the other hand, are largely overlooked. This leads us to the following questions: can we achieve the same performance as that of expensive prior models by designing a more expressive transform together with simple prior models? And if so, how much more complexity in the transform is required? ", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Interestingly, we show that by leveraging and adapting the recent development of vision transformers, not only can we build neural codecs with simple prior models that can outperform ones built on expensive spatial autoregressive priors, but do so with smaller transform complexity compared to its convolutional counterparts, attaining a strictly better ratedistortion-complexity trade-off. As can be seen in Figure 1, our proposed neural image codec SwinT-ChARM can outperform VTM-12.1 at comparable decoding time, which, to the best of our knowledge, is a first in the neural compression literature. ", + "bbox": [ + 174, + 166, + 483, + 359 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/9b4c6f997837dcdde529df12bf2f7cf57a0573288416f18acb674c0c84b370ef.jpg", + "image_caption": [ + "Figure 1: BD-rate (smaller is better) vs decoding time. Our Swin transformer based image compression models land in a favorable region of rate-distortion-computation trade-off that has never been achieved before. See Section 4.2 for more results and evaluation setup. " + ], + "image_footnote": [], + "bbox": [ + 503, + 167, + 816, + 366 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As main contributions, we 1) extend SwinTransformer (Liu et al., 2021) to a decoder setting and build Swin-transformer based neural image codecs that attain better rate-distortion performance with lower complexity compared with existing solutions, 2) verify its effectiveness in video compression by enhancing scalespace-flow, a popular neural P-frame codec, ", + "bbox": [ + 174, + 368, + 483, + 478 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "and 3) conduct extensive analysis and ablation study to explore differences between convolution and transformers, and investigate potential source of coding gain. ", + "bbox": [ + 173, + 479, + 823, + 507 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND & RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 529, + 490, + 544 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Conv-Hyperprior The seminal hyperprior architecture (Balle et al., 2018; Minnen et al., 2018) is ´ a two-level hierarchical variational autoencoder, consisting of a pair of encoder/decoder $g _ { a } , g _ { s }$ , and a pair of hyper-encoder/hyper-decoder $h _ { a } , h _ { s }$ . Given an input image $\\mathbf { x }$ , a pair of latent ${ \\bf y } = g _ { a } ( { \\bf x } )$ and hyper-latent ${ \\bf z } = h _ { a } ( { \\bf y } )$ is computed. The quantized hyper-latent ${ \\hat { \\mathbf { z } } } = Q ( \\mathbf { z } )$ is modeled and entropycoded with a learned factorized prior. The latent $\\mathbf { y }$ is modeled with a factorized Gaussian distribution $p ( \\mathbf { y } | \\hat { \\mathbf { z } } ) = \\mathcal { N } ( \\mu , \\mathrm { d i a g } ( { \\pmb \\sigma } ) )$ whose parameter is given by the hyper-decoder $( \\pmb { \\mu } , \\pmb { \\sigma } ) = h _ { s } ( \\hat { \\bf z } )$ . The quantized version of the latent $\\hat { \\mathbf { y } } = \\bar { Q } ( \\mathbf { y } - \\pmb { \\mu } ) + \\bar { \\pmb { \\mu } }$ is then entropy coded and passed through decoder $g _ { s }$ to derive reconstructed image $\\hat { \\mathbf { x } } = g _ { s } ( \\hat { \\mathbf { y } } )$ . The tranforms $g _ { a } , g _ { s } , h _ { a } , h _ { s }$ are all parameterized as ConvNets (for details, see Appendix A.1). ", + "bbox": [ + 173, + 560, + 825, + 685 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Conv-ChARM (Minnen & Singh, 2020) extends the baseline hyperprior architecture with a channel-wise auto-regressive model $( { \\mathrm { C h A R M } } ) ^ { 2 }$ , in which latent $\\mathbf { y }$ is split along channel dimension into $S$ groups (denoted as $\\mathbf { y } _ { 1 } , \\ldots , \\mathbf { y } _ { S } )$ , and the Gaussian prior $p ( \\mathbf { y } _ { s } | \\hat { \\mathbf { z } } , \\hat { \\mathbf { y } } _ { < s } )$ is made autoregressive across groups where the mean/scale of ${ \\bf y } _ { s }$ depends on quantized latent in the previous groups $\\hat { \\mathbf { y } } _ { < s }$ . In practice, $S = 1 0$ provides a good balance of performance and complexity and is adopted here. ", + "bbox": [ + 173, + 702, + 825, + 772 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Spatial AR models Most of recent performance advancements of neural image compression is driven by the use of spatial auto-regressive/context models. Variants include causal global prediction (Guo et al., 2021), 3D context (Ma et al., 2021), block-level context (Wu et al., 2020), nonlocal context (Li et al., 2020; Qian et al., 2021). One common issue with these designs is that decoding cannot be parallelized along spatial dimensions, leading to impractical3decoding latency, especially for large resolution images. ", + "bbox": [ + 173, + 787, + 825, + 872 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "ConvNet-based transforms While the design space of prior models is extensively explored, nonlinear transforms, as an important component, have received less attention. A standard convolution encoder-decoder with GDN (Balle et al., 2016; 2017) as activation is widely adopted in the literature. ´ Later works introduce new transform designs, such as residual blocks with smaller kernels (Cheng et al., 2020), nonlocal (sigmoid gating) layers (Zhou et al., 2019; Chen et al., 2021), invertible neural networks (Xie et al., 2021), and PReLU as an efficient replacement of GDN (Egilmez et al., 2021). ", + "bbox": [ + 173, + 103, + 825, + 188 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Vision transformers Although many transform networks are proposed, they are still mainly based on ConvNets. Recently transformers (Vaswani et al., 2017) have been introduced to the vision domain and have shown performance competitive with ConvNets in many tasks, e.g. object detection (Carion et al., 2020), classification (Dosovitskiy et al., 2021), image enhancement (Chen et al., 2020), and semantic segmentation (Zheng et al., 2021). Inspired by their success, in this work we explore how vision transformers work as nonlinear transforms for image and video compression. ", + "bbox": [ + 174, + 204, + 825, + 287 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 SWIN-TRANSFORMER BASED TRANSFORM CODING ", + "text_level": 1, + "bbox": [ + 176, + 309, + 633, + 325 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Among the large number of vision transformer variants, we choose Swin Transformer (Liu et al., 2021) (hereafter referred to as SwinT) to build the nonlinear transforms, mainly because of 1) its linear complexity w.r.t. input resolution due to local window attention, and 2) its flexibility in handling varying input resolutions at test time, enabled by relative position bias and hierarchical architecture. ", + "bbox": [ + 173, + 342, + 825, + 398 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/bccf6a5f871e0f46f865c4f24cd3adf543cc067900925f28b4ff6121eb00b122.jpg", + "image_caption": [ + "Figure 2: Network architecture of our proposed SwinT-ChARM model4. In the SwinT-Hyperprior model, the ChARM component is removed and instead $\\pmb { \\mu }$ and $\\sigma$ are directly output by the hyperdecoder $h _ { s }$ (For details see Figure 13 in Appendix A.1). " + ], + "image_footnote": [], + "bbox": [ + 171, + 411, + 826, + 628 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 SWINT ENCODER AND DECODER", + "text_level": 1, + "bbox": [ + 176, + 710, + 441, + 724 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The original SwinT is proposed as a vision backbone, i.e. an encoder transform with downsampling. As shown in Figure 2, the SwinT encoder $g _ { a }$ contains SwinT blocks interleaved with Patch Merge blocks. The Patch Merge block contains Space-to-Depth (for downsampling), LayerNorm, and Linear layers sequentially. SwinT block performs local self-attention within each non-overlapping window of the feature maps and preserves feature size. Consecutive SwinT blocks at the same feature size shift the window partitioning with respect to the previous block to promote information propagation across nearby windows in the previous block. ", + "bbox": [ + 174, + 737, + 825, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We adopt SwinT encoder as the encoder transform $g _ { a }$ in our model, and extend it to SwinT decoder $g _ { s }$ by reversing the order of blocks in $g _ { a }$ , and replacing the Patch Merge block with a Patch Split block, which contains Linear, LayerNorm, Depth-to-Space (for upsampling) layers in sequence. The architectures for hyper transforms $h _ { a } , h _ { s }$ are similar to $g _ { a } , g _ { s }$ with different configurations. ", + "bbox": [ + 174, + 840, + 825, + 897 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "With these four SwinT transforms, we propose two image compression models, SwinT-Hyperprior and SwinT-ChARM, whose prior and hyper prior models are respectively the same as in ConvHyperprior and Conv-ChARM introduced in Section 2. The full model architectures are shown in Figure 2 and Figure 13. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 EXTENSION TO P-FRAME COMPRESSION ", + "text_level": 1, + "bbox": [ + 178, + 183, + 495, + 196 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To investigate the effectiveness of SwinT transforms for video compression, we study one popular P-frame compression model called Scale-Space Flow (SSF) (Agustsson et al., 2020). There are three instances of Conv-Hyperprior in SSF, which are respectively for compressing I-frames, scale-space flow and residual. We propose a SwinT variant, referred to as SwinT-SSF, which is obtained by replacing Conv transforms $g _ { a } , g _ { s }$ in the flow codec and residual codec of SSF with SwinT tranforms. To stabilize training of flow codec in SwinT-SSF, we need to remove all LayerNorm layers and reduce the window size (e.g. from 8 to 4). The baseline SSF model will be referred as Conv-SSF. Even though we build our solution on top of SSF, we believe this general extension can be applied to other ConvNet-based video compression models (Rippel et al., 2021; Hu et al., 2021) as well. ", + "bbox": [ + 174, + 212, + 825, + 337 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTS AND ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 364, + 457, + 380 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 EXPERIMENT SETUP ", + "text_level": 1, + "bbox": [ + 176, + 398, + 356, + 412 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Training All image compression models are trained on the CLIC2020 training set. ConvHyperprior and SwinT-Hyperprior are trained with 2M batches. Conv-ChARM and SwinT-ChARM are trained with 3.5M and 3.1M steps. Each batch contains 8 random $2 5 6 \\times 2 5 6$ crops from training images. Training loss $L = D + \\beta R$ is a weighted combination of distortion $D$ and bitrate $R$ , with $\\beta$ being the Lagrangian multiplier steering rate-distortion trade-off. Distortion $D$ is MSE in RGB color space. To cover a wide range of rate and distortion, for each solution, we train 5 models with $\\beta \\in \\{ \\bar { 0 . } 0 0 3 , 0 . 0 0 1 , 0 . 0 0 0 3 , 0 . 0 0 0 \\bar { 1 } , 0 . 0 0 0 0 3 \\}$ . The detailed training schedule is in Appendix B.1. ", + "bbox": [ + 174, + 428, + 825, + 525 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For P-frame compression models, we follow the training setup of SSF. Both Conv-SSF and SwinTSSF are trained on Vimeo-90k Dataset (Xue et al., 2019) for 1M steps with learning rate $1 0 ^ { - 4 }$ , batch size of 8, crop size of $2 5 6 \\times 2 5 6$ , followed by 50K steps of training with learning rate $1 0 ^ { - 5 }$ and crop size $3 8 4 \\times 2 5 6$ . The models are trained with 8 $\\beta$ values $2 ^ { \\gamma } \\times 1 0 ^ { - \\bar { 4 } } : \\gamma \\in \\{ 0 , 1 , \\bar { . . . , 7 } \\}$ . We adopt one critical trick to stablize the training from (Jaegle et al., 2021; Meister et al., 2018), i.e. to forward each video sequence twice during one optimization step (mini-batch), once in the original frame order, once in the reversed frame order. Finally we add flow $\\mathrm { l o s s } ^ { 5 }$ only between 0 and 200K steps, which we found not critical for stable training but improves the RD. ", + "bbox": [ + 174, + 531, + 825, + 642 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Evaluation We evaluate image compression models on 4 datasets: Kodak (Kodak, 1999), CLIC2021 testset (CLIC, 2021), Tecnick testset (Asuni & Giachetti, 2014), and JPEG-AI testset (JPEG-AI, 2020). We use BPG and VTM-12.1 to code the images in YUV444 mode, and then calculate PSNR in RGB. For a fair comparison all images are cropped to multiples of 256 to avoid padding for neural codecs. ", + "bbox": [ + 174, + 665, + 825, + 734 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We evaluated P-frame models on UVG (Mercat et al., 2020) 6 and MCL-JCV (Wang et al., 2016), and compare them with the test model implementation of HEVC, referred to as HEVC (HM), and open source library implementation of HEVC, refered to as HEVC $\\left( \\mathrm { x } 2 6 5 \\right)$ . To align configuration, all video codecs are evaluated in low-delay-P model with a fixed GOP size of 12. ", + "bbox": [ + 174, + 741, + 825, + 797 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Besides rate-distortion curves, we also evaluate different models using BD-rate (Tan et al., 2016), which represents the average bitrate savings for the same reconstruction quality. For image codecs, BD-rate is computed for each image and then averaged across all images; for video codecs, BD-rate is computed for each video and then averaged across all videos. More details on testset preprocessing, and traditional codecs configurations can be found in Appendix B.2. ", + "bbox": [ + 176, + 804, + 823, + 875 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/4b3b6a9139045c351c32278f20ebdda0599254c2ee3d526c7e6551263623e34c.jpg", + "image_caption": [ + "Figure 3: Comparison of compression efficiency on Kodak8. Note that the encoding time of VTM12.1 is much longer than all neural codecs, as shown in Table 4 and Table 7 in the Appendix. " + ], + "image_footnote": [], + "bbox": [ + 181, + 102, + 810, + 366 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 438, + 277, + 452 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "RD and BD-rate for image codecs The RD curves for all compared image codecs evaluated on Kodak are shown in Figure 3a, and the relative rate reduction of each codec compared to VTM-12.1 at a range of PSNR levels is shown in Figure 3b 7. ", + "bbox": [ + 173, + 465, + 823, + 508 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As can be seen from Figure 3, SwinT transform consistently outperforms its convolutional counterpart; the RD-performance of SwinT-Hyperprior is on-par with Conv-ChARM, despite the simpler prior; SwinT-ChARM outperforms VTM-12.1 across a wide PSNR range. In the Appendix (Figure 28 and Figure 30), we further incorporate the results from existing literature known to us for a complete comparison. Particularly, our Conv-Hyperprior is much better than the results reported in (Minnen et al., 2018) (no context), and Conv-ChARM is on par with (Minnen & Singh, 2020). ", + "bbox": [ + 173, + 515, + 825, + 599 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/200a6611038b619cb905588b65b145817339c8bda243c3a6c4f9dc635b7278c3.jpg", + "table_caption": [ + "Table 1: BD-rate of image codecs relative to VTM-12.1 (smaller is better). " + ], + "table_footnote": [], + "table_body": "
Image CodecKodakCLIC2021TecnickJPEG-AI
BPG44420.87%28.45%27.74%27.14%
Conv-Hyperprior11.65%12.23%12.49%20.98%
Conv-ChARM3.44%4.14%3.50%9.59%
SwinT-Hyperprior1.69%0.83%-0.15%6.86%
SwinT-ChARM-3.68%-5.46%-7.10%0.69%
", + "bbox": [ + 271, + 616, + 725, + 723 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In Table 1, we summarize the BD-rate of image codecs across all four dataset with VTM-12.1 as anchor. On average SwinT-ChARM is able to achieve $3 . 8 \\%$ rate reduction compared to VTM12.1. The relative gain from Conv-Hyperprior to SwinT-Hyperprior is on-average $12 \\%$ and that from Conv-ChARM to SwinT-ChARM is on-average $9 \\%$ . Further gain over VTM-12.1 can be obtained by test-time latent optimization (Campos et al., 2019) or full model instance adaptation (van Rozendaal et al., 2021), which are out of the scope of this work. ", + "bbox": [ + 173, + 767, + 825, + 851 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/10e69c89b3c233a7ae4a3c441e524b823d81c407b5038521dbeacaac16cf474c.jpg", + "image_caption": [ + "Figure 4: PSNR vs bitrate curves of P-frame codecs on UVG and MCL-JCV datasets. " + ], + "image_footnote": [], + "bbox": [ + 178, + 98, + 815, + 308 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "RD and BD-rate for video codecs For P-frame compression, we evaluated SwinT-SSF on UVG and MCL-JCV, with RD comparison shown in Figure 4. Again, SwinT transform leads to consistently better RD. Table 2 summarizes BD-rate with our reproduced Conv-SSF model as anchor. We can see that SwinT-SSF achieves an average of $11 \\%$ rate saving over Conv-SSF. Additionally, we show that if SwinT transform is only applied to residual-autoencoder (labeled as SwinT-SSF-Res), it can only get about $4 . 6 \\%$ gain, which indicates ", + "bbox": [ + 174, + 364, + 506, + 517 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/c9a5993c9cb22fa9dd32d4fa50319c55312c5137004d2a0f5353deb198dec113.jpg", + "table_caption": [ + "Table 2: BD-rate of video codecs, with ConvSSF (reproduced) as anchor. " + ], + "table_footnote": [ + "that both flow and residual compression benefit from SwinT as encoder and decoder transforms. Note that SwinT-SSF still lags behind HM, suggesting lots of room for improvement in neural video compression. For per-video breakdown of BD-rate, see Figure 18 and Figure 17 in the Appendix. " + ], + "table_body": "
Video CodecUVGMCL-JCV
HEVC (x265)25.97%25.83%
HEVC (HM)-15.80%-24.96%
SwinT-SSF-12.35%-10.03%
SwinT-SSF-Res-5.08%-4.16%
", + "bbox": [ + 517, + 405, + 813, + 498 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/0822aca873b44b6e44a7026d0a2e8e40955f25a9013ce213dd5579ea82309b2c.jpg", + "table_caption": [ + "Table 3: Decoding complexity. All models are trained with $\\beta = 0 . 0 0 1$ , evaluated on $7 6 8 \\times 5 1 2$ images (average $0 . 7 \\ \\mathrm { b p p }$ ). Decoding time is broken down into inference time of hyper-decoder $h _ { s }$ and decoder $g _ { s }$ , entropy decoding time of hyper-code $\\hat { \\mathbf { z } }$ and code $\\hat { \\mathbf { y } }$ (including inference time of the prior model if it is ChARM). " + ], + "table_footnote": [], + "table_body": "
CodecTime (ms)GMACsPeak memoryModel params
Zhsy9stotal
Conv-Hyperprior5.54.038.2168.9219.33500.50GB21.4M
Conv-ChARM5.34.182.9168.5264.03620.53GB29.3M
SwinT-Hyperprior6.04.838.159.6114.0991.44GB24.7M
SwinT-ChARM5.94.890.760.1167.31111.47GB32.6M
", + "bbox": [ + 197, + 657, + 797, + 763 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Decoding complexity We evaluate the decoding complexity of 4 image codecs on 100 images of size $7 6 8 \\times 5 1 2$ and show the metrics in Table 3, including decoding time, GMACs and GPU peak memory during decoding and total model parameters. The models run with PyTorch 1.9.0 on a workstation with one RTX 2080 Ti GPU. From the table, the inference time of SwinT decoder is less than that of Conv decoder. The entropy decoding time of ChARM prior is about twice than the factorized prior. The total decoding time of SwinT-based models is less than Conv-based models. In ablation study A5, we show a smaller SwinT-Hyperprior with 20.6M parameters has almost the same RD as the SwinT-Hyperprior profiled here. For details on encoding complexity, profiling setup, scaling to image resolution, please refer to Table 4 and Section D.3 in the Appendix. ", + "bbox": [ + 173, + 781, + 825, + 907 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Scaling behavior To see how the BD-rate varies with model size, we scale SwinTHyperprior and Conv-Hyperprior to be twice or half of the size of the base models (i.e. medium size)9. The result is shown in Figure 5. For both types of models, as we reduce the base model size, there is a sharp drop in performance, while doubling model size only leads to marginal gain. Noticeably, SwinT-Hyperpriorsmall is on-par with Conv-Hyperprior-medium even with half of the parameters, and SwinT transforms in general incur fewer MACs per parameter. ", + "bbox": [ + 174, + 104, + 485, + 284 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Figure 1, we further consolidate the decoding latency and scaling behavior study into a single plot and show that SwinT-ChARM runs at comparable speed as VTM-12.1 while achiev", + "bbox": [ + 174, + 291, + 483, + 347 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/83b5f8b09e765aaa304f362254faa6176e752c240ac9e86af21e4c546d40f748.jpg", + "image_caption": [ + "Figure 5: Model size scaling. " + ], + "image_footnote": [], + "bbox": [ + 503, + 111, + 815, + 299 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "ing better performance,10 as opposed to state-of-the-art neural codecs with spatial autoregressive prior that decodes orders of magnitude slower. ", + "bbox": [ + 174, + 347, + 826, + 375 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 393, + 285, + 407 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Latent correlation One of the motivating principles of transform coding is that simple coding can be made more effective in the transform domain than in the original signal space (Goyal, 2001; Balle et al., 2021). A desirable transform would decorrelate the source signal so that simple scalar ´ quantization and factorized entropy model can be applied without constraining coding performance. In most mature neural compression solutions, uniform scalar quantization is adopted together with a learned factorized or conditionally factorized Gaussian prior distribution. It is critical, then, to effectively factorize and Gaussianize the source distribution so that coding overhead can be minimized. ", + "bbox": [ + 173, + 420, + 826, + 518 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/3ea711cac4732af5275d0e783c49841da96e7da69872c0da6824f7d1773adecb.jpg", + "image_caption": [ + "Figure 6: Spatial correlation11of $( \\mathbf { y } - \\pmb { \\mu } ) / \\pmb { \\sigma }$ with models trained at $\\beta = 0 . 0 0 1$ . SwinT-Hyperprior (right) achieves uniformly smaller correlation than Conv-Hyperprior (left). " + ], + "image_footnote": [], + "bbox": [ + 302, + 531, + 699, + 683 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Specifically, in hyperprior based models (Balle et al., 2018),´ ${ \\bar { \\mathbf { y } } } \\triangleq ( \\mathbf { y } - { \\pmb { \\mu } } ) / \\sigma$ is modeled as a standard spherical normal vector. The effectiveness of the analysis transform $g _ { a }$ can then be evaluated by measuring how much correlation there is among different elements in $\\bar { \\mathbf { y } }$ . We are particularly interested in measuring the correlation between nearby spatial positions, which are heavily correlated in the source domain for natural images. In Figure 6, we visualize the normalized spatial correlation of $\\bar { \\mathbf { y } }$ averaged over all latent channels, and compare Conv-Hyperprior with SwinT-Hyperprior at $\\beta = 0 . 0 0 1$ . It can be observed that while both lead to small cross-correlations, Swin-Transformer does a much better job with uniformly smaller correlation values, and the observation is consistent with other $\\beta$ values, which are provided in Figure 20 in the Appendix. This suggests that transformer based transforms incur less redundancy across different spatial latent locations compared with convolutional ones, leading to an overall better rate-distortion trade-off. The larger spatial correlation (and thus redundancy) in Conv-Hyperprior also explains why a compute-heavy spatial auto-regressive model is often needed to improve RD with convolutional based transforms (Minnen et al., 2018; Lee et al., 2019; Ma et al., 2021; Guo et al., 2021; Wu et al., 2020). Figure 6 also reveals that most of the correlation of a latent comes from the four elements surrounding it. This suggests that a checkerboard-based conditional prior model (He et al., 2021) may yield further coding gain. ", + "bbox": [ + 173, + 731, + 825, + 843 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/98f589949ffaa9a174cd05d3842dd7b09177655ebbe15e92154085910ba1e599.jpg", + "image_caption": [ + "Figure 7: Comparison of effective receptive field (ERF) of the encoders $g _ { a }$ , which is visualized as absolution gradients of the center pixel in the latent (i.e. $d \\mathbf { y } / d \\mathbf { x } )$ with respect to the input image. The plot shows the close up of the gradient maps averaged over all channels in each input of test images/videos. Check Figure 21 for the ERF of the composed encoding transform $h _ { a } \\circ g _ { a }$ . " + ], + "image_footnote": [], + "bbox": [ + 183, + 224, + 816, + 439 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Effective receptive field Intra prediction in HEVC or AV1 only rely on left and top boarders of the current coding block (Sullivan et al., 2012; Chen et al., 2018), except for intra block copy for screen content $\\mathrm { { X u } }$ et al., 2016). We would like to see how large the effective receptive field (ERF) (Luo et al., 2017) of SwinT encoders compared to Conv encoders. The theoretical receptive field of the encoders $( g _ { a } , h _ { a } \\circ g _ { a } )$ in SwinT-based codecs is much larger than that of Conv-based codecs. However comparing Figure $\\mathrm { 7 a }$ with 7e and Figure 7b with 7f, the ERF of SwinT encoders after training is even smaller than Conv encoders. When we examine the ERF of the released Swin transformers for classification, detection and segmentation tasks, they are all spanning the whole input image. This contrast suggests that (natural) image compression with rate-distortion objective is a local task, even with transformer-based nonlinear transforms. We further look into P-frame compression models, particularly the ERF of two types of transforms in flow codec and residual codec, as shown in Figure 7d & 7h, and Figure 7c & $7 \\mathrm { g }$ . Clearly for flow codec, SwinT transform has much larger ERF than the convolution counterpart. For residual codec, the ERF of SwinT transforms is similar to image (I-frame) compression case. This shows of flexibility of SwinT encoders to attend to longer or shorter range depending on the tasks. To get a better picture of the behavior of attention layers in SwinT transforms, we also show the attention distance in each layer in Figure 22. ", + "bbox": [ + 174, + 518, + 825, + 741 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Progressive decoding The ERF in the previous section shows the behavior of the encoder transforms, here we further investigate the decoder transforms through the lens of progressive decoding (Rippel et al., 2014; Minnen & Singh, 2020; Lu et al., 2021). Initialized with the prior mean, the input to the decoder is progressively updated with the dequantized latent $\\hat { \\mathbf { y } }$ in terms of coding units, leading to gradually improved reconstruction quality. For the latent with shape $( C , H , W )$ , we consider three types of coding units, i.e. per channel $( 1 , H , W )$ , per pixel $( C , 1 , 1 )$ , per element $( 1 , 1 , 1 )$ . The coding units are ordered by the sum of prior std of all elements within each unit. The RD curves of progressive decoding for SwinT-Hyperprior and Conv-Hyperprior are shown in Figure 8a, which closely follow each other when ordered by channel or element, but significantly apart when ordered by pixel (spatial dim). Particularly, we show an extreme case when the half pixels in the latent (masked by checkerboard pattern) are updated with dequantized values, corresponding to the two scatter points in Figure 8a. One visual example (CLIC2021 test) is shown in Figure 8b under this setup, where we can clearly see SwinT decoder achieves better reconstruction quality than the Conv decoder, mainly in terms of more localized response to a single latent pixel. This is potentially useful for region-of-interest decompression. More visual examples are shown in Figure 26. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 826, + 146 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/f6676a10d3d6d6c46c1baa40f109ee455e71dff1f0f3da4fb9b130557ad86082.jpg", + "image_caption": [ + "Figure 8: (Left) Progressive decoding according to the order of the sum of prior std of all elements within each latent coding unit, which can be one latent channel, pixel or element. SwinT-Hyperprior archives about 2dB better reconstruction at the same bitrate than Conv-Hyperprior for per-pixel progressive decoding. (Right) The reconstructions with SwinT-Hyperprior (top) and Conv-Hyperprior (bottom) for the latent masked with checkerboard pattern corresponding to the two plus markers on the left subfigure. The SwinT decoder has more localized reconstruction to a single latent pixel. " + ], + "image_footnote": [], + "bbox": [ + 220, + 157, + 764, + 357 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.4 ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 174, + 452, + 339, + 464 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Relative position bias There are two sources of positional information in SwinT transforms, namely the Space-to-Depth modules and the additive relative position bias (RPB). Even when the RPB is removed, SwinT-Hyperprior still outperforms Conv-Hyperprior across all bitrates, which indicates image compression may not require accurate relative position. ", + "bbox": [ + 174, + 477, + 483, + 588 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Shifted window The original motivation of shifted window design is to promotes the inter-layer feature propagation across nonoverlapping windows. Image compression performance drops slightly when there is no shifted window at all. This further suggests image compression requires local information. ", + "bbox": [ + 174, + 603, + 483, + 700 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The details of ablations A3-A5 in Figure 9 can be found in Section F of the appendix. ", + "bbox": [ + 173, + 707, + 483, + 734 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/45bffad6da55352065d348e6246a8a57c4f6d63bce3b9f324c37dda44d58ef58.jpg", + "image_caption": [ + "Figure 9: Ablation study " + ], + "image_footnote": [], + "bbox": [ + 504, + 459, + 812, + 723 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 753, + 318, + 770 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work we propose Swin transformer based transforms for image and video compression. In the image compression setting, SwinT transform consistently outperforms its convolutional counterpart. Particularly, the proposed SwinT-ChARM model outperforms VTM-12.1 at comparable decoding speed, which, to the best of our knowledge, is the first in learning-based methods. We also show the effectiveness of SwinT transforms when extended to the P-frame compression setting. Compared with convolution transforms, SwinT transforms can spatially decorrelate the latent better, have more flexible receptive field to adapt to tasks that requires either short-range (image) and long-range (motion) information, and better progressive decoding of latent pixels. While pushing the neural image compression to a new level in terms of rate-distortion-computation trade-off, we believe it is only the starting point for developing more efficient transformer-based image and video codecs. ", + "bbox": [ + 173, + 785, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 326, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank Amir Said for developing entropy coding and great advice on data compression in general. We would also appreciate the helpful discussions from Reza Pourreza and Hoang Le, and draft reviews from Auke Wiggers and Johann Brehmer. ", + "bbox": [ + 174, + 127, + 823, + 170 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Eirikur Agustsson, David Minnen, Nick Johnston, Johannes Balle, Sung Jin Hwang, and George Toderici. Scale-space flow for end-to-end optimized video compression. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8503–8512, 2020. 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", + "bbox": [ + 176, + 895, + 821, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Appendix ", + "text_level": 1, + "bbox": [ + 176, + 131, + 316, + 159 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table of Contents ", + "text_level": 1, + "bbox": [ + 174, + 183, + 352, + 202 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A Models 15 ", + "text_level": 1, + "bbox": [ + 212, + 210, + 789, + 223 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1 Convolution baselines . 15 \nA.2 Swin-Transformer based compression models . 15 \nA.3 Model configurations for model size scaling study 16 ", + "bbox": [ + 230, + 226, + 787, + 270 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B Training and Evaluation ", + "text_level": 1, + "bbox": [ + 214, + 282, + 390, + 296 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "17 ", + "text_level": 1, + "bbox": [ + 233, + 284, + 785, + 295 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.1 Training 17 \nB.2 Traditional codec evaluation 19 ", + "bbox": [ + 230, + 299, + 785, + 328 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C BD rate computation 19 ", + "text_level": 1, + "bbox": [ + 214, + 340, + 787, + 353 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C.1 BD rate for image codec . 20 \nC.2 BD rate for video codec 20 ", + "bbox": [ + 222, + 356, + 787, + 385 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D More Results 21 ", + "text_level": 1, + "bbox": [ + 214, + 398, + 785, + 410 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D.1 Image compression 21 \nD.2 Video Compression 22 \nD.3 Coding complexity 22 ", + "bbox": [ + 228, + 414, + 787, + 458 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "E Analysis 24 ", + "text_level": 1, + "bbox": [ + 210, + 470, + 787, + 483 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "E.1 Spatial correlation of latent 24 \nE.2 Effective Receptive Field 25 \nE.3 Rate distribution across latent channels 25 \nE.4 Centered kernel alignment . 25 \nE.5 Progressive decoding 25 ", + "bbox": [ + 233, + 486, + 787, + 561 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "F More ablation studies 26 ", + "text_level": 1, + "bbox": [ + 200, + 574, + 785, + 587 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A MODELS ", + "text_level": 1, + "bbox": [ + 176, + 625, + 284, + 641 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1 CONVOLUTION BASELINES ", + "text_level": 1, + "bbox": [ + 176, + 656, + 405, + 670 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Conv-Hyperprior and Conv-ChARM The architecture of Conv-Hyperprior and ConvChARM are shown in Figure 10 and Figure 11. For both architecture, our base model (i.e. medium size) has the following hyperparameters: $\\begin{array} { r l } { ( C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , C _ { 5 } , C _ { 6 } , C _ { 7 } ) } & { { } = } \\end{array}$ (320, 320, 320, 320, 192, 192, 192). ", + "bbox": [ + 174, + 681, + 825, + 739 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 SWIN-TRANSFORMER BASED COMPRESSION MODELS ", + "text_level": 1, + "bbox": [ + 174, + 756, + 588, + 770 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "SwinT-Hyperprior, SwinT-ChARM For both SwinT-Hyperprior and SwinT-ChARM, we use the same configurations: $( w _ { g } , w _ { h } ) = ( 8 , 4 )$ , $( C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , \\bar { C _ { 5 } } , \\bar { C _ { 6 } } ) = ( 1 2 8 , 1 9 2 , 2 5 6 , 3 2 0 , 1 9 2 , 1 9 2 ) _ { \\mathrm { { t } } }$ , $( d _ { 1 } , d _ { 2 } , d _ { 3 } , d _ { 4 } , d _ { 5 } , d _ { 6 } ) \\stackrel { } { = } ( 2 , 2 , 6 , 2 , 5 , 1 )$ where $C , d .$ , and $w$ are defined in Figure 13 and Figure 2. The head dim is 32 for all attention layers in SwinT-based models. ", + "bbox": [ + 174, + 781, + 825, + 838 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "SwinT-SSF For SwinT transforms used in SSF variant, the first Patch Merge block is with downsample rate of 4 and two other Patch Merge blocks with downsampling rate of 2. Thus the downsampling rate for the encoder is still 16, the same as the image compression models. There are only 3 transformer stages with depths 2, 4, 2. The embedding dim is 96. The number of latent and hyper latent channels are all 192. The window size is 4 for flow codec and 8 for residual codec. ", + "bbox": [ + 174, + 853, + 823, + 922 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/6c74f8cf5f2ec87f6e7bc71fb99a4df061adf60a57a748a53aa9ec1400682875.jpg", + "image_caption": [ + "Figure 10: Conv-Hyperprior " + ], + "image_footnote": [], + "bbox": [ + 171, + 99, + 826, + 318 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/d3188d84b029cbec50c21fa7bd92fbef221ed662cf2e33681017ca7ed5707bc2.jpg", + "image_caption": [ + "Figure 11: Conv-ChARM " + ], + "image_footnote": [], + "bbox": [ + 171, + 354, + 826, + 573 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "SwinT-SSF-Res This is a variant where only residual autoencoder uses SwinT transforms. Same architecture as the residual autoencoder in SwinT-SSF. ", + "bbox": [ + 173, + 625, + 825, + 652 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.3 MODEL CONFIGURATIONS FOR MODEL SIZE SCALING STUDY ", + "bbox": [ + 174, + 670, + 637, + 684 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.3.1 SWINT-HYPERPRIOR ", + "text_level": 1, + "bbox": [ + 176, + 695, + 379, + 710 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Set of model hyperparameters that are common to all experiments: $( d _ { 1 } , d _ { 2 } , d _ { 3 } , d _ { 4 } , d _ { 5 } , d _ { 6 } ) \\ =$ $( 2 , 2 , 6 , 2 , 5 , 1 )$ $( w _ { g } , w _ { h } ) = ( 8 , 4 )$ ", + "bbox": [ + 176, + 719, + 828, + 750 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/818163b165be1ef4b74abb9c188c46f48392588129222252a17f710de3e3cd7c.jpg", + "text": "$$\n( C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , C _ { 5 } , C _ { 6 } ) = ( 9 6 , 1 2 8 , 1 6 0 , 1 9 2 , 9 6 , 1 2 8 )\n$$", + "text_format": "latex", + "bbox": [ + 370, + 763, + 736, + 780 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "SwinT-Hyperprior (medium) $( C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , C _ { 5 } , C _ { 6 } ) = ( 1 2 8 , 1 9 2 , 2 5 6 , 3 2 0 , 1 9 2 , 1 9 2 )$ ", + "bbox": [ + 173, + 792, + 774, + 810 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "SwinT-Hyperprior (large) $( C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , C _ { 5 } , C _ { 6 } ) = ( 1 6 0 , 2 5 6 , 3 5 2 , 4 4 8 , 1 9 2 , 2 5 6 )$ ", + "bbox": [ + 173, + 823, + 754, + 840 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.3.2 CONV-HYPERPRIOR ", + "bbox": [ + 176, + 853, + 372, + 869 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Conv-Hyperprior (small) $( C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , C _ { 5 } , C _ { 6 } , C _ { 7 } ) = ( 1 9 2 , 1 9 2 , 1 9 2 , 1 9 2 , 1 2 8 , 1 2 8 , 1 2 8 )$ ", + "bbox": [ + 171, + 877, + 807, + 895 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Conv-Hyperprior (medium) $( C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , C _ { 5 } , C _ { 6 } , C _ { 7 } ) = ( 3 2 0 , 3 2 0 , 3 2 0 , 3 2 0 , 1 9 2 , 1 9 2 , 1 9 2 )$ ", + "bbox": [ + 174, + 907, + 823, + 925 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/55b8f680fe09a318620b958a7a3d29d6b9e32849b94df29b9507eb533034489c.jpg", + "image_caption": [ + "Figure 12: ChARM architecture. Since $y _ { i }$ can only be decoded after $\\mu _ { i }$ and $\\sigma _ { i }$ is obtained, the $S$ ChARM-blocks are executed sequentially. We use $S = 1 0$ in all our experiments, which is consistent with (Minnen & Singh, 2020). " + ], + "image_footnote": [], + "bbox": [ + 171, + 99, + 825, + 321 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/85ae68c5c4ee814dc45835263e47119f2e733e7749b72163efe572039d0c8568.jpg", + "image_caption": [ + "Figure 13: SwinT-Hyperprior " + ], + "image_footnote": [], + "bbox": [ + 171, + 388, + 826, + 607 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/0e9f22d961baeef973d99d8f1db3e3f5061d2a78f18b1b29621a223ec9fbf0c4.jpg", + "text": "$$\n( C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , C _ { 5 } , C _ { 6 } , C _ { 7 } ) = ( 4 4 8 , 4 4 8 , 4 4 8 , 4 4 8 , 2 5 6 , 2 5 6 , 2 5 6 )\n$$", + "text_format": "latex", + "bbox": [ + 362, + 660, + 802, + 678 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B TRAINING AND EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 700, + 449, + 718 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.1 TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 734, + 289, + 750 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "All image compression models are trained on CLIC2020 training set, which contains both professional and mobile training sets, in total 1,633 high resolution natural images. Conv-Hyperprior and SwinT-Hyperprior are trained with 2M batches. Each batch contains 8 patches of size $2 5 6 \\times 2 5 6$ randomly cropped from the training images. Learning rate starts at $1 0 ^ { - 4 }$ and is reduced to $1 0 ^ { - 5 }$ at 1.8M step. ", + "bbox": [ + 174, + 762, + 825, + 833 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For Conv-ChARM, we first train a model Conv-ChARM at $\\beta = 0 . 0 0 0 1$ from scratch for 2M steps, with it as the starting point, we continue to train other beta values Conv-ChARM- $\\beta$ , $\\beta \\in B$ for $1 . 5 \\mathbf { M }$ steps. For SwinT-ChARM- $\\beta$ , we load the transform weights from the checkpoint at 2M step of the pretrained SwinT-Hyperprior- $\\beta$ , then finetune the transforms together with the random initialized ChARM prior for 1.1M steps. Learning rate starts at $1 0 ^ { - 4 }$ and is reduced to $1 0 ^ { - 5 }$ for the last 100K steps. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Training loss $L = D + \\beta R$ is a weighted combination of distortion $D$ and bitrate $R$ , with $\\beta$ being the Lagrangian multiplier steering rate-distortion trade-off. Distortion $D$ is MSE in RGB color space. To cover a wide range of rate and distortion, for each solution, we train 5 models with $\\bar { \\beta } \\in B = \\{ 0 . 0 0 3 , 0 . 0 0 1 , 0 . 0 0 \\bar { 0 } 3 , 0 . 0 0 0 1 , 0 . 0 0 0 0 3 \\}$ . ", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Usually we need to train longer for the model with larger bitrates (i.e. smaller $\\beta$ ) to converge. Particularly for the results presented in this paper, We train SwinT-Hyperprior-0.00003 for $2 . 5 \\mathbf { M }$ steps instead of 2M steps for the other 4 lower bitrates. ", + "bbox": [ + 176, + 166, + 823, + 208 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For P-frame compression models, we follow the training setup of SSF. Both Conv-SSF and SwinTSSF are trained on Vimeo-90k Dataset (Xue et al., 2019) for 1M steps with learning rate $1 0 ^ { - 4 }$ , batch size of 8, crop size of $2 5 6 \\times 2 5 6$ , followed by 50K steps of training with learning rate $1 0 ^ { - 5 }$ and crop size12 $3 8 4 \\times 2 5 6$ . The models are trained with 8 $\\beta$ values $2 ^ { \\gamma } \\times 1 0 ^ { - 4 } : \\gamma \\in \\{ 0 , 1 , . . . , 7 \\}$ . We adopt one critical trick to stablize the training from (Jaegle et al., 2021; Meister et al., 2018), i.e. to forward each video sequence twice during one optimization step (mini-batch), once in the original frame order, once in the reversed frame order. When this trick is used, we set the batch size to be 4 instead of 8. Finally we add flow loss only between 0 and 200K steps, which we found not critical for stable training but helps improve the RD. ", + "bbox": [ + 174, + 215, + 825, + 340 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For all model training, Adam optimizer is used without weighted decay. Training for 2M steps takes about 10 days and 14 days respectively for Conv-Hyperprior and SwinT-Hyperprior on a single Nvidia V100 GPU. Total training time is about 7.5 days on a single Nvidia V100 GPU. ", + "bbox": [ + 174, + 348, + 825, + 390 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For all models, we use mixed quantization during training (Minnen & Singh, 2020), i.e. adding uniform noise to the continuous latent before passing to the prior model, subtracting prior mean from the continuous latent followed by rounding before passing to the decoder transform. ", + "bbox": [ + 174, + 397, + 825, + 439 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2 TRADITIONAL CODEC EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 103, + 462, + 117 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this section, we provide evaluation script used to generate results for traditional codecs. ", + "bbox": [ + 176, + 128, + 761, + 145 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.2.1 IMAGE CODECS ", + "text_level": 1, + "bbox": [ + 174, + 159, + 339, + 172 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "VTM-12.1: VTM-12.1 software is built from https://vcgit.hhi.fraunhofer.de/ jvet/VVCSoftware_VTM/-/tags/VTM-12.1 and we use the script from CompressAI (https://github.com/InterDigitalInc/CompressAI/tree/efc69ea24) for dataset evaluation. Specifically, the following command is issued to gather VTM-12.1 image compression evaluation results: ", + "bbox": [ + 173, + 184, + 823, + 253 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "python -m compressai.utils.bench vtm [path to image folder] -c [path to VVCSoftware_VTM folder]/cfg/encoder_intra_vtm.cfg -b [path to VVCSoftware_VTM folder]/bin -q 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40 ", + "bbox": [ + 174, + 265, + 774, + 321 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "BPG: BPG software is obtained from https://bellard.org/bpg/ and the following commands are used for encoding and decoding. ", + "bbox": [ + 174, + 335, + 821, + 364 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "bpgenc -e x265 -q [0 to 51] -f 444 -o [encoded heic file] [original png file] bpgdec $- \\bigcirc$ [decoded png file] [encoded heic file] ", + "bbox": [ + 173, + 376, + 642, + 417 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.2.2 VIDEO CODECS ", + "text_level": 1, + "bbox": [ + 174, + 433, + 339, + 446 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "HEVC (x265) ", + "text_level": 1, + "bbox": [ + 174, + 457, + 271, + 472 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "ffmpeg -y -pix_fmt yuv420p -s [resolution] -r [frame-rate] -i [input yuv420 raw video] -c:v libx265 -preset medium -crf [9, 12, 15, 18, 21, 24, 27, 30] -tune zerolatency -x265-params \"keyint=12:min-keyint=12:verbose $^ { = 1 }$ \" [output mkv file path] ", + "bbox": [ + 176, + 483, + 740, + 553 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "AVC (x264) ", + "text_level": 1, + "bbox": [ + 174, + 568, + 258, + 582 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "ffmpeg -y -pix_fmt yuv420p -s [resolution] -r [frame-rate] -i [input yuv420 raw video] -c:v libx264 -preset medium -crf [9, 12, 15, 18, 21, 24, 27, 30] -tune zerolatency -x264-params \"keyint $= 1 2$ :min-keyint $= 1 2$ :verbose $^ { = 1 }$ \" [output mkv file path] ", + "bbox": [ + 176, + 594, + 740, + 665 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "HEVC (HM) ", + "text_level": 1, + "bbox": [ + 174, + 679, + 266, + 693 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "[Path to HM folder]/bin/TAppEncoderStatic -c [Path to HM folder]/cfg/encoder_lowdelay_P_main.cfg -i [input yuv raw video] --InputBitDepth $^ { \\underline { { { \\boldsymbol { \\ b } } } } } = 8$ -wdt [width] -hgt [height] -fr [frame-rate] -f [number of frames] -o [output yuv video] -b [encoded bitstream bin file] -ip 12 -q [12, 17, 22, 27, 32, 37, 42] ", + "bbox": [ + 176, + 705, + 712, + 790 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C BD RATE COMPUTATION ", + "text_level": 1, + "bbox": [ + 176, + 809, + 411, + 824 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "def Bjontegaard_Delta_Rate( # rate and psnr in ascending order rate_ref, psnr_ref, # reference rate_new, psnr_new, # new result ): min_psnr $=$ max(psnr_ref[0], psnr_new[0], 30) ", + "bbox": [ + 173, + 840, + 643, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "max_psnr $=$ min(psnr_ref[-1], psnr_new[-1], 44) \nlog_rate_ref $=$ log(rate_ref) \nlog_rate_new $=$ log(rate_new) \nspline_ref $=$ scipy.interpolate.CubicSpline( psnr_ref, log_rate_ref, bc_type $= ^ { \\prime }$ not-a-knot’, extrapolate $=$ True, \n) \nspline_new $=$ scipy.interpolate.CubicSpline( psnr_new, log_rate_new, bc_type $= ^ { \\prime }$ not-a-knot’, extrapolate $=$ True, \n) \ndelta_log_rate $=$ ( spline_new.integrate(min_psnr, max_psnr) - spline_ref.integrate(min_psnr, max_psnr) \n) \ndelta_rate $=$ exp(delta_log_rate / (max_psnr - min_psnr)) \nreturn 100 $\\star$ (delta_rate - 1) ", + "bbox": [ + 210, + 102, + 754, + 410 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.1 BD RATE FOR IMAGE CODEC ", + "text_level": 1, + "bbox": [ + 174, + 440, + 416, + 453 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "# Evaluate BD-rate on an image dataset \nbd_rates $=$ list() \nfor image in image_dataset: # evaluate rate and psnr on reference and new codec # for this image with different qualities rate_ref, psnr_ref $=$ ReferenceCodec(image, $\\mathrm { { q } } \\mathrm { { p } } \\mathrm { { = } } [ \\ \\dots \\ ]$ ) rate_new, psnr_new $=$ NewImageCodec(image, bet $\\mathsf { a } = [ \\hdots ]$ ) bd_rates.append( Bjontegaard_Delta_Rate( rate_ref, psnr_ref, rate_new, psnr_new, ) ) \n# BD is computed per image and then averaged \nbd_rate $=$ bd_rates.mean() ", + "bbox": [ + 174, + 467, + 732, + 672 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.2 BD RATE FOR VIDEO CODEC ", + "text_level": 1, + "bbox": [ + 176, + 690, + 415, + 703 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "# Evaluate BD-rate on a video dataset \nbd_rates $=$ list() \nfor video in video_dataset: # evaluate rate and psnr on reference and new codec # for this video with different qualities rate_ref, psnr_ref $=$ ReferenceCodec(video, $\\mathrm { { q } } \\mathrm { { p } } \\mathrm { { = } } [ \\ \\dots \\ ]$ ) rate_new, psnr_new $=$ NewVideoCodec(video, bet $\\mathsf { a } = [ \\hdots ]$ ) bd_rates.append( Bjontegaard_Delta_Rate( rate_ref, psnr_ref, rate_new, psnr_new, ) ) \n# BD is computed per video and then averaged \nbd_rate $=$ bd_rates.mean() ", + "bbox": [ + 173, + 717, + 732, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D MORE RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 142, + 344, + 159 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "D.1 IMAGE COMPRESSION ", + "text_level": 1, + "bbox": [ + 176, + 175, + 370, + 190 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Additional rate-distortion results on CLIC2021, Tecnick, and JPEG-AI are provided in Figure 14, Figure 15, and Figure 16. ", + "bbox": [ + 173, + 203, + 823, + 232 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For a complete comparison of results from existing literatures, we provide a summary RD plot of all neural image codec solutions known to us in Figure 28 on Kodak. In Figure 29 and Figure 30, we plot the percentage rate saving with BPG444 and VTM-12.1 as reference, respectively. ", + "bbox": [ + 173, + 238, + 825, + 281 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/7ab7602f540c02539c9f95d4b8795e3ef9038a651599df83d481292576ebbd42.jpg", + "image_caption": [ + "Figure 14: Comparison of compression efficiency on CLIC Test 2021. " + ], + "image_footnote": [], + "bbox": [ + 173, + 297, + 803, + 560 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/8a12ecc6fcff5fc655a74c203e81c98e50d1b20e436c74b412124c008a3ad9f2.jpg", + "image_caption": [ + "Figure 15: Comparison of compression efficiency on Tecnick Test. " + ], + "image_footnote": [], + "bbox": [ + 173, + 614, + 802, + 882 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/4efe5cc91732f98f03d5246316e55bfa6106ccd9eda7c7d91a5b06cc93b8a571.jpg", + "image_caption": [ + "Figure 16: Comparison of compression efficiency on JPEG AI Test. " + ], + "image_footnote": [], + "bbox": [ + 173, + 102, + 795, + 375 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "D.2 VIDEO COMPRESSION ", + "text_level": 1, + "bbox": [ + 174, + 428, + 372, + 443 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In Figure 17 and Figure 18, we provide performance comparison of Conv-SSF, SwinT-SSF, HEVC $\\left( \\mathbf { \\boldsymbol { x } } 2 6 5 \\right)$ , and AVC $\\left( \\mathrm { x } 2 6 4 \\right)$ with per-video breakdown. ", + "bbox": [ + 166, + 454, + 825, + 483 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/3a40efe0bdab280351af34cae86c47e674e911c6c4a85725941323750e59c743.jpg", + "image_caption": [ + "Figure 17: Rate savings for each video in the MCL-JCV dataset. Values represent the file size relative to H.264 as estimated by BD rate. [18, 20, 24, 25] are animated sequences. " + ], + "image_footnote": [], + "bbox": [ + 171, + 494, + 826, + 606 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/286511207b71fbc9f7800fcbfdd63750a73b34c42a0337baa2adf3d417780041.jpg", + "image_caption": [ + "Figure 18: Rate savings for each video in the UVG dataset. Values represent the file size relative to H.264 as estimated by BD rate. " + ], + "image_footnote": [], + "bbox": [ + 300, + 662, + 694, + 787 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "D.3 CODING COMPLEXITY ", + "text_level": 1, + "bbox": [ + 174, + 856, + 370, + 869 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We evaluate the decoding complexity of all neural image codecs in terms of the time for network inference and entropy coding, peak GPU memory, model size, etc. We select 100 high resolution images from CLIC testset and center crop them to three resolutions $( 7 6 8 \\times 5 1 2 , 1 2 8 0 \\times 7 6 8 , 1 7 9 2 \\times$ ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "1024) to see how those metrics scale with image size. Batch size is one for all model inference. We run the experiment on a local workstation with one RTX 2080 Ti GPU, with PyTorch 1.9.0 and Cuda toolkit 11.1. For bit-exact entropy coding, we need to use deterministic13 convolution. The neural networks run on the single GPU and entropy coding runs on CPU with 8 threads. We follow the standard protocols to measure the inference time and peak memory of neural nets, such as GPU warm up and synchronization. File open/close is excluded from coding time measurement. MACs, GPU peak memory, and model parameter count are profiled using get model profile function of deepspeed profiler14. ", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We show more details on the coding complexity of neural image codecs in Figure 19, particularly the linear scaling to image resolution of both SwinT-based and Conv-based models. The break-down of encoding complexity is shown in Table 4. ", + "bbox": [ + 174, + 222, + 825, + 263 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/afbbf183fdfe4427a2ff051f55ae522c31d1043de1e4075297d35cb9124c8726.jpg", + "image_caption": [ + "Figure 19: Detailed profiling on a single RTX 2080 Ti with deterministic conv. Note that while Conv encoder and Conv decoder are of symmetric architectures, Conv decoder takes more time than Conv encoder mainly because Transposed Conv layers in the decoder take much longer than Conv layers in the encoder. " + ], + "image_footnote": [], + "bbox": [ + 200, + 281, + 795, + 608 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "For completeness, we also report the profiling for CPU coding time in Table 5 and Table 6. The evaluation setup is the same as the GPU profiling case, except models are run on the CPU instead (same host machine with Intel(R) Xeon(R) W-2123 CPU $\\textcircled { a } 3 . 6 0 \\mathrm { G H z }$ ). ", + "bbox": [ + 174, + 695, + 825, + 737 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Table 7 reports encoding and decoding time of VTM-12.1 under different quantization parameters (QPs), evaluated on an Intel Core i9-9940 CPU $@$ $) ~ 3 . 3 0 \\mathrm { G H z }$ , averaged over 24 Kodak images. As can be seen from the table, decoding time of VTM-12.1 is a function of reconstruction quality, where longer decoding time is observed for higher quality reconstruction. In Figure 1, the reported VTM12.1 decoding speed corresponds to a QP value of 28, where the bpp value is similar to that obtained by models trained with $\\beta = 0 . 0 0 1$ . It is worth pointing out that VTM-12.1 encoding process is much slower, ranging anywhere from 1 to 5 minutes per image, whereas neural codec runs much faster. ", + "bbox": [ + 174, + 744, + 825, + 842 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/5d489b4b08806257b464a6ba01957413c40f91d863f9944f1258e0f09c68c8f0.jpg", + "table_caption": [ + "Table 4: GPU encoding complexity. All models are trained with $\\beta ~ = ~ 0 . 0 0 1$ , evaluated on the resolution $7 6 8 \\times 5 1 2$ (the average bitrate is around $0 . 7 \\ \\mathsf { b p p }$ ). For the encoding time, we show the network inference time for the encoder $g _ { a }$ , the hyper encoder $h _ { a }$ , the hyper decoder $h _ { s }$ , the entropy encoding time for the hyper latent $\\hat { \\mathbf { z } }$ and the latent $\\hat { \\mathbf { y } }$ (including network inference time for the prior model if it is ChARM). GMACs and GPU peak memory during encoding and the parameter count of the entire model are also listed. " + ], + "table_footnote": [], + "table_body": "
CodecTime (ms)GMACsMemory (GB)Params (M)
gahaZhsytotal
Conv-Hyperprior22.72.32.84.029.862.01020.6221.4
SwinT-Hyperprior57.51.22.44.834.3100.51001.4724.7
Conv-ChARM22.92.42.64.163.495.71140.6629.3
SwinT-ChARM57.81.32.44.868.7135.31121.5132.6
", + "bbox": [ + 179, + 195, + 813, + 303 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/10e48af6d73c305e83933ef6baafc76217648a69f14b34b448aa7a866affa87a.jpg", + "table_caption": [ + "Table 5: CPU decoding complexity. All models are trained with $\\beta = 0 . 0 0 1$ , evaluated on $7 6 8 \\times 5 1 2$ images (average $0 . 7 \\ \\mathrm { b p p }$ ). Decoding time is broken down into inference time of hyper-decoder $h _ { s }$ and decoder $g _ { s }$ , entropy decoding time of hyper-code $\\hat { \\mathbf { z } }$ and code $\\hat { \\mathbf { y } }$ (including inference time of the prior model if it is ChARM). " + ], + "table_footnote": [], + "table_body": "
CodecCPU decoding ' Time (ms)GMACs
Zhsy9stotal
Conv-Hyperprior5.120.930.51636.21703.3350
SwinT-Hyperprior5.718.330.31912.81979.599
Conv-ChARM4.720.9154.61634.31825.6362
SwinT-ChARM5.818.2141.11877.32054.7111
", + "bbox": [ + 250, + 385, + 741, + 491 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/2ea05e828ecb040e4147200f5d9161c3186e469bf66a64bc85c0c1dc3b730742.jpg", + "table_caption": [ + "Table 6: CPU encoding complexity. All models are trained with $\\beta ~ = ~ 0 . 0 0 1$ , evaluated on the resolution $7 6 8 \\times 5 1 2$ (the average bitrate is around $0 . 7 \\ \\mathsf { b p p }$ ). For the encoding time, we show the network inference time for the encoder $g _ { a }$ , the hyper encoder $h _ { a }$ , the hyper decoder $h _ { s }$ , the entropy encoding time for the hyper latent $\\hat { \\mathbf { z } }$ and the latent $\\hat { \\mathbf { y } }$ (including network inference time for the prior model if it is ChARM). " + ], + "table_footnote": [], + "table_body": "
CodecCPU encoding Time (ms)GMACs
gahaZhsytotal
Conv-Hyperprior896.08.92.720.931.1960.9102
SwinT-Hyperprior1781.614.52.618.329.71847.9100
Conv-ChARM895.78.92.520.9146.31075.5114
SwinT-ChARM1697.114.52.618.2133.51867.1112
", + "bbox": [ + 227, + 588, + 766, + 694 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/3431dedb916eaf096c36a289ac12736e18f6a077e9f171f3ff5416eded01bdc9.jpg", + "table_caption": [ + "Table 7: Decoding and encoding time of VTM-12.1 averaged over 24 Kodak images. " + ], + "table_footnote": [], + "table_body": "
QPbits per pixelPSNR (dB)Decoding time(s)Encoding time (s)
162.544144.090.284300.47
280.784436.760.249118.85
400.155729.510.14971.10
", + "bbox": [ + 240, + 736, + 753, + 808 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "E ANALYSIS ", + "text_level": 1, + "bbox": [ + 173, + 834, + 294, + 851 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "E.1 SPATIAL CORRELATION OF LATENT ", + "text_level": 1, + "bbox": [ + 174, + 868, + 459, + 882 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We visualize the spatial correlation map for Conv-Hyperprior and SwinT-Hyperprior at different $\\beta$ in Figure 20. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "E.2 EFFECTIVE RECEPTIVE FIELD ", + "text_level": 1, + "bbox": [ + 176, + 103, + 424, + 117 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "See Figure 21 for the effective receptive field for the composed encoding tranforms $h _ { a } \\circ g _ { a }$ and Figure 22 for the mean attention distance visualization of each head within each transformer layer. ", + "bbox": [ + 173, + 130, + 825, + 159 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "E.3 RATE DISTRIBUTION ACROSS LATENT CHANNELS ", + "text_level": 1, + "bbox": [ + 174, + 178, + 557, + 190 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "It is generally believed that ConvNets learn to extract various features and store them in each channel of the activations. Here we look into the features in the latent channels which are to be quantized and entropy coded to bitstreams. Particularly we order the total bitrate of each channel averaged over Kodak dataset $2 4 7 6 8 \\times 2 5 6$ images). The result is shown in Figure 23. We find an interesting phenomenon across models under different bitrates: there is a cutoff point of the bitrate-vs-channel curve where the bitrate suddenly drops to zero, which manifest the rate constraint in the loss function. As expected, the cutoff index decreases for the model trained for smaller bitrate (larger $\\beta$ ). ", + "bbox": [ + 174, + 203, + 825, + 300 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "E.4 CENTERED KERNEL ALIGNMENT ", + "text_level": 1, + "bbox": [ + 176, + 319, + 441, + 333 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "To investigate the difference or similarity between latent features of Conv-based and SwinT-based models, we resort to a commonly used tool in representation learning called centered kernel alignment (CKA) (Kornblith et al., 2019). We evaluate CKA between each of the Conv latent channel and SwinT latent channel (both models are trained under the same $\\beta$ ) over the 24 Kodak images. There are 320 channels for both Conv latent and SwinT latent, resulting a $3 2 0 \\times 3 2 0$ CKA matrix. The result is shown in Figure 24. The latent channel is ordered by the averaged bitrate of each channel over Kodak images (same as in Section E.3). The CKA matrix has clear block structure, where high similarity region corresponds to the latent channels before the bitrate cutoff in the rate distribution curve (Figure 23). ", + "bbox": [ + 174, + 344, + 825, + 469 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Identification of SwinT and Conv latent channels with CKA Within the block of high similarity (from the CKA matrix), we identify the ‘less’ similar SwinT latent channels with lowest CKA values between this SwinT channel and all other Conv latent channels. For each of the identified SwinT channel, we find the Conv latent channel with the largest CKA value between the two. This way, we are able to identify latent channels of two different models with high similarity. We show the identified top 8 channels in Figure 25. The channels are indeed highly similar, up to a sign flip, even through the two model architectures are quite different. This empirical result is relevant to the literature on the identifiability of generative models (Khemakhem et al., 2020). ", + "bbox": [ + 174, + 486, + 825, + 598 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "E.5 PROGRESSIVE DECODING ", + "text_level": 1, + "bbox": [ + 174, + 616, + 392, + 631 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "More visual examples of reconstructions of checkerboard (spatially) masked latent are provided in Figure 26. ", + "bbox": [ + 174, + 643, + 823, + 671 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Channel-wise progressive decoding Here we visualize the behavior of channel-wise progressive decoding of Conv and SwinT models. For both models, we order the latent channels with a heuristic importance metric: the reduction of distortion over the increase of rate if one channel is included for decoding. We start with the order of bitrate per channel, we pass the leading channels of bitrate order (zero out all rest channels) to the decoder to obtain reconstruction and calculate distortion. We plot the top 8 channels following this importance order. For each channel, we show 6 maps from top to bottom: latent values, mean prediction from the hyper decoder, standard deviation prediction from the hyper decoder, the bitmap, the reconstruction with only current channel, the reconstruction with up to current channel (all leading channels). The result is shown in Figure 27. For Conv models, usually the top 3 important channels are responsible for lightness, and two color components (blueyellow, red-green), similar to the LAB colorspace. The rest of the latent channels are responsible for adding details like texture and edges. For the Swint models, at low bitrate, there is one significantly different channel (the first column in Figure 27b), which is in a coarse scale (with smooth blocks) and responsible for the reconstruction of a nearly constant image with value close to 120 (the mean value of natural image dataset). This latent channel costs extremely small bitrate but reaches PSNR of 13dB. We tried remove this first channel, the progressive reconstruction with the rest leading 7 channels only leads to PSNR around 16dB, instead of 26dB shown in the figure. ", + "bbox": [ + 174, + 688, + 825, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "F MORE ABLATION STUDIES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 426, + 117 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Local self-attention To see if local self-attention is the most important component in transformers, we replace it by depthwise separable convolution block15 (Han et al., 2021; El-Nouby et al., 2021), which performs similar spatial feature aggregation as self-attention, while keeping all other components the same as in the SwinT. We found this change only leads to minor degradation in RD. This suggests other components in transformers such as MLPs and skip connections may also play a big role, other than just self-attention, for the leading performance in our work and many other tasks (Dong et al., 2021). ", + "bbox": [ + 173, + 133, + 825, + 231 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Small depths Upon investigating the mean attention distance as shown in Figure 22, we find the last block in each of the last two encoder stages has about half of its attention heads degenerate to attending to fixed nearly pixels. This suggests redundant transformer blocks at that stage, so we remove those two blocks, i.e. from depths [2, 2, 6, 2] to [2, 2, 5, 1]. The resulting SwinT-Hyperprior has even less parameters (20.6M) than Conv-Hyperprior (21.4M) while with almost no RD loss compared to the larger model. We expect more hyperparameter search will identify models with better RD-complexity trade-off than we currently show in this work. ", + "bbox": [ + 174, + 246, + 825, + 344 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Deeper Conv encoder Deeper models are usually more expressive (Raghu et al., 2017) and the state-of-the-art Conv-based compression models typically use much deeper layers than the encoder in the original Hyperprior model (Balle et al., 2018). As a sanity check on whether deeper con- ´ volutional transforms can outperform SwinT-based encoder transforms with 12 blocks, we take an existing design (Chen et al., 2021) with residual blocks and attention (sigmoid gating) layers, which has over 50 conv layers in either encoder or decoder, and more parameters than conv baseline. It indeed improves the RD in lower bitrate, but still worse than SwinT-Hyperprior, and gets much worse in higher bitrate. This is probably the reason that compression models based on this type of transforms did not report results at higher bitrates. ", + "bbox": [ + 174, + 359, + 825, + 484 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/5ece8611c78cfb8b31d7c22e595b1ace3757dff67e6292719fac0c9d5b0b11be.jpg", + "image_caption": [ + "Figure 20: Spatial correlation of $( \\mathbf y - \\pmb \\mu ( \\hat { \\mathbf z } ) ) / \\pmb \\sigma ( \\hat { \\mathbf z } )$ , averaged across all latent elements of all images on Kodak. The value with index $( i , j )$ corresponds to the normalized cross-correlation of latents at spatial location $( w , h )$ and $( w + i , h + j )$ . Left column corresponds to Conv-Hyperprior and right columns corresponds to SwinT-Hyperprior. Each row shows a pair of models trained at the same $\\beta$ , where the $\\beta$ values from top to bottom are 0.0001, 0.0003, 0.001, 0.003. A consistent observation across all these models is that SwinT-Hyperprior achieves uniformly smaller correlation than its convolutional counterpart. As $\\beta$ gets smaller (rate becomes lower), the correlation in both models increase, with SwinT-Hyperprior increasing much slower than Conv-Hyperprior. " + ], + "image_footnote": [], + "bbox": [ + 284, + 98, + 712, + 785 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/8dc14536a3f979b2f5a6f5a9e1c3bd15857d4ad29ec6965a5ec1f138b141ec38.jpg", + "image_caption": [ + "Figure 21: Comparison of effective receptive field (ERF) of the composed encoders $h _ { a } \\circ g _ { a }$ . The ERF is visualized as the absolution gradients of the center pixel in the hyper latent (i.e. $d z / d x )$ with respect to the input images/videos, specifically 24 Kodak images cropped to $5 1 2 \\times 5 1 2$ for image codecs (the left four models, all with $\\beta = 0 . 0 0 3$ ), and 24 randomly selected batches from UVG video dataset cropped to $7 6 8 \\times 7 6 8$ for P-frame codecs (the right four models, all with $\\beta = 0 . 0 0 0 8 )$ . " + ], + "image_footnote": [], + "bbox": [ + 184, + 107, + 816, + 319 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "images/c5039bf6854bf4d81268d94d405759346e2b1c860b31a6c5cb51272aa26b25bc.jpg", + "image_caption": [ + "Figure 22: Mean attention distance of SwinT-Hyperprior models evaluated on Kodak. It is calculated as the average relative distance between each query and key weighted by the attention weight for each query, in each head of each layer. Each vertical color bar in the figure shows the mean and 1-std for the mean attention distance of one head, dashed black bar separates heads from different transformer stages (feature resolutions). The order of input to output is from left to right within each figure. Lower bitrate models have smaller attention distance. " + ], + "image_footnote": [], + "bbox": [ + 204, + 465, + 794, + 801 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "images/5f7c0ab1dd30745d88a2920fd1c5c847c8f3230d0f51de87f37fd767d31a8a5e.jpg", + "image_caption": [ + "Figure 23: Rate distribution of latent channels. Five lines with the same colormap from bright to dark corresponds to the model trained under five $\\beta$ values from small to large (i.e. overall bitrate from large to small). The cutoff index decreases as the total rate goes down. " + ], + "image_footnote": [], + "bbox": [ + 204, + 132, + 794, + 493 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/1cb5bd9eb4c17c6f31f423cf32f51366899fe2f31ef884a77bf1709644b54efb.jpg", + "image_caption": [ + "Figure 24: Centered Kernel Alignment between 320 SwinT latent channels and 320 Conv latent channels. Blocks with high similarity correspond to latent channels before the cutoff in Figure 23. " + ], + "image_footnote": [], + "bbox": [ + 133, + 617, + 866, + 851 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/c901556d465b19fc0756842613c7d394905d48e27bf0fe0ba0da6e4a35528ada.jpg", + "image_caption": [ + "Figure 25: Identification of latent features from SwinT and Conv models with CKA (on 23rd image in Kodak). For each subfigure, top row shows 8 channels of SwinT latents with the smallest mean CKA values between this channel and all Conv latent channels (before the rate cutoff in Figure 23). The corresponding image on the bottom shows the Conv latent channel with the largest CKA with the top SwinT channel. The identified channels are highly similar, up to sign flip. " + ], + "image_footnote": [], + "bbox": [ + 178, + 285, + 823, + 657 + ], + "page_idx": 29 + }, + { + "type": "image", + "img_path": "images/d8a8ec45f471df232b9a64901dc865fa4b8cf431b8a4a87acc3ef484cc334c6a.jpg", + "image_caption": [ + "Figure 26: More examples of reconstructions with SwinT-Hyperprior (top row) and ConvHyperprior (bottom row) for the latent masked with checkerboard pattern where half of the latent dequantized values are replaced with the corresponding prior mean before passing to the decoder $g _ { s }$ . The example image shown in Figure 8b is andy-kelly-0E vhMVqL9g-unsplash from CLIC2021 test, same as the two images shown here. " + ], + "image_footnote": [], + "bbox": [ + 200, + 303, + 797, + 636 + ], + "page_idx": 30 + }, + { + "type": "image", + "img_path": "images/6dc9e2834f1aa66964d2bd3084e432f186f229a8e2533c6ab9be89184c3981ff.jpg", + "image_caption": [ + "Figure 27: Channel-wise progressive decoding. The 3 leading latent channels following an importance order are responsible for grayscale, and two color components of the reconstruction (second to the last row). For SwinT models, we find a coarse-scale latent channel (the left most channel) which costs negligible bits but is important for reconstruction (without it, the 7 other leading channels can only decode to a reconstruction with PSNR around 16dB). For each subfigure, from top to bottom: latent, prior mean, prior std, bitmap, reconstruction with current channel, reconstruction with leading channels (up to current channel). " + ], + "image_footnote": [], + "bbox": [ + 173, + 196, + 823, + 710 + ], + "page_idx": 31 + }, + { + "type": "image", + "img_path": "images/f6bd7b5719f559fbc2984a63fa81a51b0956488dc35d9737c8c62bd49b5963bf.jpg", + "image_caption": [ + "Figure 28: Rate-distortion performance on Kodak, comparing with existing works (Cheng et al., 2020; Xie et al., 2021; Ma et al., 2021; Guo et al., 2021; Balle et al., 2018; Minnen & Singh, 2020; ´ Lee et al., 2019; Wu et al., 2020; Minnen et al., 2018). Dashed line-style indicates the use of spatial autoregressive model (block-wise or pixel-wise) as prior model. " + ], + "image_footnote": [], + "bbox": [ + 173, + 123, + 821, + 845 + ], + "page_idx": 32 + }, + { + "type": "image", + "img_path": "images/2eb11b18fc99d2f1b506ac8ce75a6913a7baad268392a0090c4a65e45b47765e.jpg", + "image_caption": [ + "Figure 29: Percentage of rate-saving over BPG444 evaluated on Kodak (extended version of Figure 3b), comparing with existing works (Cheng et al., 2020; Xie et al., 2021; Ma et al., 2021; Guo et al., 2021; Balle et al., 2018; Minnen & Singh, 2020; Lee et al., 2019; Wu et al., 2020; Minnen ´ et al., 2018). Dashed line-style indicates the use of spatial autoregressive model (block-wise or pixel-wise) as prior model. " + ], + "image_footnote": [], + "bbox": [ + 173, + 116, + 821, + 829 + ], + "page_idx": 33 + }, + { + "type": "image", + "img_path": "images/9af6113360108da2dd1cec305a98ec52eeafe9ad35b84e8274fb9925230f9ee0.jpg", + "image_caption": [ + "Figure 30: Percentage of rate-saving over VTM-12.1 evaluated on Kodak (extended version of Figure 3b), comparing with existing works (Cheng et al., 2020; Xie et al., 2021; Ma et al., 2021; Guo et al., 2021; Balle et al., 2018; Minnen & Singh, 2020; Lee et al., 2019; Wu et al., 2020; Minnen ´ et al., 2018). Dashed line-style indicates the use of spatial autoregressive model (block-wise or pixel-wise) as prior model. 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Specifically, we show that nonlinear transforms built on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 280, + 469, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 469, + 293 + ], + "score": 1.0, + "content": "Swin-transformers can achieve better compression efficiency than transforms built", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 291, + 469, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 469, + 304 + ], + "score": 1.0, + "content": "on convolutional neural networks (ConvNets), while requiring fewer parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 303, + 469, + 314 + ], + "spans": [ + { + "bbox": [ + 142, + 303, + 469, + 314 + ], + "score": 1.0, + "content": "and shorter decoding time. 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Codecs based on transform coding divide the task of lossy compression into three", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "modularized components: transform, quantization, and entropy coding. All three components can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "be enhanced by deep neural networks: autoencoder networks are adopted as flexible nonlinear trans-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "forms, deep generative models are used as powerful learnable entropy models, and various differen-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 511, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 526 + ], + "score": 1.0, + "content": "tiable quantization schemes are proposed to aid end-to-end training. 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In the SwinT-Hyperprior", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 517, + 504, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 333, + 531 + ], + "score": 1.0, + "content": "model, the ChARM component is removed and instead", + "type": "text" + }, + { + "bbox": [ + 333, + 519, + 342, + 529 + ], + "score": 0.8, + "content": "\\pmb { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 517, + 360, + 531 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 361, + 519, + 370, + 528 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 517, + 504, + 531 + ], + "score": 1.0, + "content": "are directly output by the hyper-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 528, + 332, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 140, + 541 + ], + "score": 1.0, + "content": "decoder", + "type": "text" + }, + { + "bbox": [ + 140, + 529, + 152, + 540 + ], + "score": 0.87, + "content": "h _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 528, + 332, + 541 + ], + "score": 1.0, + "content": "(For details see Figure 13 in Appendix A.1).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 563, + 270, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 272, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 272, + 575 + ], + "score": 1.0, + "content": "3.1 SWINT ENCODER AND DECODER", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 597 + ], + "score": 1.0, + "content": "The original SwinT is proposed as a vision backbone, i.e. an encoder transform with downsampling.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 277, + 607 + ], + "score": 1.0, + "content": "As shown in Figure 2, the SwinT encoder", + "type": "text" + }, + { + "bbox": [ + 278, + 596, + 289, + 606 + ], + "score": 0.84, + "content": "g _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "contains SwinT blocks interleaved with Patch Merge", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "blocks. The Patch Merge block contains Space-to-Depth (for downsampling), LayerNorm, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 615, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 631 + ], + "score": 1.0, + "content": "Linear layers sequentially. SwinT block performs local self-attention within each non-overlapping", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 627, + 504, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 504, + 640 + ], + "score": 1.0, + "content": "window of the feature maps and preserves feature size. Consecutive SwinT blocks at the same", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "feature size shift the window partitioning with respect to the previous block to promote information", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 650, + 340, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 340, + 662 + ], + "score": 1.0, + "content": "propagation across nearby windows in the previous block.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 582, + 505, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 310, + 679 + ], + "score": 1.0, + "content": "We adopt SwinT encoder as the encoder transform", + "type": "text" + }, + { + "bbox": [ + 310, + 668, + 321, + 678 + ], + "score": 0.85, + "content": "g _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "in our model, and extend it to SwinT decoder", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 117, + 689 + ], + "score": 0.82, + "content": "g _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 678, + 262, + 690 + ], + "score": 1.0, + "content": "by reversing the order of blocks in", + "type": "text" + }, + { + "bbox": [ + 262, + 679, + 273, + 689 + ], + "score": 0.83, + "content": "g _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 678, + 506, + 690 + ], + "score": 1.0, + "content": ", and replacing the Patch Merge block with a Patch Split", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "block, which contains Linear, LayerNorm, Depth-to-Space (for upsampling) layers in sequence. The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 699, + 470, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 244, + 712 + ], + "score": 1.0, + "content": "architectures for hyper transforms", + "type": "text" + }, + { + "bbox": [ + 244, + 700, + 270, + 711 + ], + "score": 0.91, + "content": "h _ { a } , h _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 699, + 326, + 712 + ], + "score": 1.0, + "content": "are similar to", + "type": "text" + }, + { + "bbox": [ + 327, + 701, + 351, + 711 + ], + "score": 0.89, + "content": "g _ { a } , g _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 699, + 470, + 712 + ], + "score": 1.0, + "content": "with different configurations.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 666, + 506, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "With these four SwinT transforms, we propose two image compression models, SwinT-Hyperprior", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "and SwinT-ChARM, whose prior and hyper prior models are respectively the same as in Conv-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "Hyperprior and Conv-ChARM introduced in Section 2. The full model architectures are shown in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 203, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 203, + 127 + ], + "score": 1.0, + "content": "Figure 2 and Figure 13.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 109, + 145, + 303, + 156 + ], + "lines": [ + { + "bbox": [ + 106, + 145, + 304, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 304, + 158 + ], + "score": 1.0, + "content": "3.2 EXTENSION TO P-FRAME COMPRESSION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "To investigate the effectiveness of SwinT transforms for video compression, we study one popular", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 179, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 505, + 191 + ], + "score": 1.0, + "content": "P-frame compression model called Scale-Space Flow (SSF) (Agustsson et al., 2020). There are three", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "score": 1.0, + "content": "instances of Conv-Hyperprior in SSF, which are respectively for compressing I-frames, scale-space", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 214 + ], + "score": 1.0, + "content": "flow and residual. We propose a SwinT variant, referred to as SwinT-SSF, which is obtained by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 211, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 213, + 226 + ], + "score": 1.0, + "content": "replacing Conv transforms", + "type": "text" + }, + { + "bbox": [ + 214, + 213, + 238, + 223 + ], + "score": 0.89, + "content": "g _ { a } , g _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 211, + 506, + 226 + ], + "score": 1.0, + "content": "in the flow codec and residual codec of SSF with SwinT tranforms.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "To stabilize training of flow codec in SwinT-SSF, we need to remove all LayerNorm layers and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 504, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 504, + 245 + ], + "score": 1.0, + "content": "reduce the window size (e.g. from 8 to 4). The baseline SSF model will be referred as Conv-SSF.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Even though we build our solution on top of SSF, we believe this general extension can be applied", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 492, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 492, + 268 + ], + "score": 1.0, + "content": "to other ConvNet-based video compression models (Rippel et al., 2021; Hu et al., 2021) as well.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 289, + 280, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 282, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 282, + 303 + ], + "score": 1.0, + "content": "4 EXPERIMENTS AND ANALYSIS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 316, + 218, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 219, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 219, + 328 + ], + "score": 1.0, + "content": "4.1 EXPERIMENT SETUP", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "Training All image compression models are trained on the CLIC2020 training set. Conv-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "Hyperprior and SwinT-Hyperprior are trained with 2M batches. Conv-ChARM and SwinT-ChARM", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 382, + 374 + ], + "score": 1.0, + "content": "are trained with 3.5M and 3.1M steps. Each batch contains 8 random", + "type": "text" + }, + { + "bbox": [ + 383, + 361, + 425, + 371 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "crops from training", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 196, + 384 + ], + "score": 1.0, + "content": "images. Training loss", + "type": "text" + }, + { + "bbox": [ + 196, + 372, + 253, + 383 + ], + "score": 0.92, + "content": "L = D + \\beta R", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 372, + 415, + 384 + ], + "score": 1.0, + "content": "is a weighted combination of distortion", + "type": "text" + }, + { + "bbox": [ + 415, + 372, + 425, + 382 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 372, + 471, + 384 + ], + "score": 1.0, + "content": "and bitrate", + "type": "text" + }, + { + "bbox": [ + 472, + 372, + 481, + 382 + ], + "score": 0.78, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 372, + 505, + 384 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 114, + 394 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 383, + 426, + 395 + ], + "score": 1.0, + "content": "being the Lagrangian multiplier steering rate-distortion trade-off. Distortion", + "type": "text" + }, + { + "bbox": [ + 426, + 383, + 436, + 393 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "is MSE in RGB", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "color space. To cover a wide range of rate and distortion, for each solution, we train 5 models with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 405, + 495, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 287, + 417 + ], + "score": 0.83, + "content": "\\beta \\in \\{ \\bar { 0 . } 0 0 3 , 0 . 0 0 1 , 0 . 0 0 0 3 , 0 . 0 0 0 \\bar { 1 } , 0 . 0 0 0 0 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 405, + 495, + 417 + ], + "score": 1.0, + "content": ". The detailed training schedule is in Appendix B.1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "For P-frame compression models, we follow the training setup of SSF. Both Conv-SSF and SwinT-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 455, + 445 + ], + "score": 1.0, + "content": "SSF are trained on Vimeo-90k Dataset (Xue et al., 2019) for 1M steps with learning rate", + "type": "text" + }, + { + "bbox": [ + 456, + 432, + 477, + 443 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 432, + 506, + 445 + ], + "score": 1.0, + "content": ", batch", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 442, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 193, + 457 + ], + "score": 1.0, + "content": "size of 8, crop size of", + "type": "text" + }, + { + "bbox": [ + 193, + 444, + 234, + 454 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 442, + 445, + 457 + ], + "score": 1.0, + "content": ", followed by 50K steps of training with learning rate", + "type": "text" + }, + { + "bbox": [ + 446, + 443, + 467, + 454 + ], + "score": 0.9, + "content": "1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 442, + 506, + 457 + ], + "score": 1.0, + "content": "and crop", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 124, + 468 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 455, + 167, + 465 + ], + "score": 0.9, + "content": "3 8 4 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 453, + 291, + 468 + ], + "score": 1.0, + "content": ". The models are trained with 8", + "type": "text" + }, + { + "bbox": [ + 292, + 455, + 300, + 466 + ], + "score": 0.72, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 453, + 328, + 468 + ], + "score": 1.0, + "content": "values", + "type": "text" + }, + { + "bbox": [ + 328, + 454, + 444, + 466 + ], + "score": 0.93, + "content": "2 ^ { \\gamma } \\times 1 0 ^ { - \\bar { 4 } } : \\gamma \\in \\{ 0 , 1 , \\bar { . . . , 7 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 453, + 506, + 468 + ], + "score": 1.0, + "content": ". We adopt one", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "critical trick to stablize the training from (Jaegle et al., 2021; Meister et al., 2018), i.e. to forward", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "score": 1.0, + "content": "each video sequence twice during one optimization step (mini-batch), once in the original frame", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 351, + 500 + ], + "score": 1.0, + "content": "order, once in the reversed frame order. Finally we add flow", + "type": "text" + }, + { + "bbox": [ + 352, + 487, + 372, + 498 + ], + "score": 0.81, + "content": "\\mathrm { l o s s } ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "only between 0 and 200K steps,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 379, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 379, + 510 + ], + "score": 1.0, + "content": "which we found not critical for stable training but improves the RD.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "Evaluation We evaluate image compression models on 4 datasets: Kodak (Kodak, 1999),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "CLIC2021 testset (CLIC, 2021), Tecnick testset (Asuni & Giachetti, 2014), and JPEG-AI test-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "set (JPEG-AI, 2020). We use BPG and VTM-12.1 to code the images in YUV444 mode, and then", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "calculate PSNR in RGB. For a fair comparison all images are cropped to multiples of 256 to avoid", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 215, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 215, + 585 + ], + "score": 1.0, + "content": "padding for neural codecs.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "We evaluated P-frame models on UVG (Mercat et al., 2020) 6 and MCL-JCV (Wang et al., 2016),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "and compare them with the test model implementation of HEVC, referred to as HEVC (HM), and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 378, + 623 + ], + "score": 1.0, + "content": "open source library implementation of HEVC, refered to as HEVC", + "type": "text" + }, + { + "bbox": [ + 379, + 610, + 405, + 621 + ], + "score": 0.5, + "content": "\\left( \\mathrm { x } 2 6 5 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 610, + 505, + 623 + ], + "score": 1.0, + "content": ". To align configuration,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 432, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 432, + 632 + ], + "score": 1.0, + "content": "all video codecs are evaluated in low-delay-P model with a fixed GOP size of 12.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 108, + 637, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 651 + ], + "score": 1.0, + "content": "Besides rate-distortion curves, we also evaluate different models using BD-rate (Tan et al., 2016),", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "which represents the average bitrate savings for the same reconstruction quality. For image codecs,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "BD-rate is computed for each image and then averaged across all images; for video codecs, BD-rate", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 669, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 506, + 685 + ], + "score": 1.0, + "content": "is computed for each video and then averaged across all videos. More details on testset preprocess-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 398, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 398, + 694 + ], + "score": 1.0, + "content": "ing, and traditional codecs configurations can be found in Appendix B.2.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 116, + 710, + 494, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 707, + 429, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 707, + 429, + 724 + ], + "score": 1.0, + "content": "5We did not observe RD improvement when applying flow loss to Conv-SSF training.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 719, + 493, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 493, + 734 + ], + "score": 1.0, + "content": "6We use the original 7 UVG sequences that are commonly used in other works (Agustsson et al., 2020).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "With these four SwinT transforms, we propose two image compression models, SwinT-Hyperprior", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "and SwinT-ChARM, whose prior and hyper prior models are respectively the same as in Conv-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "Hyperprior and Conv-ChARM introduced in Section 2. The full model architectures are shown in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 203, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 203, + 127 + ], + "score": 1.0, + "content": "Figure 2 and Figure 13.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 127 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 145, + 303, + 156 + ], + "lines": [ + { + "bbox": [ + 106, + 145, + 304, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 304, + 158 + ], + "score": 1.0, + "content": "3.2 EXTENSION TO P-FRAME COMPRESSION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "To investigate the effectiveness of SwinT transforms for video compression, we study one popular", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 179, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 505, + 191 + ], + "score": 1.0, + "content": "P-frame compression model called Scale-Space Flow (SSF) (Agustsson et al., 2020). There are three", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "score": 1.0, + "content": "instances of Conv-Hyperprior in SSF, which are respectively for compressing I-frames, scale-space", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 214 + ], + "score": 1.0, + "content": "flow and residual. We propose a SwinT variant, referred to as SwinT-SSF, which is obtained by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 211, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 213, + 226 + ], + "score": 1.0, + "content": "replacing Conv transforms", + "type": "text" + }, + { + "bbox": [ + 214, + 213, + 238, + 223 + ], + "score": 0.89, + "content": "g _ { a } , g _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 211, + 506, + 226 + ], + "score": 1.0, + "content": "in the flow codec and residual codec of SSF with SwinT tranforms.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "To stabilize training of flow codec in SwinT-SSF, we need to remove all LayerNorm layers and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 504, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 504, + 245 + ], + "score": 1.0, + "content": "reduce the window size (e.g. from 8 to 4). The baseline SSF model will be referred as Conv-SSF.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Even though we build our solution on top of SSF, we believe this general extension can be applied", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 492, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 492, + 268 + ], + "score": 1.0, + "content": "to other ConvNet-based video compression models (Rippel et al., 2021; Hu et al., 2021) as well.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 168, + 506, + 268 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 289, + 280, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 282, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 282, + 303 + ], + "score": 1.0, + "content": "4 EXPERIMENTS AND ANALYSIS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 316, + 218, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 219, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 219, + 328 + ], + "score": 1.0, + "content": "4.1 EXPERIMENT SETUP", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "Training All image compression models are trained on the CLIC2020 training set. Conv-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "Hyperprior and SwinT-Hyperprior are trained with 2M batches. Conv-ChARM and SwinT-ChARM", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 382, + 374 + ], + "score": 1.0, + "content": "are trained with 3.5M and 3.1M steps. Each batch contains 8 random", + "type": "text" + }, + { + "bbox": [ + 383, + 361, + 425, + 371 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "crops from training", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 372, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 196, + 384 + ], + "score": 1.0, + "content": "images. Training loss", + "type": "text" + }, + { + "bbox": [ + 196, + 372, + 253, + 383 + ], + "score": 0.92, + "content": "L = D + \\beta R", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 372, + 415, + 384 + ], + "score": 1.0, + "content": "is a weighted combination of distortion", + "type": "text" + }, + { + "bbox": [ + 415, + 372, + 425, + 382 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 372, + 471, + 384 + ], + "score": 1.0, + "content": "and bitrate", + "type": "text" + }, + { + "bbox": [ + 472, + 372, + 481, + 382 + ], + "score": 0.78, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 372, + 505, + 384 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 114, + 394 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 383, + 426, + 395 + ], + "score": 1.0, + "content": "being the Lagrangian multiplier steering rate-distortion trade-off. Distortion", + "type": "text" + }, + { + "bbox": [ + 426, + 383, + 436, + 393 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "is MSE in RGB", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "color space. To cover a wide range of rate and distortion, for each solution, we train 5 models with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 405, + 495, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 287, + 417 + ], + "score": 0.83, + "content": "\\beta \\in \\{ \\bar { 0 . } 0 0 3 , 0 . 0 0 1 , 0 . 0 0 0 3 , 0 . 0 0 0 \\bar { 1 } , 0 . 0 0 0 0 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 405, + 495, + 417 + ], + "score": 1.0, + "content": ". The detailed training schedule is in Appendix B.1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 338, + 506, + 417 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 421, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "For P-frame compression models, we follow the training setup of SSF. Both Conv-SSF and SwinT-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 455, + 445 + ], + "score": 1.0, + "content": "SSF are trained on Vimeo-90k Dataset (Xue et al., 2019) for 1M steps with learning rate", + "type": "text" + }, + { + "bbox": [ + 456, + 432, + 477, + 443 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 432, + 506, + 445 + ], + "score": 1.0, + "content": ", batch", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 442, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 193, + 457 + ], + "score": 1.0, + "content": "size of 8, crop size of", + "type": "text" + }, + { + "bbox": [ + 193, + 444, + 234, + 454 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 442, + 445, + 457 + ], + "score": 1.0, + "content": ", followed by 50K steps of training with learning rate", + "type": "text" + }, + { + "bbox": [ + 446, + 443, + 467, + 454 + ], + "score": 0.9, + "content": "1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 442, + 506, + 457 + ], + "score": 1.0, + "content": "and crop", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 453, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 124, + 468 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 455, + 167, + 465 + ], + "score": 0.9, + "content": "3 8 4 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 453, + 291, + 468 + ], + "score": 1.0, + "content": ". The models are trained with 8", + "type": "text" + }, + { + "bbox": [ + 292, + 455, + 300, + 466 + ], + "score": 0.72, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 453, + 328, + 468 + ], + "score": 1.0, + "content": "values", + "type": "text" + }, + { + "bbox": [ + 328, + 454, + 444, + 466 + ], + "score": 0.93, + "content": "2 ^ { \\gamma } \\times 1 0 ^ { - \\bar { 4 } } : \\gamma \\in \\{ 0 , 1 , \\bar { . . . , 7 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 453, + 506, + 468 + ], + "score": 1.0, + "content": ". We adopt one", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "critical trick to stablize the training from (Jaegle et al., 2021; Meister et al., 2018), i.e. to forward", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "score": 1.0, + "content": "each video sequence twice during one optimization step (mini-batch), once in the original frame", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 351, + 500 + ], + "score": 1.0, + "content": "order, once in the reversed frame order. Finally we add flow", + "type": "text" + }, + { + "bbox": [ + 352, + 487, + 372, + 498 + ], + "score": 0.81, + "content": "\\mathrm { l o s s } ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "only between 0 and 200K steps,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 498, + 379, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 379, + 510 + ], + "score": 1.0, + "content": "which we found not critical for stable training but improves the RD.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 421, + 506, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "Evaluation We evaluate image compression models on 4 datasets: Kodak (Kodak, 1999),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "CLIC2021 testset (CLIC, 2021), Tecnick testset (Asuni & Giachetti, 2014), and JPEG-AI test-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "set (JPEG-AI, 2020). We use BPG and VTM-12.1 to code the images in YUV444 mode, and then", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "calculate PSNR in RGB. For a fair comparison all images are cropped to multiples of 256 to avoid", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 571, + 215, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 215, + 585 + ], + "score": 1.0, + "content": "padding for neural codecs.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 527, + 506, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "We evaluated P-frame models on UVG (Mercat et al., 2020) 6 and MCL-JCV (Wang et al., 2016),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "and compare them with the test model implementation of HEVC, referred to as HEVC (HM), and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 378, + 623 + ], + "score": 1.0, + "content": "open source library implementation of HEVC, refered to as HEVC", + "type": "text" + }, + { + "bbox": [ + 379, + 610, + 405, + 621 + ], + "score": 0.5, + "content": "\\left( \\mathrm { x } 2 6 5 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 610, + 505, + 623 + ], + "score": 1.0, + "content": ". To align configuration,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 432, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 432, + 632 + ], + "score": 1.0, + "content": "all video codecs are evaluated in low-delay-P model with a fixed GOP size of 12.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 587, + 505, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 637, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 651 + ], + "score": 1.0, + "content": "Besides rate-distortion curves, we also evaluate different models using BD-rate (Tan et al., 2016),", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "which represents the average bitrate savings for the same reconstruction quality. For image codecs,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "BD-rate is computed for each image and then averaged across all images; for video codecs, BD-rate", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 669, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 669, + 506, + 685 + ], + "score": 1.0, + "content": "is computed for each video and then averaged across all videos. More details on testset preprocess-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 398, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 398, + 694 + ], + "score": 1.0, + "content": "ing, and traditional codecs configurations can be found in Appendix B.2.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 636, + 506, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 81, + 496, + 290 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 81, + 496, + 290 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 81, + 496, + 290 + ], + "spans": [ + { + "bbox": [ + 111, + 81, + 496, + 290 + ], + "score": 0.97, + "type": "image", + "image_path": "4b3b6a9139045c351c32278f20ebdda0599254c2ee3d526c7e6551263623e34c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 81, + 496, + 150.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 150.66666666666669, + 496, + 220.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 220.33333333333337, + 496, + 290.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 103, + 299, + 503, + 322 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "Figure 3: Comparison of compression efficiency on Kodak8. Note that the encoding time of VTM-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 310, + 478, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 478, + 323 + ], + "score": 1.0, + "content": "12.1 is much longer than all neural codecs, as shown in Table 4 and Table 7 in the Appendix.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 107, + 347, + 170, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 171, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 171, + 360 + ], + "score": 1.0, + "content": "4.2 RESULTS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "RD and BD-rate for image codecs The RD curves for all compared image codecs evaluated on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "Kodak are shown in Figure 3a, and the relative rate reduction of each codec compared to VTM-12.1", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 392, + 308, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 308, + 403 + ], + "score": 1.0, + "content": "at a range of PSNR levels is shown in Figure 3b 7.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 408, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 420 + ], + "score": 1.0, + "content": "As can be seen from Figure 3, SwinT transform consistently outperforms its convolutional counter-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "part; the RD-performance of SwinT-Hyperprior is on-par with Conv-ChARM, despite the simpler", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "prior; SwinT-ChARM outperforms VTM-12.1 across a wide PSNR range. In the Appendix (Fig-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "ure 28 and Figure 30), we further incorporate the results from existing literature known to us for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "a complete comparison. 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Image CodecKodakCLIC2021TecnickJPEG-AI
BPG44420.87%28.45%27.74%27.14%
Conv-Hyperprior11.65%12.23%12.49%20.98%
Conv-ChARM3.44%4.14%3.50%9.59%
SwinT-Hyperprior1.69%0.83%-0.15%6.86%
SwinT-ChARM-3.68%-5.46%-7.10%0.69%
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Image CodecKodakCLIC2021TecnickJPEG-AI
BPG44420.87%28.45%27.74%27.14%
Conv-Hyperprior11.65%12.23%12.49%20.98%
Conv-ChARM3.44%4.14%3.50%9.59%
SwinT-Hyperprior1.69%0.83%-0.15%6.86%
SwinT-ChARM-3.68%-5.46%-7.10%0.69%
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Video CodecUVGMCL-JCV
HEVC (x265)25.97%25.83%
HEVC (HM)-15.80%-24.96%
SwinT-SSF-12.35%-10.03%
SwinT-SSF-Res-5.08%-4.16%
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CodecTime (ms)GMACsPeak memoryModel params
Zhsy9stotal
Conv-Hyperprior5.54.038.2168.9219.33500.50GB21.4M
Conv-ChARM5.34.182.9168.5264.03620.53GB29.3M
SwinT-Hyperprior6.04.838.159.6114.0991.44GB24.7M
SwinT-ChARM5.94.890.760.1167.31111.47GB32.6M
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Video CodecUVGMCL-JCV
HEVC (x265)25.97%25.83%
HEVC (HM)-15.80%-24.96%
SwinT-SSF-12.35%-10.03%
SwinT-SSF-Res-5.08%-4.16%
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CodecTime (ms)GMACsPeak memoryModel params
Zhsy9stotal
Conv-Hyperprior5.54.038.2168.9219.33500.50GB21.4M
Conv-ChARM5.34.182.9168.5264.03620.53GB29.3M
SwinT-Hyperprior6.04.838.159.6114.0991.44GB24.7M
SwinT-ChARM5.94.890.760.1167.31111.47GB32.6M
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The models run with PyTorch 1.9.0 on a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 652, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 506, + 666 + ], + "score": 1.0, + "content": "workstation with one RTX 2080 Ti GPU. From the table, the inference time of SwinT decoder is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "less than that of Conv decoder. The entropy decoding time of ChARM prior is about twice than the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "factorized prior. The total decoding time of SwinT-based models is less than Conv-based models.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "In ablation study A5, we show a smaller SwinT-Hyperprior with 20.6M parameters has almost the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 695, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 695, + 506, + 711 + ], + "score": 1.0, + "content": "same RD as the SwinT-Hyperprior profiled here. For details on encoding complexity, profiling setup,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 446, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 446, + 720 + ], + "score": 1.0, + "content": "scaling to image resolution, please refer to Table 4 and Section D.3 in the Appendix.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 618, + 506, + 720 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 297, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 297, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 297, + 94 + ], + "score": 1.0, + "content": "Scaling behavior To see how the BD-rate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 297, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 297, + 105 + ], + "score": 1.0, + "content": "varies with model size, we scale SwinT-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 298, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 298, + 117 + ], + "score": 1.0, + "content": "Hyperprior and Conv-Hyperprior to be twice or", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 297, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 297, + 127 + ], + "score": 1.0, + "content": "half of the size of the base models (i.e. medium", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 298, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 298, + 138 + ], + "score": 1.0, + "content": "size)9. The result is shown in Figure 5. For", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 297, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 297, + 149 + ], + "score": 1.0, + "content": "both types of models, as we reduce the base", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 297, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 297, + 160 + ], + "score": 1.0, + "content": "model size, there is a sharp drop in perfor-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 297, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 297, + 171 + ], + "score": 1.0, + "content": "mance, while doubling model size only leads to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 297, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 297, + 182 + ], + "score": 1.0, + "content": "marginal gain. Noticeably, SwinT-Hyperprior-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 297, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 297, + 192 + ], + "score": 1.0, + "content": "small is on-par with Conv-Hyperprior-medium", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 297, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 297, + 204 + ], + "score": 1.0, + "content": "even with half of the parameters, and SwinT", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 297, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 297, + 217 + ], + "score": 1.0, + "content": "transforms in general incur fewer MACs per pa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 216, + 141, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 141, + 225 + ], + "score": 1.0, + "content": "rameter.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6 + }, + { + "type": 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A desirable transform would decorrelate the source signal so that simple scalar ´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "quantization and factorized entropy model can be applied without constraining coding performance.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "In most mature neural compression solutions, uniform scalar quantization is adopted together with a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "learned factorized or conditionally factorized Gaussian prior distribution. 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The effectiveness of the analysis transform", + "type": "text" + }, + { + "bbox": [ + 403, + 594, + 414, + 603 + ], + "score": 0.83, + "content": "g _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "can then be evaluated", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 399, + 615 + ], + "score": 1.0, + "content": "by measuring how much correlation there is among different elements in", + "type": "text" + }, + { + "bbox": [ + 400, + 604, + 407, + 614 + ], + "score": 0.68, + "content": "\\bar { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 603, + 505, + 615 + ], + "score": 1.0, + "content": ". We are particularly in-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "score": 1.0, + "content": "terested in measuring the correlation between nearby spatial positions, which are heavily correlated", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "in the source domain for natural images. 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It can be observed that while both lead to small cross-correlations, Swin-Transformer", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 658, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 669 + ], + "score": 1.0, + "content": "does a much better job with uniformly smaller correlation values, and the observation is consistent", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 55.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 680, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 678, + 343, + 691 + ], + "spans": [ + { + "bbox": [ + 118, + 678, + 343, + 691 + ], + "score": 1.0, + "content": "9detailed model configurations are provided in Appendix A.3.", + "type": "text" + } + ] + }, + { + "bbox": [ + 115, + 686, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 115, + 686, + 506, + 705 + ], + "score": 1.0, + "content": "10Note that it is difficult to fairly compare the decoding time of VTM and neural codecs since they run on", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 699, + 361, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 361, + 713 + ], + "score": 1.0, + "content": "different hardware. For more discussion please refer to Appendix D.3.", + "type": "text" + } + ] + }, + { + "bbox": [ + 114, + 708, + 506, + 726 + ], + "spans": [ + { + "bbox": [ + 114, + 708, + 203, + 726 + ], + "score": 1.0, + "content": "11The value with index", + "type": "text" + }, + { + "bbox": [ + 204, + 711, + 223, + 722 + ], + "score": 0.9, + "content": "( i , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 708, + 506, + 726 + ], + "score": 1.0, + "content": "corresponds to the normalized cross-correlation of latents at spatial location", + "type": "text" + } + ] + }, + { + "bbox": [ + 107, + 720, + 419, + 733 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 131, + 732 + ], + "score": 0.91, + "content": "( w , h )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 720, + 147, + 733 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 148, + 721, + 201, + 732 + ], + "score": 0.91, + "content": "( w + i , h + j )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 720, + 419, + 733 + ], + "score": 1.0, + "content": ", averaged across all latent elements of all images on Kodak.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 297, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 297, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 297, + 94 + ], + "score": 1.0, + "content": "Scaling behavior To see how the BD-rate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 297, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 297, + 105 + ], + "score": 1.0, + "content": "varies with model size, we scale SwinT-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 298, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 298, + 117 + ], + "score": 1.0, + "content": "Hyperprior and Conv-Hyperprior to be twice or", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 297, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 297, + 127 + ], + "score": 1.0, + "content": "half of the size of the base models (i.e. medium", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 298, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 298, + 138 + ], + "score": 1.0, + "content": "size)9. The result is shown in Figure 5. For", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 297, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 297, + 149 + ], + "score": 1.0, + "content": "both types of models, as we reduce the base", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 297, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 297, + 160 + ], + "score": 1.0, + "content": "model size, there is a sharp drop in perfor-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 297, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 297, + 171 + ], + "score": 1.0, + "content": "mance, while doubling model size only leads to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 297, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 297, + 182 + ], + "score": 1.0, + "content": "marginal gain. 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We are particularly in-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "score": 1.0, + "content": "terested in measuring the correlation between nearby spatial positions, which are heavily correlated", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "in the source domain for natural images. 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This suggests that trans-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "former based transforms incur less redundancy across different spatial latent locations compared", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "with convolutional ones, leading to an overall better rate-distortion trade-off. The larger spatial", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "correlation (and thus redundancy) in Conv-Hyperprior also explains why a compute-heavy spatial", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "auto-regressive model is often needed to improve RD with convolutional based transforms (Minnen", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "et al., 2018; Lee et al., 2019; Ma et al., 2021; Guo et al., 2021; Wu et al., 2020). Figure 6 also reveals", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "that most of the correlation of a latent comes from the four elements surrounding it. This suggests", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 500, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 500, + 173 + ], + "score": 1.0, + "content": "that a checkerboard-based conditional prior model (He et al., 2021) may yield further coding gain.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 55.5, + "bbox_fs": [ + 105, + 579, + 506, + 669 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 151, + 95 + ], + "score": 1.0, + "content": "with other", + "type": "text" + }, + { + "bbox": [ + 151, + 83, + 159, + 94 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "values, which are provided in Figure 20 in the Appendix. This suggests that trans-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "former based transforms incur less redundancy across different spatial latent locations compared", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "with convolutional ones, leading to an overall better rate-distortion trade-off. The larger spatial", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "correlation (and thus redundancy) in Conv-Hyperprior also explains why a compute-heavy spatial", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "auto-regressive model is often needed to improve RD with convolutional based transforms (Minnen", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "et al., 2018; Lee et al., 2019; Ma et al., 2021; Guo et al., 2021; Wu et al., 2020). Figure 6 also reveals", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "that most of the correlation of a latent comes from the four elements surrounding it. This suggests", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 500, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 500, + 173 + ], + "score": 1.0, + "content": "that a checkerboard-based conditional prior model (He et al., 2021) may yield further coding gain.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "image", + "bbox": [ + 112, + 178, + 500, + 348 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 178, + 500, + 348 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 178, + 500, + 348 + ], + "spans": [ + { + "bbox": [ + 112, + 178, + 500, + 348 + ], + "score": 0.976, + "type": "image", + "image_path": "98f589949ffaa9a174cd05d3842dd7b09177655ebbe15e92154085910ba1e599.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 112, + 178, + 500, + 234.66666666666666 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 112, + 234.66666666666666, + 500, + 291.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 112, + 291.3333333333333, + 500, + 348.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 357, + 505, + 398 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 370, + 370 + ], + "score": 1.0, + "content": "Figure 7: Comparison of effective receptive field (ERF) of the encoders", + "type": "text" + }, + { + "bbox": [ + 371, + 360, + 380, + 369 + ], + "score": 0.84, + "content": "g _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 356, + 506, + 370 + ], + "score": 1.0, + "content": ", which is visualized as absolution", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 269, + 381 + ], + "score": 1.0, + "content": "gradients of the center pixel in the latent (i.e.", + "type": "text" + }, + { + "bbox": [ + 269, + 368, + 298, + 379 + ], + "score": 0.75, + "content": "d \\mathbf { y } / d \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "with respect to the input image. The plot shows the close", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "up of the gradient maps averaged over all channels in each input of test images/videos. Check Figure 21 for the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 386, + 293, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 259, + 402 + ], + "score": 1.0, + "content": "ERF of the composed encoding transform", + "type": "text" + }, + { + "bbox": [ + 259, + 388, + 287, + 398 + ], + "score": 0.92, + "content": "h _ { a } \\circ g _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 386, + 293, + 402 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + } + ], + "index": 10.75 + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "Effective receptive field Intra prediction in HEVC or AV1 only rely on left and top boarders", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "score": 1.0, + "content": "of the current coding block (Sullivan et al., 2012; Chen et al., 2018), except for intra block copy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 186, + 446 + ], + "score": 1.0, + "content": "for screen content", + "type": "text" + }, + { + "bbox": [ + 186, + 434, + 200, + 444 + ], + "score": 0.39, + "content": "\\mathrm { { X u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "et al., 2016). We would like to see how large the effective receptive field", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "(ERF) (Luo et al., 2017) of SwinT encoders compared to Conv encoders. The theoretical receptive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 193, + 468 + ], + "score": 1.0, + "content": "field of the encoders", + "type": "text" + }, + { + "bbox": [ + 193, + 456, + 246, + 467 + ], + "score": 0.91, + "content": "( g _ { a } , h _ { a } \\circ g _ { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "in SwinT-based codecs is much larger than that of Conv-based", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 252, + 479 + ], + "score": 1.0, + "content": "codecs. However comparing Figure", + "type": "text" + }, + { + "bbox": [ + 253, + 467, + 264, + 477 + ], + "score": 0.31, + "content": "\\mathrm { 7 a }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "with 7e and Figure 7b with 7f, the ERF of SwinT encoders", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "after training is even smaller than Conv encoders. When we examine the ERF of the released Swin", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "transformers for classification, detection and segmentation tasks, they are all spanning the whole", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "input image. This contrast suggests that (natural) image compression with rate-distortion objective", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 504, + 522 + ], + "score": 1.0, + "content": "is a local task, even with transformer-based nonlinear transforms. We further look into P-frame", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "compression models, particularly the ERF of two types of transforms in flow codec and residual", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 312, + 544 + ], + "score": 1.0, + "content": "codec, as shown in Figure 7d & 7h, and Figure 7c &", + "type": "text" + }, + { + "bbox": [ + 312, + 532, + 324, + 543 + ], + "score": 0.42, + "content": "7 \\mathrm { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 532, + 505, + 544 + ], + "score": 1.0, + "content": ". Clearly for flow codec, SwinT transform has", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "much larger ERF than the convolution counterpart. For residual codec, the ERF of SwinT transforms", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 565 + ], + "score": 1.0, + "content": "is similar to image (I-frame) compression case. This shows of flexibility of SwinT encoders to attend", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "to longer or shorter range depending on the tasks. To get a better picture of the behavior of attention", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 576, + 469, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 469, + 588 + ], + "score": 1.0, + "content": "layers in SwinT transforms, we also show the attention distance in each layer in Figure 22.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 504, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 504, + 612 + ], + "score": 1.0, + "content": "Progressive decoding The ERF in the previous section shows the behavior of the encoder trans-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "forms, here we further investigate the decoder transforms through the lens of progressive decod-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "ing (Rippel et al., 2014; Minnen & Singh, 2020; Lu et al., 2021). 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The coding units are ordered by the sum of prior std of all elements within each unit. The", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 675, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 691 + ], + "score": 1.0, + "content": "RD curves of progressive decoding for SwinT-Hyperprior and Conv-Hyperprior are shown in Fig-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "ure 8a, which closely follow each other when ordered by channel or element, but significantly apart", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "when ordered by pixel (spatial dim). Particularly, we show an extreme case when the half pixels in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "the latent (masked by checkerboard pattern) are updated with dequantized values, corresponding to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "the two scatter points in Figure 8a. 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The plot shows the close", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "up of the gradient maps averaged over all channels in each input of test images/videos. Check Figure 21 for the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 386, + 293, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 259, + 402 + ], + "score": 1.0, + "content": "ERF of the composed encoding transform", + "type": "text" + }, + { + "bbox": [ + 259, + 388, + 287, + 398 + ], + "score": 0.92, + "content": "h _ { a } \\circ g _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 386, + 293, + 402 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + } + ], + "index": 10.75 + }, + { + "type": "text", + "bbox": [ + 107, + 411, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "Effective receptive field Intra prediction in HEVC or AV1 only rely on left and top boarders", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "score": 1.0, + "content": "of the current coding block (Sullivan et al., 2012; Chen et al., 2018), except for intra block copy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 186, + 446 + ], + "score": 1.0, + "content": "for screen content", + "type": "text" + }, + { + "bbox": [ + 186, + 434, + 200, + 444 + ], + "score": 0.39, + "content": "\\mathrm { { X u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "et al., 2016). We would like to see how large the effective receptive field", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "(ERF) (Luo et al., 2017) of SwinT encoders compared to Conv encoders. The theoretical receptive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 193, + 468 + ], + "score": 1.0, + "content": "field of the encoders", + "type": "text" + }, + { + "bbox": [ + 193, + 456, + 246, + 467 + ], + "score": 0.91, + "content": "( g _ { a } , h _ { a } \\circ g _ { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "in SwinT-based codecs is much larger than that of Conv-based", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 252, + 479 + ], + "score": 1.0, + "content": "codecs. However comparing Figure", + "type": "text" + }, + { + "bbox": [ + 253, + 467, + 264, + 477 + ], + "score": 0.31, + "content": "\\mathrm { 7 a }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "with 7e and Figure 7b with 7f, the ERF of SwinT encoders", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "after training is even smaller than Conv encoders. When we examine the ERF of the released Swin", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "transformers for classification, detection and segmentation tasks, they are all spanning the whole", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "input image. This contrast suggests that (natural) image compression with rate-distortion objective", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 504, + 522 + ], + "score": 1.0, + "content": "is a local task, even with transformer-based nonlinear transforms. We further look into P-frame", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "compression models, particularly the ERF of two types of transforms in flow codec and residual", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 312, + 544 + ], + "score": 1.0, + "content": "codec, as shown in Figure 7d & 7h, and Figure 7c &", + "type": "text" + }, + { + "bbox": [ + 312, + 532, + 324, + 543 + ], + "score": 0.42, + "content": "7 \\mathrm { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 532, + 505, + 544 + ], + "score": 1.0, + "content": ". Clearly for flow codec, SwinT transform has", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "much larger ERF than the convolution counterpart. For residual codec, the ERF of SwinT transforms", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 565 + ], + "score": 1.0, + "content": "is similar to image (I-frame) compression case. This shows of flexibility of SwinT encoders to attend", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "to longer or shorter range depending on the tasks. To get a better picture of the behavior of attention", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 576, + 469, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 469, + 588 + ], + "score": 1.0, + "content": "layers in SwinT transforms, we also show the attention distance in each layer in Figure 22.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 412, + 506, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 504, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 504, + 612 + ], + "score": 1.0, + "content": "Progressive decoding The ERF in the previous section shows the behavior of the encoder trans-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "forms, here we further investigate the decoder transforms through the lens of progressive decod-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "ing (Rippel et al., 2014; Minnen & Singh, 2020; Lu et al., 2021). Initialized with the prior mean,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 418, + 646 + ], + "score": 1.0, + "content": "the input to the decoder is progressively updated with the dequantized latent", + "type": "text" + }, + { + "bbox": [ + 419, + 633, + 426, + 644 + ], + "score": 0.83, + "content": "\\hat { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "in terms of coding", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 457, + 657 + ], + "score": 1.0, + "content": "units, leading to gradually improved reconstruction quality. 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The coding units are ordered by the sum of prior std of all elements within each unit. The", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 675, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 691 + ], + "score": 1.0, + "content": "RD curves of progressive decoding for SwinT-Hyperprior and Conv-Hyperprior are shown in Fig-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "ure 8a, which closely follow each other when ordered by channel or element, but significantly apart", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "when ordered by pixel (spatial dim). Particularly, we show an extreme case when the half pixels in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "the latent (masked by checkerboard pattern) are updated with dequantized values, corresponding to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "the two scatter points in Figure 8a. One visual example (CLIC2021 test) is shown in Figure 8b under", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "this setup, where we can clearly see SwinT decoder achieves better reconstruction quality than the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "Conv decoder, mainly in terms of more localized response to a single latent pixel. This is potentially", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 471, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 471, + 117 + ], + "score": 1.0, + "content": "useful for region-of-interest decompression. More visual examples are shown in Figure 26.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 600, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "this setup, where we can clearly see SwinT decoder achieves better reconstruction quality than the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "Conv decoder, mainly in terms of more localized response to a single latent pixel. This is potentially", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 471, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 471, + 117 + ], + "score": 1.0, + "content": "useful for region-of-interest decompression. More visual examples are shown in Figure 26.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "image", + "bbox": [ + 135, + 125, + 468, + 283 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 135, + 125, + 468, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 125, + 468, + 283 + ], + "spans": [ + { + "bbox": [ + 135, + 125, + 468, + 283 + ], + "score": 0.973, + "type": "image", + "image_path": "f6676a10d3d6d6c46c1baa40f109ee455e71dff1f0f3da4fb9b130557ad86082.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 135, + 125, + 468, + 177.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 135, + 177.66666666666666, + 468, + 230.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 135, + 230.33333333333331, + 468, + 283.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 293, + 506, + 353 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "Figure 8: (Left) Progressive decoding according to the order of the sum of prior std of all elements within", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 304, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 315 + ], + "score": 1.0, + "content": "each latent coding unit, which can be one latent channel, pixel or element. 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Note that while", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "Conv encoder and Conv decoder are of symmetric architectures, Conv decoder takes more time than", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "score": 1.0, + "content": "Conv encoder mainly because Transposed Conv layers in the decoder take much longer than Conv", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 524, + 193, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 193, + 536 + ], + "score": 1.0, + "content": "layers in the encoder.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + } + ], + "index": 13.75 + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 564 + ], + "score": 1.0, + "content": "For completeness, we also report the profiling for CPU coding time in Table 5 and Table 6. The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "evaluation setup is the same as the GPU profiling case, except models are run on the CPU instead", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 573, + 392, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 336, + 585 + ], + "score": 1.0, + "content": "(same host machine with Intel(R) Xeon(R) W-2123 CPU", + "type": "text" + }, + { + "bbox": [ + 337, + 573, + 386, + 584 + ], + "score": 0.29, + "content": "\\textcircled { a } 3 . 6 0 \\mathrm { G H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 573, + 392, + 585 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 550, + 506, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "Table 7 reports encoding and decoding time of VTM-12.1 under different quantization parameters", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 302, + 613 + ], + "score": 1.0, + "content": "(QPs), evaluated on an Intel Core i9-9940 CPU", + "type": "text" + }, + { + "bbox": [ + 302, + 601, + 312, + 611 + ], + "score": 0.63, + "content": "@", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 601, + 351, + 611 + ], + "score": 0.33, + "content": ") ~ 3 . 3 0 \\mathrm { G H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 600, + 505, + 613 + ], + "score": 1.0, + "content": ", averaged over 24 Kodak images. As", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "can be seen from the table, decoding time of VTM-12.1 is a function of reconstruction quality, where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "longer decoding time is observed for higher quality reconstruction. In Figure 1, the reported VTM-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "12.1 decoding speed corresponds to a QP value of 28, where the bpp value is similar to that obtained", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 199, + 657 + ], + "score": 1.0, + "content": "by models trained with", + "type": "text" + }, + { + "bbox": [ + 199, + 645, + 242, + 656 + ], + "score": 0.91, + "content": "\\beta = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 645, + 505, + 657 + ], + "score": 1.0, + "content": ". It is worth pointing out that VTM-12.1 encoding process is much", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 655, + 496, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 496, + 668 + ], + "score": 1.0, + "content": "slower, ranging anywhere from 1 to 5 minutes per image, whereas neural codec runs much faster.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 589, + 506, + 668 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 110, + 155, + 498, + 240 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 79, + 505, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 381, + 92 + ], + "score": 1.0, + "content": "Table 4: GPU encoding complexity. 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GMACs and GPU peak memory during encoding and the parameter count", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 135, + 244, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 244, + 147 + ], + "score": 1.0, + "content": "of the entire model are also listed.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "table_body", + "bbox": [ + 110, + 155, + 498, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 155, + 498, + 240 + ], + "spans": [ + { + "bbox": [ + 110, + 155, + 498, + 240 + ], + "score": 0.979, + "html": "
CodecTime (ms)GMACsMemory (GB)Params (M)
gahaZhsytotal
Conv-Hyperprior22.72.32.84.029.862.01020.6221.4
SwinT-Hyperprior57.51.22.44.834.3100.51001.4724.7
Conv-ChARM22.92.42.64.163.495.71140.6629.3
SwinT-ChARM57.81.32.44.868.7135.31121.5132.6
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CodecCPU decoding ' Time (ms)GMACs
Zhsy9stotal
Conv-Hyperprior5.120.930.51636.21703.3350
SwinT-Hyperprior5.718.330.31912.81979.599
Conv-ChARM4.720.9154.61634.31825.6362
SwinT-ChARM5.818.2141.11877.32054.7111
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CodecCPU encoding Time (ms)GMACs
gahaZhsytotal
Conv-Hyperprior896.08.92.720.931.1960.9102
SwinT-Hyperprior1781.614.52.618.329.71847.9100
Conv-ChARM895.78.92.520.9146.31075.5114
SwinT-ChARM1697.114.52.618.2133.51867.1112
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QPbits per pixelPSNR (dB)Decoding time(s)Encoding time (s)
162.544144.090.284300.47
280.784436.760.249118.85
400.155729.510.14971.10
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CodecCPU decoding ' Time (ms)GMACs
Zhsy9stotal
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CodecCPU encoding Time (ms)GMACs
gahaZhsytotal
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SwinT-Hyperprior1781.614.52.618.329.71847.9100
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Here we look into the features in the latent channels which are to be quantized", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 195 + ], + "score": 1.0, + "content": "and entropy coded to bitstreams. Particularly we order the total bitrate of each channel averaged", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 193, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 188, + 208 + ], + "score": 1.0, + "content": "over Kodak dataset", + "type": "text" + }, + { + "bbox": [ + 188, + 194, + 244, + 205 + ], + "score": 0.86, + "content": "2 4 7 6 8 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 193, + 505, + 208 + ], + "score": 1.0, + "content": "images). The result is shown in Figure 23. We find an interesting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "phenomenon across models under different bitrates: there is a cutoff point of the bitrate-vs-channel", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "curve where the bitrate suddenly drops to zero, which manifest the rate constraint in the loss func-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 227, + 490, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 476, + 239 + ], + "score": 1.0, + "content": "tion. As expected, the cutoff index decreases for the model trained for smaller bitrate (larger", + "type": "text" + }, + { + "bbox": [ + 477, + 227, + 484, + 239 + ], + "score": 0.77, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 228, + 490, + 239 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 253, + 270, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 272, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 272, + 265 + ], + "score": 1.0, + "content": "E.4 CENTERED KERNEL ALIGNMENT", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "To investigate the difference or similarity between latent features of Conv-based and SwinT-based", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "score": 1.0, + "content": "models, we resort to a commonly used tool in representation learning called centered kernel align-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "ment (CKA) (Kornblith et al., 2019). We evaluate CKA between each of the Conv latent channel and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 357, + 318 + ], + "score": 1.0, + "content": "SwinT latent channel (both models are trained under the same", + "type": "text" + }, + { + "bbox": [ + 357, + 307, + 365, + 318 + ], + "score": 0.73, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 307, + 505, + 318 + ], + "score": 1.0, + "content": ") over the 24 Kodak images. There", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 383, + 330 + ], + "score": 1.0, + "content": "are 320 channels for both Conv latent and SwinT latent, resulting a", + "type": "text" + }, + { + "bbox": [ + 383, + 318, + 428, + 329 + ], + "score": 0.89, + "content": "3 2 0 \\times 3 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "CKA matrix. The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "result is shown in Figure 24. The latent channel is ordered by the averaged bitrate of each channel", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "over Kodak images (same as in Section E.3). The CKA matrix has clear block structure, where high", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "similarity region corresponds to the latent channels before the bitrate cutoff in the rate distribution", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 361, + 181, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 181, + 374 + ], + "score": 1.0, + "content": "curve (Figure 23).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "Identification of SwinT and Conv latent channels with CKA Within the block of high similarity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "(from the CKA matrix), we identify the ‘less’ similar SwinT latent channels with lowest CKA values", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "score": 1.0, + "content": "between this SwinT channel and all other Conv latent channels. For each of the identified SwinT", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "channel, we find the Conv latent channel with the largest CKA value between the two. This way,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "we are able to identify latent channels of two different models with high similarity. We show the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 440, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 455 + ], + "score": 1.0, + "content": "identified top 8 channels in Figure 25. The channels are indeed highly similar, up to a sign flip,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "even through the two model architectures are quite different. This empirical result is relevant to the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 462, + 423, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 423, + 475 + ], + "score": 1.0, + "content": "literature on the identifiability of generative models (Khemakhem et al., 2020).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 488, + 240, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 241, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 241, + 501 + ], + "score": 1.0, + "content": "E.5 PROGRESSIVE DECODING", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 504, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 523 + ], + "score": 1.0, + "content": "More visual examples of reconstructions of checkerboard (spatially) masked latent are provided in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 520, + 150, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 150, + 533 + ], + "score": 1.0, + "content": "Figure 26.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "Channel-wise progressive decoding Here we visualize the behavior of channel-wise progressive", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "score": 1.0, + "content": "decoding of Conv and SwinT models. For both models, we order the latent channels with a heuristic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "importance metric: the reduction of distortion over the increase of rate if one channel is included for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "decoding. We start with the order of bitrate per channel, we pass the leading channels of bitrate order", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "(zero out all rest channels) to the decoder to obtain reconstruction and calculate distortion. We plot", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "the top 8 channels following this importance order. For each channel, we show 6 maps from top to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "bottom: latent values, mean prediction from the hyper decoder, standard deviation prediction from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "the hyper decoder, the bitmap, the reconstruction with only current channel, the reconstruction with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "up to current channel (all leading channels). The result is shown in Figure 27. For Conv models,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "usually the top 3 important channels are responsible for lightness, and two color components (blue-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "yellow, red-green), similar to the LAB colorspace. The rest of the latent channels are responsible for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "adding details like texture and edges. For the Swint models, at low bitrate, there is one significantly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "different channel (the first column in Figure 27b), which is in a coarse scale (with smooth blocks)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "and responsible for the reconstruction of a nearly constant image with value close to 120 (the mean", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "value of natural image dataset). This latent channel costs extremely small bitrate but reaches PSNR", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "of 13dB. We tried remove this first channel, the progressive reconstruction with the rest leading 7", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 428, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 428, + 734 + ], + "score": 1.0, + "content": "channels only leads to PSNR around 16dB, instead of 26dB shown in the figure.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 40 + } + ], + "page_idx": 24, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 260, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 261, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 261, + 95 + ], + "score": 1.0, + "content": "E.2 EFFECTIVE RECEPTIVE FIELD", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 453, + 116 + ], + "score": 1.0, + "content": "See Figure 21 for the effective receptive field for the composed encoding tranforms", + "type": "text" + }, + { + "bbox": [ + 453, + 104, + 486, + 115 + ], + "score": 0.92, + "content": "h _ { a } \\circ g _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 500, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 500, + 127 + ], + "score": 1.0, + "content": "Figure 22 for the mean attention distance visualization of each head within each transformer layer.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 103, + 505, + 127 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 141, + 341, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 342, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 342, + 153 + ], + "score": 1.0, + "content": "E.3 RATE DISTRIBUTION ACROSS LATENT CHANNELS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 161, + 505, + 238 + ], + "lines": [ + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "It is generally believed that ConvNets learn to extract various features and store them in each chan-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "nel of the activations. Here we look into the features in the latent channels which are to be quantized", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 195 + ], + "score": 1.0, + "content": "and entropy coded to bitstreams. Particularly we order the total bitrate of each channel averaged", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 193, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 188, + 208 + ], + "score": 1.0, + "content": "over Kodak dataset", + "type": "text" + }, + { + "bbox": [ + 188, + 194, + 244, + 205 + ], + "score": 0.86, + "content": "2 4 7 6 8 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 193, + 505, + 208 + ], + "score": 1.0, + "content": "images). The result is shown in Figure 23. We find an interesting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "phenomenon across models under different bitrates: there is a cutoff point of the bitrate-vs-channel", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "curve where the bitrate suddenly drops to zero, which manifest the rate constraint in the loss func-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 227, + 490, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 476, + 239 + ], + "score": 1.0, + "content": "tion. As expected, the cutoff index decreases for the model trained for smaller bitrate (larger", + "type": "text" + }, + { + "bbox": [ + 477, + 227, + 484, + 239 + ], + "score": 0.77, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 228, + 490, + 239 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 161, + 505, + 239 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 253, + 270, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 272, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 272, + 265 + ], + "score": 1.0, + "content": "E.4 CENTERED KERNEL ALIGNMENT", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "To investigate the difference or similarity between latent features of Conv-based and SwinT-based", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "score": 1.0, + "content": "models, we resort to a commonly used tool in representation learning called centered kernel align-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "ment (CKA) (Kornblith et al., 2019). We evaluate CKA between each of the Conv latent channel and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 357, + 318 + ], + "score": 1.0, + "content": "SwinT latent channel (both models are trained under the same", + "type": "text" + }, + { + "bbox": [ + 357, + 307, + 365, + 318 + ], + "score": 0.73, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 307, + 505, + 318 + ], + "score": 1.0, + "content": ") over the 24 Kodak images. There", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 383, + 330 + ], + "score": 1.0, + "content": "are 320 channels for both Conv latent and SwinT latent, resulting a", + "type": "text" + }, + { + "bbox": [ + 383, + 318, + 428, + 329 + ], + "score": 0.89, + "content": "3 2 0 \\times 3 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "CKA matrix. The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "result is shown in Figure 24. The latent channel is ordered by the averaged bitrate of each channel", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "over Kodak images (same as in Section E.3). The CKA matrix has clear block structure, where high", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "similarity region corresponds to the latent channels before the bitrate cutoff in the rate distribution", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 361, + 181, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 181, + 374 + ], + "score": 1.0, + "content": "curve (Figure 23).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 273, + 506, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "Identification of SwinT and Conv latent channels with CKA Within the block of high similarity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "(from the CKA matrix), we identify the ‘less’ similar SwinT latent channels with lowest CKA values", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "score": 1.0, + "content": "between this SwinT channel and all other Conv latent channels. For each of the identified SwinT", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "channel, we find the Conv latent channel with the largest CKA value between the two. This way,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "we are able to identify latent channels of two different models with high similarity. We show the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 440, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 455 + ], + "score": 1.0, + "content": "identified top 8 channels in Figure 25. The channels are indeed highly similar, up to a sign flip,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "even through the two model architectures are quite different. This empirical result is relevant to the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 462, + 423, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 423, + 475 + ], + "score": 1.0, + "content": "literature on the identifiability of generative models (Khemakhem et al., 2020).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 385, + 506, + 475 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 488, + 240, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 241, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 241, + 501 + ], + "score": 1.0, + "content": "E.5 PROGRESSIVE DECODING", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 504, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 523 + ], + "score": 1.0, + "content": "More visual examples of reconstructions of checkerboard (spatially) masked latent are provided in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 520, + 150, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 150, + 533 + ], + "score": 1.0, + "content": "Figure 26.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 508, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "Channel-wise progressive decoding Here we visualize the behavior of channel-wise progressive", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 568 + ], + "score": 1.0, + "content": "decoding of Conv and SwinT models. For both models, we order the latent channels with a heuristic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "importance metric: the reduction of distortion over the increase of rate if one channel is included for", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "decoding. We start with the order of bitrate per channel, we pass the leading channels of bitrate order", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "(zero out all rest channels) to the decoder to obtain reconstruction and calculate distortion. We plot", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "the top 8 channels following this importance order. For each channel, we show 6 maps from top to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "bottom: latent values, mean prediction from the hyper decoder, standard deviation prediction from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "the hyper decoder, the bitmap, the reconstruction with only current channel, the reconstruction with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "up to current channel (all leading channels). The result is shown in Figure 27. For Conv models,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "usually the top 3 important channels are responsible for lightness, and two color components (blue-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "yellow, red-green), similar to the LAB colorspace. The rest of the latent channels are responsible for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "adding details like texture and edges. For the Swint models, at low bitrate, there is one significantly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "different channel (the first column in Figure 27b), which is in a coarse scale (with smooth blocks)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "and responsible for the reconstruction of a nearly constant image with value close to 120 (the mean", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "value of natural image dataset). This latent channel costs extremely small bitrate but reaches PSNR", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "of 13dB. We tried remove this first channel, the progressive reconstruction with the rest leading 7", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 428, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 428, + 734 + ], + "score": 1.0, + "content": "channels only leads to PSNR around 16dB, instead of 26dB shown in the figure.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 544, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 261, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 262, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 262, + 95 + ], + "score": 1.0, + "content": "F MORE ABLATION STUDIES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 504, + 119 + ], + "score": 1.0, + "content": "Local self-attention To see if local self-attention is the most important component in transform-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 104, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "ers, we replace it by depthwise separable convolution block15 (Han et al., 2021; El-Nouby et al.,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "2021), which performs similar spatial feature aggregation as self-attention, while keeping all other", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 505, + 151 + ], + "score": 1.0, + "content": "components the same as in the SwinT. We found this change only leads to minor degradation in RD.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 149, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 104, + 149, + 505, + 164 + ], + "score": 1.0, + "content": "This suggests other components in transformers such as MLPs and skip connections may also play", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 174 + ], + "score": 1.0, + "content": "a big role, other than just self-attention, for the leading performance in our work and many other", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 209, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 209, + 184 + ], + "score": 1.0, + "content": "tasks (Dong et al., 2021).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 195, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "Small depths Upon investigating the mean attention distance as shown in Figure 22, we find the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "last block in each of the last two encoder stages has about half of its attention heads degenerate to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "attending to fixed nearly pixels. This suggests redundant transformer blocks at that stage, so we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "remove those two blocks, i.e. from depths [2, 2, 6, 2] to [2, 2, 5, 1]. The resulting SwinT-Hyperprior", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 504, + 251 + ], + "score": 1.0, + "content": "has even less parameters (20.6M) than Conv-Hyperprior (21.4M) while with almost no RD loss", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "compared to the larger model. 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The example image shown in Figure 8b is andy-kelly-0E vhMVqL9g-unsplash from CLIC2021", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 558, + 272, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 272, + 570 + ], + "score": 1.0, + "content": "test, same as the two images shown here.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 156, + 504, + 563 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 156, + 504, + 563 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 156, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 504, + 563 + ], + "score": 0.97, + "type": "image", + "image_path": "6dc9e2834f1aa66964d2bd3084e432f186f229a8e2533c6ab9be89184c3981ff.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 156, + 504, + 291.66666666666663 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 291.66666666666663, + 504, + 427.33333333333326 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 427.33333333333326, + 504, + 562.9999999999999 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 573, + 505, + 651 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "Figure 27: Channel-wise progressive decoding. 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For SwinT models, we find a coarse-scale latent channel (the left most channel)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "which costs negligible bits but is important for reconstruction (without it, the 7 other leading chan-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "nels can only decode to a reconstruction with PSNR around 16dB). 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However, due to the massive number of parameters and model design, e.g., attention mechanism, ViT-based models are generally times slower than lightweight convolutional networks. Therefore, the deployment of ViT for real-time applications is particularly challenging, especially on resource-constrained hardware such as mobile devices. Recent efforts try to reduce the computation complexity of ViT through network architecture search or hybrid design with MobileNet block, yet the inference speed is still unsatisfactory. This leads to an important question: can transformers run as fast as MobileNet while obtaining high performance? To answer this, we first revisit the network architecture and operators used in ViT-based models and identify inefficient designs. Then we introduce a dimension-consistent pure transformer (without MobileNet blocks) as a design paradigm. Finally, we perform latencydriven slimming to get a series of final models dubbed EfficientFormer. Extensive experiments show the superiority of EfficientFormer in performance and speed on mobile devices. Our fastest model, EfficientFormer-L1, achieves $7 9 . 2 \%$ top-1 accuracy on ImageNet-1K with only $1 . 6 \mathrm { m s }$ inference latency on iPhone 12 (compiled with CoreML), which runs as fast as MobileNet ${ \tt V } 2 { \times } 1 . 4$ $\mathrm { 1 . 6 m s }$ , $7 4 . 7 \%$ top-1), and our largest model, EfficientFormer-L7, obtains $8 3 . 3 \%$ accuracy with only $7 . 0 \mathrm { m s }$ latency. Our work proves that properly designed transformers can reach extremely low latency on mobile devices while maintaining high performance1. + +# 1 Introduction + +The transformer architecture [1], initially designed for Natural Language Processing (NLP) tasks, introduces the Multi-Head Self Attention (MHSA) mechanism that allows the network to model long-term dependencies and is easy to parallelize. In this context, Dosovitskiy et al. $\mathbb { \left[ \sum \right] }$ adapt the attention mechanism to 2D images and propose Vision Transformer (ViT): the input image is divided into non-overlapping patches, and the inter-patch representations are learned through MHSA without inductive bias. ViTs demonstrate promising results compared to convolutional neural networks (CNNs) on computer vision tasks. Following this success, several efforts explore the potential of ViT by improving training strategies [3, 4, 5], introducing architecture changes [6, 7], redesigning attention mechanisms [8, 9], and elevating the performance of various vision tasks such as classification [10, 11, 12], segmentation [13, 14], and detection [15, 16]. + +On the downside, transformer models are usually times slower than competitive CNNs [17, 18]. There are many factors that limit the inference speed of ViT, including the massive number of parameters, quadratic-increasing computation complexity with respect to token length, non-foldable normalization layers, and lack of compiler level optimizations (e.g., Winograd for CNN [19]). The high latency makes transformers impractical for real-world applications on resource-constrained hardware, such as augmented or virtual reality applications on mobile devices and wearables. As a result, lightweight CNNs [20, 21, 22] remain the default choice for real-time inference. + +![](images/69013ffaf9879e4028a840dc250a58fb123722e9df8aafdb2ad8688d71498da4.jpg) +Figure 1: Inference Speed vs. Accuracy. All models are trained on ImageNet-1K and measured by iPhone 12 with CoreMLTools to get latency. Compared to CNNs, EfficientFormer-L1 runs $4 0 \%$ faster than EfficientNet-B0, while achieves $2 . 1 \%$ higher accuracy. For the latest MobileViT-XS, EfficientFormer-L7 runs $0 . 2 \mathrm { m s }$ faster with $8 . 5 \%$ higher accuracy. + +To alleviate the latency bottleneck of transformers, many approaches have been proposed. For instance, some efforts consider designing new architectures or operations by changing the linear layers with convolutional layers (CONV) $[ [ 2 3 ] ]$ , combining self-attention with MobileNet blocks $\pmb { \bigtriangledown } 2 4 \mathbf { \| }$ , or introducing sparse attention [25, 26, 27], to reduce the computational cost, while other efforts leverage network searching algorithm $\left. 2 8 \right.$ or pruning $[ [ 2 9 ]$ to improve efficiency. Although the computation-performance trade-off has been improved by existing works, the fundamental question that relates to the applicability of transformer models remains unanswered: Can powerful vision transformers run at MobileNet speed and become a default option for edge applications? This work provides a study towards the answer through the following contributions: + +• First, we revisit the design principles of ViT and its variants through latency analysis (Sec. $3 )$ . Following existing work $\bar { \lVert 1 8 \rVert }$ , we utilize iPhone 12 as the testbed and publicly available CoreML $\pmb { \mathbb { B } } \pmb { \mathrm { O } } \Vert$ as the compiler, since the mobile device is widely used and the results can be easily reproduced. • Second, based on our analysis, we identify inefficient designs and operators in ViT and propose a new dimension-consistent design paradigm for vision transformers (Sec. 4.1). • Third, starting from a supernet with the new design paradigm, we propose a simple yet effective latency-driven slimming method to obtain a new family of models, namely, EfficientFormers (Sec. 4.2). We directly optimize for inference speed instead of MACs or number of parameters [31, 32, 33]. + +Our fastest model, EfficientFormer-L1, achieves $7 9 . 2 \%$ top-1 accuracy on ImageNet-1K [34] classification task with only $1 . 6 ~ \mathrm { m s }$ inference time (averaged over 1, 000 runs), which runs as fast as MobileNet $V 2 { \times } 1 . 4$ and wields $4 . 5 \%$ higher top-1 accuracy (more results in Fig. 1 and Tab. 1). The promising results demonstrate that latency is no longer an obstacle for the widespread adoption of vision transformers. Our largest model, EfficientFormer-L7, achieves $8 3 . 3 \%$ accuracy with only $7 . 0 \mathrm { m s }$ latency, outperforms ViT $\mathrm { \Phi } ^ { \prime } \times \mathrm { \Phi }$ MobileNet hybrid designs (MobileViT-XS, $7 4 . 8 \%$ , $7 . 2 \mathrm { m s }$ ) by a large margin. Additionally, we observe superior performance by employing EfficientFormer as the backbone in image detection and segmentation benchmarks (Tab. 2). We provide a preliminary answer to the aforementioned question, ViTs can achieve ultra fast inference speed and wield powerful performance at the same time. We hope our EfficientFormer can serve as a strong baseline and inspire followup works on the edge deployment of vision transformers. + +# 2 Related Work + +Transformers are initially proposed to handle the learning of long sequences in NLP tasks [1]. Dosovitskiy et al. $\left[ \left[ 2 \right] \right]$ and Carion et al. $\mathbb { \left. \overline { { 1 5 } } \right. }$ adapt the transformer architecture to classification and detection, respectively, and achieve competitive performance against CNN counterparts with stronger training techniques and larger-scale datasets. DeiT $\mathbb { \left[ 3 \right] }$ further improves the training pipeline with the aid of distillation, eliminating the need for large-scale pretraining [35]. Inspired by the competitive performance and global receptive field of transformer models, follow-up works are proposed to refine the architecture [36, 37], explore the relationship between CONV nets and ViT [38, 39, 40], and adapt ViT to different computer vision tasks [13, 41, 42, 43, 44, 45, 46]. Other research efforts explore the essence of attention mechanism and propose insightful variants of token mixer, e.g., local attention $\pmb { \mathbb { B } } ] \mathbf { l }$ , spatial MLP [47, 48], and pooling-mixer [6]. + +Despite the success in most vision tasks, ViT-based models cannot compete with the well-studied lightweight CNNs [21, 49] when the inference speed is the major concern [50, 51, 52], especially on resource-constrained edge devices [17]. To accelerate ViT, many approaches have been introduced with different methodologies, such as proposing new architectures or modules [53, 54, 55, 56, 57, 58], re-thinking self-attention and sparse-attention mechanisms [59, 60, 61, 62, 63, 64, 65], and utilizing search algorithms that are widely explored in CNNs to find smaller and faster ViTs [66, 28, 29, 67]. Recently, LeViT $\mathbb { \left[ \left. 2 3 \right] \right. }$ proposes a CONV-clothing design to accelerate vision transformer. However, in order to perform MHSA, the 4D features need to be frequently reshaped into flat patches, which is still expensive to compute on edge resources (Fig. 2). Likewise, MobileViT $\boxed { 1 8 }$ introduces a hybrid architecture that combines lightweight MobileNet blocks (with point-wise and depth-wise CONV) and MHSA blocks; the former is placed at early stages in the network pipeline to extract low-level features, while the latter is placed in late stages to enjoy the global receptive field. Similar approach has been explored by several works $ { \mathbb { P } } ^ { \smash { 2 4 , \sqrt { 2 8 } } \rvert }$ as a straightforward strategy to reduce computation. + +Different from existing works, we aim at pushing the latency-performance boundary of pure vision transformers instead of relying on hybrid designs, and directly optimize for mobile latency. Through our detailed analysis $( \mathrm { S e c . } \boxed { 3 } )$ , we propose a new design paradigm (Sec. 4.1), which can be further elevated through architecture search (Sec. 4.2). + +# 3 On-Device Latency Analysis of Vision Transformers + +Most existing approaches optimize the inference speed of transformers through computation complexity (MACs) or throughput (images/sec) obtained from server GPU [23, 28]. While such metrics do not reflect the real on-device latency. To have a clear understanding of which operations and design choices slow down the inference of ViTs on edge devices, we perform a comprehensive latency analysis over a number of models and operations, as shown in Fig. $\bigtriangledown$ whereby the following observations are drawn. + +# Observation 1: Patch embedding with large kernel and stride is a speed bottleneck on mobile devices. + +Patch embedding is often implemented with a non-overlapping convolution layer that has large kernel size and stride [3, 55]. A common belief is that the computation cost of the patch embedding layer in a transformer network is unremarkable or negligible [2, 6]. However, our comparison in Fig. 2 between models with large kernel and stride for patch embedding, i.e., DeiT-S $\pmb { \mathbb { B } } \|$ and PoolFormer-S24 [6], and the models without it, i.e., LeViT-256 $\pmb { \mathbb { Z } } 3 \|$ and EfficientFormer, shows that patch embedding is instead a speed bottleneck on mobile devices. + +Large-kernel convolutions are not well supported by most compilers and cannot be accelerated through existing algorithms like Winograd $\mathbb { I m }$ . Alternatively, the non-overlapping patch embedding can be replaced by a convolution stem with fast downsampling [68, 69, 23] that consists of several hardware-efficient $3 \times 3$ convolutions (Fig. 3). + +![](images/62de6711ec53ab7af829b490ab1cc474211661ae30de5649a9164283c077f358.jpg) +Figure 2: Latency profiling. Results are obtained on iPhone 12 with CoreML. The on-device speed for CNN (MobileNet ${ \mathrm { V } } 2 { \times } 1 . 4$ , ResNet50, and EfficientNet-B0), ViT-based models (DeiT-Small, LeViT-256, PoolFormer-S24, and EfficientFormer), and various operators are reported. The latency of models and operations are denoted with different color. ( ) is the top-1 accuracy on ImageNet-1K. †LeViT uses HardSwish which is not well supported by CoreML, we replace it with GeLU for fair comparison. + +Observation 2: Consistent feature dimension is important for the choice of token mixer. MHSA is not necessarily a speed bottleneck. + +Recent work extends ViT-based models to the MetaFormer architecture $\pmb { \Vert 6 \Vert }$ consisting of MLP blocks and unspecified token mixers. Selecting a token mixer is an essential design choice when building ViT-based models. The options are many—the conventional MHSA mixer with a global receptive field, more sophisticated shifted window attention $\pmb { \Vert 8 \Vert }$ , or a non-parametric operator like pooling [6]. + +We narrow the comparison to the two token mixers, pooling and MHSA, where we choose the former for its simplicity and efficiency, while the latter for better performance. More complicated token mixers like shifted window $\textcircled { 8 } \textcircled { 1 8 }$ are currently not supported by most public mobile compilers and we leave them outside our scope. Furthermore, we do not use depth-wise convolution to replace pooling $ { \mathbb { I } } ^ { { \mathbb { Z } } 0 \| }$ as we focus on building architecture without the aid of lightweight convolutions. + +To understand the latency of the two token mixers, we perform the following two comparisons: + +• First, by comparing PoolFormer-s24 $\textcircled { 6 }$ and LeViT-256 $\mathbb { \left[ \left. 2 3 \right] \right. }$ , we observe that the Reshape operation is a bottleneck for LeViT-256. The majority of LeViT-256 is implemented with CONV on 4D tensor, requiring frequent reshaping operations when forwarding features into MHSA since the attention has to be performed on patchified 3D tensor (discarding the extra dimension of attention heads). The extensive usage of Reshape limits the speed of LeViT on mobile devices (Fig. 2). On the other hand, pooling naturally suits the 4D tensor when the network primarily consists of CONV-based implementations, e.g., CONV $1 \times 1$ as MLP implementation and CONV stem for downsampling. As a result, PoolFormer exhibits faster inference speed. • Second, by comparing DeiT-Small $\textcircled { 1 3 } \textcircled { 1 }$ and LeViT-256 $\mathbb { \left[ \left. 2 3 \right] \right. }$ , we find that MHSA does not bring significant overhead on mobiles if the feature dimensions are consistent and Reshape is not required. Though much more computation intensive, DeiT-Small with a consistent 3D feature can achieve comparable speed to the new ViT variant, i.e., LeViT-256. + +In this work, we propose a dimension-consistent network $( \mathsf { S e c . 4 . 1 } )$ with both 4D feature implementation and 3D MHSA, but the inefficient frequent Reshape operations are eliminated. + +Observation 3: CONV-BN is more latency-favorable than LN (GN)-Linear and the accuracy drawback is generally acceptable. + +![](images/217b3bdcaec421eb6f1a51d1837b74f1a81762c6338312ed2971e23cd2cef64c.jpg) +Figure 3: Overview of EfficientFormer. The network starts with a convolution stem as patch embedding, followed by MetaBlock (MB). The $\tt M B ^ { 4 D }$ and $\mathtt { M B } ^ { 3 D }$ contain different token mixer configurations, i.e., local pooling or global multi-head self-attention, arranged in a dimension-consistent manner. + +Choosing the MLP implementation is another essential design choice. Usually, one of the two options is selected: layer normalization (LN) with 3D linear projection (proj.) and CONV $1 \times 1$ with batch normalization (BN). CONV-BN is more latency favorable because BN can be folded into the preceding convolution for inference speedup, while dynamic normalizations, such as LN and GN, still collects running statistics at the inference phase, thus contributing to latency. From the analysis of DeiT-Small and PoolFormer-S24 in Fig. $\bigtriangledown$ and previous work [17], the latency introduced by LN constitutes around $1 0 \% - 2 0 \%$ latency of the whole network. + +Based on our ablation study in Appendix Tab. 3, CONV-BN only slightly downgrades performance compared to GN and achieves comparable results to channel-wise LN. In this work, we apply CONVBN as much as possible (in all latent 4D features) for the latency gain with a negligible performance drop, while using LN for the 3D features, which aligns with the original MHSA design in ViT and yields better accuracy. + +# Observation 4: The latency of nonlinearity is hardware and compiler dependent. + +Lastly, we study nonlinearity, including GeLU, ReLU, and HardSwish. Previous work $\mathbb { \lVert 1 7 \rVert }$ suggests GeLU is not efficient on hardware and slows down inference. However, we observe GeLU is well supported by iPhone 12 and hardly slower than its counterpart, ReLU. On the contrary, HardSwish is surprisingly slow in our experiments and may not be well supported by the compiler (LeViT-256 latency with HardSwish is $4 4 . 5 \mathrm { m s }$ while with GeLU $1 1 . 9 \mathrm { m s }$ ). We conclude that nonlinearity should be determined on a case-by-case basis given specific hardware and compiler at hand. We believe that most of the activations will be supported in the future. In this work, we employ GeLU activations. + +# 4 Design of EfficientFormer + +Based on the latency analysis, we propose the design of EfficientFormer, demonstrated in Fig. 3. The network consists of a patch embedding (PatchEmbed) and stack of meta transformer blocks, denoted as MB: + +$$ +\mathcal { Y } = \prod _ { i } ^ { m } \mathtt { M B } _ { i } \big ( \mathtt { P a t c h E m b e d } ( \mathcal { X } _ { 0 } ^ { B , 3 , H , W } ) \big ) , +$$ + +where $\mathcal { X } _ { 0 }$ is the input image with batch size as $B$ and spatial size as $[ H , W ]$ , $\mathcal { V }$ is the desired output, and $m$ is the total number of blocks (depth). MB consists of unspecified token mixer (TokenMixer) followed by a MLP block and can be expressed as follows: + +$$ +\begin{array} { r } { \mathcal { X } _ { i + 1 } = \mathtt { M B } _ { i } \big ( \mathcal { X } _ { i } \big ) = \mathtt { M L P } \big ( \mathtt { T o k e n M i x e r } ( \mathcal { X } _ { i } ) \big ) , } \end{array} +$$ + +where ${ \mathcal { X } } _ { i \mid i > 0 }$ is the intermediate feature that forwarded into the $i ^ { t h } \ M \mathrm { B }$ . We further define Stage (or S) as the stack of several MetaBlocks that processes the features with the same spatial size, such as $N _ { 1 } \times$ in Fig. $\textcircled { 3 }$ denoting ${ \tt S } _ { 1 }$ has $N _ { 1 }$ MetaBlocks. The network includes 4 Stages. Among each Stage, there is an embedding operation to project embedding dimension and downsample token length, denoted as Embedding in Fig. $3 .$ With the above architecture, EfficientFormer is a fully transformer-based model without integrating MobileNet structures. Next, we dive into the details of the network design, specifically, the architecture details and the search algorithm. + +# 4.1 Dimension-Consistent Design + +With the observations in Sec. $\textcircled { 3 }$ we propose a dimension consistent design which splits the network into a 4D partition where operators are implemented in CONV-net style $( \mathrm { M B ^ { 4 D } } )$ , and a 3D partition where linear projections and attentions are performed over 3D tensor to enjoy the global modeling power of MHSA without sacrificing efficiency $( \mathrm { M B } ^ { 3 D } )$ , as shown in Fig. 3. Specifically, the network starts with 4D partition, while 3D partition is applied in the last stages. Note that Fig. $\textcircled { 3 }$ is just an instance, the actual length of 4D and 3D partition is specified later through architecture search. + +First, input images are processed by a CONV stem with two $3 \times 3$ convolutions with stride 2 as patch embedding, + +$$ +\begin{array} { r } { \mathcal { X } _ { 1 } ^ { B , C _ { j | j = 1 } , \frac { H } { 4 } , \frac { W } { 4 } } = \mathtt { P a t c h E m b e d } ( \mathcal { X } _ { 0 } ^ { B , 3 , H , W } ) , } \end{array} +$$ + +where $C _ { j }$ is the channel number (width) of the $j t h$ stage. Then the network starts with $\tt M B ^ { 4 D }$ with a simple Pool mixer to extract low level features, + +$$ +\begin{array} { r l } & { \mathcal { T } _ { i } = \mathrm { P o o 1 } ( \mathcal { X } _ { i } ^ { B , C _ { j } , \frac { H } { 2 ^ { j + 1 } } , \frac { W } { 2 ^ { j + 1 } } } ) + \mathcal { X } _ { i } ^ { B , C _ { j } , \frac { H } { 2 ^ { j + 1 } } , \frac { W } { 2 ^ { j + 1 } } } , } \\ & { \mathcal { X } _ { i + 1 } ^ { B , C _ { j } , \frac { H } { 2 ^ { j + 1 } } , \frac { W } { 2 ^ { j + 1 } } } = \mathrm { C o n v } _ { B } ( \mathrm { C o n v } _ { B , G } ( \mathcal { T } _ { i } ) ) + \mathcal { T } _ { i } , } \end{array} +$$ + +where $\mathtt { C o n v } _ { B , G }$ refers to whether the convolution is followed by BN and GeLU, respectively. Note here we do not employ Group or Layer Normalization (LN) before the Pool mixer as in $\pmb { \Vert 6 \Vert }$ , since the 4D partition is CONV-BN based design, thus there exists a BN in front of each Pool mixer. + +After processing all the $\tt M B ^ { 4 D }$ blocks, we perform a one-time reshaping to transform the features size and enter 3D partition. $\mathtt { M B } ^ { 3 D }$ follows conventional ViT structure, as in Fig. $3 .$ Formally, + +$$ +\begin{array} { r l } & { \mathcal { T } _ { i } = \mathtt { L i n e a r } \big ( \mathtt { M H S A } \big ( \mathrm { L i n e a r } \big ( \mathrm { L N } \big ( \mathcal { X } _ { i } ^ { B , \frac { H W } { 4 ^ { j + 1 } } , C _ { j } } \big ) \big ) \big ) \big ) + \mathcal { X } _ { i } ^ { B , \frac { H W } { 4 ^ { j + 1 } } , C _ { j } } , } \\ & { \qquad \mathcal { X } _ { i + 1 } ^ { B , \frac { H W } { 4 ^ { j + 1 } } , C _ { j } } = \mathtt { L i n e a r } \big ( \mathrm { L i n e a r } _ { G } \big ( \mathrm { L N } ( \mathcal { T } _ { i } ) \big ) \big ) + \mathcal { T } _ { i } , } \end{array} +$$ + +where Linear $G$ denotes the Linear followed by GeLU, and + +$$ +\mathtt { M H S A } ( Q , K , V ) = \mathtt { S o f t m a x } ( \frac { Q \cdot K ^ { T } } { \sqrt { C _ { j } } } + b ) \cdot V , +$$ + +where $Q , K , V$ represents query, key, and values learned by the linear projection, and $b$ is parameterized attention bias as position encodings. + +# 4.2 Latency Driven Slimming + +Design of Supernet. Based on the dimension-consistent design, we build a supernet for searching efficient models of the network architecture shown in Fig. $\textcircled { 3 }$ (Fig. $3$ shows an example of searched final network). In order to represent such a supernet, we define the MetaPath (MP), which is the collection of possible blocks: + +$$ +\begin{array} { r l } & { \mathtt { M P } _ { i , j = 1 , 2 } \in \{ \mathtt { M B } _ { i } ^ { 4 D } , I _ { i } \} , } \\ & { \mathtt { M P } _ { i , j = 3 , 4 } \in \{ \mathtt { M B } _ { i } ^ { 4 D } , \mathtt { M B } _ { i } ^ { 3 D } , I _ { i } \} , } \end{array} +$$ + +where $I$ represents identity path, $j$ denotes the $j ^ { t h }$ Stage, and $i$ denotes the $i ^ { t h }$ block. The supernet can be illustrated by replacing $\tt M B$ in Fig. $3$ with MP. + +As in Eqn. $^ { 7 , }$ in ${ \tt S } _ { 1 }$ and ${ \tt S } _ { 2 }$ of the supernet, each block can select from $\tt M B ^ { 4 D }$ or $I$ , while in ${ \sf S } _ { 3 }$ and ${ \tt S } _ { 4 }$ , the block can be $\mathtt { M B } ^ { 3 D }$ , $\mathtt { M B } ^ { 4 D }$ , or $I$ . We only enable $\mathtt { M B } ^ { 3 D }$ in the last two Stages for two reasons. First, since the computation of MHSA grows quadratically with respect to token length, integrating it in early Stages would largely increase the computation cost. Second, applying the global MHSA to the last Stages aligns with the intuition that early stages in the networks capture low-level features, while late layers learn long-term dependencies. + +Searching Space. Our searching space includes $C _ { j }$ (the width of each Stage), $N _ { j }$ (the number of blocks in each Stage, i.e., depth), and last $\mathbb { N }$ blocks to apply $\mathtt { M B } ^ { 3 D }$ . + +Searching Algorithm. Previous hardware-aware network searching methods generally rely on hardware deployment of each candidate in search space to obtain the latency, which is time consuming [71]. In this work, we propose a simple, fast yet effective gradient-based search algorithm to obtain a candidate network that just needs to train the supernet for once. The algorithm has three major steps. + +First, we train the supernet with Gumbel Softmax sampling $\pmb { \mathbb { Z } 2 } \|$ to get the importance score for the blocks within each $\tt M P$ , which can be expressed as + +$$ +\mathcal { X } _ { i + 1 } = \sum _ { n } \frac { e ^ { ( \alpha _ { i } ^ { n } + \epsilon _ { i } ^ { n } ) / \tau } } { \sum _ { n } e ^ { ( \alpha _ { i } ^ { n } + \epsilon _ { i } ^ { n } ) / \tau } } \cdot \mathtt { M P } _ { i , j } ( \mathcal { X } _ { i } ) , +$$ + +where $\alpha$ evaluates the importance of each block in $\tt M P$ as it represents the probability to select a block, e.g., $\mathtt { M B } ^ { 4 D }$ or $\mathtt { M B } ^ { 3 D }$ for the $i ^ { t h }$ block. $\epsilon \sim U ( 0 , 1 )$ ensures exploration, $\tau$ is the temperature, and $n$ represents the type of blocks in MP, i.e., $n \in \{ 4 D , I \}$ for ${ \tt S } _ { 1 }$ and ${ \tt S } _ { 2 }$ , and $n \in \{ 4 D , 3 D , I \}$ for ${ \tt S } _ { 3 }$ and ${ \tt S } _ { 4 }$ . By using Eqn. $\mathbb { B } ,$ the derivatives with respect to network weights and $\alpha$ can be computed easily. The training follows the standard recipe (see Sec. $\underline { { \boldsymbol { \mathsf { F . 1 } } } } \underline { { \boldsymbol { \mathsf { I } } } }$ to obtain the trained weights and architecture parameter $\alpha$ . + +Second, we build a latency lookup table by collecting the on-device latency of $\tt M B ^ { 4 D }$ and $\mathtt { M B } ^ { 3 D }$ with different widths (multiples of 16). + +Finally, we perform network slimming on the supernet obtained from the first step through latency evaluation using the lookup table. Note that a typical gradient-based searching algorithm simply select the block with largest $\alpha \left\| \overline { { \mathbb { Z } \mathrm { 2 } } } \right\|$ , which does not fit our scope as it cannot search the width $C _ { j }$ . In fact, constructing a multiple-width supernet is memory-consuming and even unrealistic given that each MP has several branches in our design. Instead of directly searching on the complex searching space, we perform a gradual slimming on the single-width supernet as follows. + +We first define the importance score for $\mathtt { M P } _ { i }$ as ↵4D $\frac { \alpha _ { i } ^ { 4 D } } { \alpha _ { i } ^ { I } }$ and $\frac { \alpha _ { i } ^ { 3 D } + \alpha _ { i } ^ { 4 D } } { \alpha _ { i } ^ { I } }$ for $\mathtt { S } _ { 1 , 2 }$ and $\mathsf { S } _ { 3 , 4 }$ , respectively. Similarly, the importance score for each Stage can be obtained by summing up the scores for all MP within the Stage. With the importance score, we define the action space that includes three options: 1) select $I$ for the least important MP, 2) remove the first $\mathtt { M B } ^ { 3 D }$ , and 3) reduce the width of the least important Stage (by multiples of 16). Then, we calculate the resulting latency of each action through lookup table, and evaluate the accuracy drop of each action. Lastly, we choose the action based on per-latency accuracy drop $\big ( \frac { - \% } { m s } \big )$ . This process is performed iteratively until target latency is achieved. We show more details of the algorithm in Appendix. + +# 5 Experiments and Discussion + +We implement EfficientFormer through PyTorch 1.11 $\mathbb { \left[ \left. \left. \right. Z 3 \right] \right. }$ and Timm library $\pmb { \Vert 7 4 \Vert }$ , which is the common practice in recent arts $\boxed { 1 8 } \boxed { 6 }$ . Our models are trained on a cluster with NVIDIA A100 and V100 GPUs. The inference speed on iPhone 12 (A14 bionic chip) is measured with iOS version 15 and averaged over $1 , 0 0 0$ runs, with all available computing resources (NPU), or CPU only. CoreMLTools is used to deploy the run-time model. In addition, we provide latency analysis on Nvidia A100 GPU with batch size 64 to exploit hardware roofline. The trained PyTorch models are deployed in ONNX format and are compiled with TensorRT. We report GPU runtime that excludes preprocessing. We provide the detailed network architecture and more ablation studies in Appendix + +# 5.1 Image Classification + +All EfficientFormer models are trained from scratch on ImageNet-1K dataset $\pmb { \mathbb { B 4 } }$ to perform the image classification task. We employ standard image size $( 2 2 4 \times 2 2 4 )$ for both training and testing. We follow the training recipe from DeiT $\mathbb { \left[ 3 \right] }$ but mainly report results with 300 training epochs to have the comparison with other ViT-based models. We use AdamW optimizer [75, 76], warm-up training with 5 epochs, and a cosine annealing learning rate schedule. The initial learning rate is set as $1 0 ^ { - 3 } \times ( b a t c h \ : s i z e / 1 0 2 4 )$ and the minimum learning rate is $1 0 ^ { - 5 }$ . The teacher model for distillation is RegNetY-16GF [77] pretrained on ImageNet with $8 2 . 9 \%$ top-1 accuracy. Results are demonstrated in Tab. 1 and Fig. 1 + +Table 1: Comparison results on ImgeNet-1K. The latency results are tested on iPhone Neural Engine (NPU), iPhone CPU and Nvidia A100 GPU correspondingly. Note that for mobile speed, we report latency per frame, while on A100 GPU, we report latency per batch size 64 to maximum resource utilization. Hybrid refers to a mixture of MobileNet blocks and ViT blocks. (-) refers to unrevealed or unsupported models. †Latency measured with GeLU activation for fair comparison, the original LeViT-256 model with HardSwish activations runs at $4 4 . 5 \mathrm { m s }$ . Different training seeds lead to less than $\pm 0 . 2 \%$ fluctuation in accuracy for EfficientFormer, and the error for latency benchmark is less than $\pm 0 . 1 \mathrm { { m s } }$ . + +
ModelTypeParams(M)GMACsTrain. EpochTop-1(%)Latency (ms)
NPUCPUA100
MobileNetV2×1.0CONV3.50.330071.81.38.05.0
MobileNetV2×1.4CONV6.10.630074.71.610.77.3
ResNet50CONV25.54.130078.53.029.49.0
EfficientNet-B0CONV5.30.435077.12.714.510.0
EfficientNet-B3CONV12.01.835081.66.652.635.0
EffcientNet-B5CONV30.09.935083.623.0258.8141.0
DeiT-TAttention5.91.2300/100074.5/76.69.216.77.1
DeiT-SAttention22.54.5300/100081.2/82.611.841.015.5
PVT-SmallAttention24.53.830079.824.489.523.8
T2T-ViT-14Attention21.54.831081.5--21.0
Swin-TinyAttention294.530081.3-■22.0
CSwin-TAttention234.330082.7--28.7
PoolFormer-s12Pool122.030077.26.159.014.5
PoolFormer-s24Pool213.630080.36.2126.728.2
PoolFormer-s36Pool315.230081.46.7192.641.2
ResMLP-S24SMLP306.030079.47.640.217.4
Convmixer-768Hybrid21.120.730080.211.629.3-
LeViT-256Hybrid18.91.1100081.611.9 †13.54.5
NASViT-A5Hybrid-0.7636081.8-■-
MobileViT-XSHybrid2.30.730074.87.226.511.7
MobileFormer-508MHybrid14.00.5145079.313.222.214.6
EfficientFormer-L1MetaBlock12.31.3300/100079.2/80.21.611.56.2
EfficientFormer-L3MetaBlock31.33.930082.43.028.213.9
EfficientFormer-L7MetaBlock82.110.230083.37.067.730.7
+ +Comparison to CNNs. Compared with the widely used CNN-based models, EfficientFormer achieves a better trade-off between accuracy and latency. On iPhone Neural Engine, EfficientFormerL1 runs at MobileNet ${ \tt V } 2 { \times } 1 . 4$ speed while achieving $4 . 5 \%$ higher top-1 accuracy. In addition, EfficientFormer-L3 runs at a similar speed to EfficientNet-B0 while achieving relative $5 . 3 \%$ higher top-1 accuracy. For the models with high performance $( > 8 3 \%$ top-1), EfficientFormer-L7 runs more than $3 \times$ faster than EfficientNet-B5, demonstrating the advantageous performance of our models. Moreover on desktop GPU (A100), EfficientFormer-L1 runs $38 \%$ faster than EfficientNet-B0 while achieving $2 . 1 \%$ higher top-1 accuracy. EfficientFormer-L7 runs $4 . 6 \times$ faster than EfficientNet-B5. These results allow us to answer the central question raised earlier; ViTs do not need to sacrifice latency to achieve good performance, and an accurate ViT can still have ultra-fast inference speed as lightweight CNNs do. + +Comparison to ViTs. Conventional ViTs are still under-performing CNNs in terms of latency. For instance, DeiT-Tiny achieves similar accuracy to EfficientNet-B0 while it runs $3 . 4 \times$ slower. However, EfficientFormer performs like other transformer models while running times faster. EfficientFormerL3 achieves higher accuracy than DeiT-Small $( 8 2 . 4 \%$ vs. $8 1 . 2 \%$ ) while being $4 \times$ faster. It is notable that though the recent transformer variant, PoolFormer $\textcircled { 6 }$ , naturally has a consistent 4D architecture and runs faster compared to typical ViTs, the absence of global MHSA greatly limits the performance upper-bound. EfficientFormer-L3 achieves $1 \%$ higher top-1 accuracy than PoolFormer-S36, while being $3 \times$ faster on Nvidia A100 GPU, $2 . 2 \times$ faster on iPhone NPU and $6 . 8 \times$ faster on iPhone CPU. + +Comparison to Hybrid Designs. Existing hybrid designs, e.g., LeViT-256 and MobileViT, still struggle with the latency bottleneck of ViTs and can hardly outperform lightweight CNNs. For example, LeViT-256 runs slower than DeiT-Small while having $1 \%$ lower top-1 accuracy. For MobileViT, which is a hybrid model with both MHSA and MobileNet blocks, we observe that it is significantly slower than CNN counterparts, e.g., MobileNetV2 and EfficientNet-B0, while the accuracy is not satisfactory either $( 2 . 3 \%$ lower than EfficientNet-B0). Thus, simply trading-off MHSA with MobileNet blocks can hardly push forward the Pareto curve, as in Fig. $i _ { \cdot }$ In contrast, EfficientFormer, as pure transformer-based model, can maintain high performance while achieving ultra-fast inference speed. EfficientFormer-L1 has $4 . 4 \%$ higher top-1 accuracy than MobileViT-XS and runs much faster across different hardware and compilers ( $1 . 9 \times$ faster on Nvidia A100 GPU Computing, $2 . 3 \times$ faster on iPhone CPU, and $4 . 5 \times$ faster on iPhone NPU). At a similar inference time, EfficientFormer-L7 outperforms MobileViT-XS by $8 . 5 \%$ top-1 accuracy on ImageNet, demonstrating the superiority of our design. + +Table 2: Comparison results using EfficientFormer as backbone. Results on object detection $\&$ instance segmentation are obtained from COCO 2017. Results on semantic segmentation are obtained from ADE20K. + +
BackboneDetection & Instance SegmentationSemantic mIoU(%)
ApboxAP5APbAPmaskAPmask
ResNet1834.054.036.731.251.0APmask 32.732.9
PoolFormer-S1237.359.040.134.655.836.937.2
EfficientFormer-L137.960.341.035.457.337.338.9
ResNet5038.058.641.434.455.136.736.7
PoolFormer-S2440.162.243.437.059.139.640.3
EfficientFormer-L341.463.944.738.161.040.443.5
ResNet10140.461.144.236.457.738.838.8
PoolFormer-S3641.063.144.837.760.140.042.0
EfficientFormer-L742.665.146.139.062.2 41.745.1
+ +# 5.2 EfficientFormer as Backbone + +Object Detection and Instance Segmentation. We follow the implementation of Mask-RCNN [78] to integrate EfficientFormer as the backbone and verify performance. We experiment over COCO2017 $\dot { \left[ \left| \begin{array} { l } { \overline { { 7 9 } } } \end{array} \right| \right] }$ which contains training and validations sets of 118K and 5K images, respectively. The EfficientFormer backbone is initialized with ImageNet-1K pretrained weights. Similar to prior work $\pmb { \mathbb { H } }$ , we use AdamW optimizer $\textcircled { 1 7 5 } , \textcircled { 7 6 } $ with initial learning rate of $2 \times 1 0 ^ { - 4 }$ , and train the model for 12 epochs. We set the input size as $1 3 3 3 \times 8 0 0$ . + +The results for detection and instance segmentation are shown in Tab. 2. EfficientFormers consistently outperform CNN (ResNet) and transformer (PoolFormer) backbones. With similar computation cost, EfficientFormer-L3 outperforms ResNet50 backbone by 3.4 box AP and 3.7 mask AP, and outperforms PoolFormer-S24 backbone with 1.3 box AP and 1.1 mask AP, proving that EfficientFormer generalizes well as a strong backbone in vision tasks. + +Semantic Segmentation. We further validate the performance of EfficientFormer on the semantic segmentation task. We use the challenging scene parsing dataset, ADE20K [80, 81], which contains 20K training images and 2K validation ones covering 150 class categories. Similar to existing work $\pmb { \mathbb { H } }$ , we build EfficientFormer as backbone along with Semantic FPN $[ \textcircled { 8 2 } ]$ as segmentation decoder for fair comparison. The backbone is initialized with pretrained weights on ImageNet-1K and the model is trained for 40K iterations with a total batch size of 32 over 8 GPUs. We follow the common practice in segmentation $[ 6 , 1 1 3 ]$ , use AdamW optimizer $\boxed { 7 5 } \boxed { 7 6 }$ , and apply a poly learning rate schedule with power 0.9, starting from a initial learning rate $2 \times 1 0 ^ { - 4 }$ . We resize and crop input images to $5 1 2 \times 5 1 2$ for training and shorter side as 512 for testing (on validation set). + +As shown in Tab. 2, EfficientFormer consistently outperforms CNN- and transformer-based backbones by a large margin under a similar computation budget. For example, EfficientFormer-L3 outperforms PoolFormer-S24 by 3.2 mIoU. We show that with global attention, EfficientFormer learns better long-term dependencies, which is beneficial in high-resolution dense prediction tasks. + +# 5.3 Discussion + +Relations to MetaFormer. The design of EfficientFormer is partly inspired by the MetaFormer concept $\textcircled { 6 }$ . Compared to PoolFormer, EfficientFormer addresses the dimension mismatch problem, which is a root cause of inefficient edge inference, thus being capable of utilizing global MHSA without sacrificing speed. Consequently, EfficientFormer exhibits advantageous accuracy performance over PoolFormer. In spite of its fully 4D design, PoolFormer employs inefficient patch embedding and group normalization (Fig. 2), leading to increased latency. Instead, our redesigned 4D partition of EfficientFormer (Fig. 3) is more hardware friendly and exhibits better performance across several tasks. + +Limitations. (i) Though most designs in EfficientFormer are general-purposed, e.g., dimensionconsistent design and 4D block with CONV-BN fusion, the actual speed of EfficientFormer may vary on other platforms. For instance, if GeLU is not well supported while HardSwish is efficiently implemented on specific hardware and compiler, the operator may need to be modified accordingly. (ii) The proposed latency-driven slimming is simple and fast. However, better results may be achieved if search cost is not a concern and an enumeration-based brute search is performed. + +# 6 Conclusion + +In this work, we show that Vision Transformer can operate at MobileNet speed on mobile devices. Starting from a comprehensive latency analysis, we identify inefficient operators in a series of ViTbased architectures, whereby we draw important observations that guide our new design paradigm. The proposed EfficientFormer complies with a dimension consistent design that smoothly leverages hardware-friendly 4D MetaBlocks and powerful 3D MHSA blocks. We further propose a fast latencydriven slimming method to derive optimized configurations based on our design space. Extensive experiments on image classification, object detection, and segmentation tasks show that EfficientFormer models outperform existing transformer models while being faster than most competitive CNNs. 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ModelTypeParams(M)GMACsTrain. EpochTop-1(%)Latency (ms)
NPUCPUA100
MobileNetV2×1.0CONV3.50.330071.81.38.05.0
MobileNetV2×1.4CONV6.10.630074.71.610.77.3
ResNet50CONV25.54.130078.53.029.49.0
EfficientNet-B0CONV5.30.435077.12.714.510.0
EfficientNet-B3CONV12.01.835081.66.652.635.0
EffcientNet-B5CONV30.09.935083.623.0258.8141.0
DeiT-TAttention5.91.2300/100074.5/76.69.216.77.1
DeiT-SAttention22.54.5300/100081.2/82.611.841.015.5
PVT-SmallAttention24.53.830079.824.489.523.8
T2T-ViT-14Attention21.54.831081.5--21.0
Swin-TinyAttention294.530081.3-■22.0
CSwin-TAttention234.330082.7--28.7
PoolFormer-s12Pool122.030077.26.159.014.5
PoolFormer-s24Pool213.630080.36.2126.728.2
PoolFormer-s36Pool315.230081.46.7192.641.2
ResMLP-S24SMLP306.030079.47.640.217.4
Convmixer-768Hybrid21.120.730080.211.629.3-
LeViT-256Hybrid18.91.1100081.611.9 †13.54.5
NASViT-A5Hybrid-0.7636081.8-■-
MobileViT-XSHybrid2.30.730074.87.226.511.7
MobileFormer-508MHybrid14.00.5145079.313.222.214.6
EfficientFormer-L1MetaBlock12.31.3300/100079.2/80.21.611.56.2
EfficientFormer-L3MetaBlock31.33.930082.43.028.213.9
EfficientFormer-L7MetaBlock82.110.230083.37.067.730.7
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BackboneDetection & Instance SegmentationSemantic mIoU(%)
ApboxAP5APbAPmaskAPmask
ResNet1834.054.036.731.251.0APmask 32.732.9
PoolFormer-S1237.359.040.134.655.836.937.2
EfficientFormer-L137.960.341.035.457.337.338.9
ResNet5038.058.641.434.455.136.736.7
PoolFormer-S2440.162.243.437.059.139.640.3
EfficientFormer-L341.463.944.738.161.040.443.5
ResNet10140.461.144.236.457.738.838.8
PoolFormer-S3641.063.144.837.760.140.042.0
EfficientFormer-L742.665.146.139.062.2 41.745.1
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0000000000000000000000000000000000000000..5eab5c6c152a574457262f80b514896cf1df708d --- /dev/null +++ b/parse/dev/TQ75Md-FqQp/TQ75Md-FqQp.md @@ -0,0 +1,852 @@ +# EFFICIENT AND MODULAR IMPLICIT DIFFERENTIATION + +# Anonymous authors + +Paper under double-blind review + +# ABSTRACT + +Automatic differentiation (autodiff) has revolutionized machine learning. It allows expressing complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently, differentiation of optimization problem solutions has attracted widespread attention with applications such as optimization layers, and in bi-level problems such as hyper-parameter optimization and meta-learning. However, so far, implicit differentiation remained difficult to use for practitioners, as it often required case-by-case tedious mathematical derivations and implementations. In this paper, we propose an efficient and modular approach for implicit differentiation of optimization problems. In our approach, the user defines directly in Python a function $F$ capturing the optimality conditions of the problem to be differentiated. Once this is done, we leverage autodiff of $F$ and implicit differentiation to automatically differentiate the optimization problem. Our approach thus combines the benefits of implicit differentiation and autodiff. It is efficient as it can be added on top of any state-of-the-art solver and modular as the optimality condition specification is decoupled from the implicit differentiation mechanism. We show that seemingly simple principles allow to recover many existing implicit differentiation methods and create new ones easily. We demonstrate the ease of formulating and solving bi-level optimization problems using our framework. We also showcase an application to the sensitivity analysis of molecular dynamics. + +# 1 INTRODUCTION + +Automatic differentiation (autodiff) is now an inherent part of machine learning software. It allows to express complex computations by composing elementary ones in creative ways and removes the tedious burden of computing their derivatives by hand. In parallel, the differentiation of optimization problem solutions has found many applications. A classical example is bi-level optimization, which typically involves computing the derivatives of a nested optimization problem in order to solve an outer one. Examples of applications in machine learning include hyper-parameter optimization (Chapelle et al., 2002; Seeger, 2008; Pedregosa, 2016; Franceschi et al., 2017; Bertrand et al., 2020; 2021), neural networks (Lorraine et al., 2020), and meta-learning (Franceschi et al., 2018; Rajeswaran et al., 2019). Another line of active research involving differentiation of optimization problem solutions are optimization layers (Kim et al., 2017; Amos & Kolter, 2017; Niculae & Blondel, 2017; Djolonga & Krause, 2017; Gould et al., 2019), which can be used to encourage structured outputs, and implicit deep networks (Bai et al., 2019; El Ghaoui et al., 2019), which have a smaller memory footprint than backprop-trained networks. + +Since optimization problem solutions typically do not enjoy an explicit formula in terms of their inputs, autodiff cannot be used directly to differentiate these functions. In recent years, two main approaches have been developed to circumvent this problem. The first one consists of unrolling the iterations of an optimization algorithm and using the final iteration as a proxy for the optimization problem solution (Wengert, 1964; Domke, 2012; Deledalle et al., 2014; Franceschi et al., 2018; Ablin et al., 2020). This allows to explicitly construct a computational graph relating the algorithm output to the inputs, on which autodiff can then be used transparently. However, this requires a reimplementation of the algorithm using the autodiff system, and not all algorithms are necessarily autodiff friendly. Moreover, forward-mode autodiff has time complexity that scales linearly with the number of variables and reverse-mode autodiff has memory complexity that scales linearly with the number of algorithm iterations. In contrast, a second approach consists in implicitly relating an optimization problem solution to its inputs using optimality conditions. In a machine learning context, such implicit differentiation has been used for stationarity conditions (Bengio, 2000; + +Lorraine et al., 2020), KKT conditions (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae & Martins, 2020) and the proximal gradient fixed point (Niculae & Blondel, 2017; Bertrand et al., 2020; 2021). An advantage of implicit differentiation is that a solver reimplementation is not needed, allowing to build upon decades of state-of-the-art software. Although implicit differentiation has a long history in numerical analysis (Griewank & Walther, 2008; Bell & Burke, 2008; Krantz & Parks, 2012; Bonnans & Shapiro, 2013), so far, it remained difficult to use for practitioners, as it required a case-by-case tedious mathematical derivation and implementation. CasADi (Andersson et al., 2019) allows to differentiate various optimization and root finding problem algorithms provided by the library. However, it does not allow to easily add implicit differentiation on top of existing solvers from optimality conditions expressed by the user, as we do. A recent tutorial explains how to implement implicit differentiation in JAX (Duvenaud et al., 2020). However, the tutorial requires the user to take care of low-level technical details and does not cover a large catalog of optimality condition mappings as we do. Other work (Agrawal et al., 2019a) attempts to address this issue by adding implicit differentiation on top of cvxpy (Diamond & Boyd, 2016). This works by reducing all convex optimization problems to a conic program and using conic programming’s optimality conditions to derive an implicit differentiation formula. While this approach is very generic, solving a convex optimization problem using a conic programming solver—an ADMM-based splitting conic solver (O’Donoghue et al., 2016) in the case of cvxpy—is rarely the state-of-the-art approach for each particular problem instance. + +In this work, we adopt a different strategy that makes it easy to add implicit differentiation on top of any existing solver. In our approach, the user defines directly in Python a mapping function $F$ capturing the optimality conditions of the problem solved by the algorithm. Once this is done, we leverage autodiff of $F$ combined with implicit differentiation to automatically differentiate the optimization problem solution. In this way, our approach is generic, yet it can exploit the efficiency of state-of-the-art solvers. It therefore combines the benefits of implicit differentiation and autodiff. To summarize, we make the following contributions. + +• We describe our framework and its JAX implementation (provided in the supplementary material). Our framework significantly lowers the barrier to use implicit differentiation thanks to the use of autodiff of the optimality conditions and the seamless integration in JAX. Our framework significantly extends JAX for numerical optimization, with low-level details all abstracted away. • We instantiate our framework on a large catalog of optimality conditions (Table 1), recovering existing schemes and obtaining new ones, such as the mirror descent fixed point based one. • On the theoretical side, we provide new bounds on the Jacobian error when the optimization problem is only solved approximately, and empirically validate them. • We implement four illustrative applications, demonstrating our framework’s ease of use. + +Beyond our software implementation, we hope this paper provides a self-contained blueprint for creating an efficient and modular implementation of implicit differentiation. + +Notation. We denote the gradient and Hessian of $f \colon { \mathbb { R } ^ { d } } \to { \mathbb { R } }$ evaluated at $x \in \mathbb { R } ^ { d }$ by $\nabla f ( x ) \in \mathbb { R } ^ { d }$ and $\nabla ^ { 2 } f ( x ) \in \mathbb { R } ^ { d \times d }$ . We denote the Jacobian of $F \colon { \mathbb { R } ^ { d } } \to { \mathbb { R } ^ { p } }$ evaluated at $x \in \mathbb { R } ^ { d }$ by $\partial F ( x ) \in$ $\mathbb { R } ^ { p \times d }$ . When $f$ or $F$ have several arguments, we denote the gradient, Hessian and Jacobian in the $i ^ { \mathrm { { t h } } }$ argument by $\nabla _ { i }$ , $\nabla _ { i } ^ { 2 }$ and $\partial _ { i }$ , respectively. The standard probability simplex is denoted by $\triangle ^ { d } : = \{ x \in \mathbb { R } ^ { d } \colon \| x \| _ { 1 } = 1 , x \geq 0 \}$ . For any set $\mathcal { C } \subset \mathbb { R } ^ { d }$ , we denote the indicator function $I _ { \mathcal { C } } : \mathbb { R } ^ { d } \overset { \cdot } { } \mathbb { R } \cup \{ + \infty \}$ where $I _ { \mathit { C } } ( x ) = 0$ if $x \in { \mathcal { C } }$ , $I _ { C } ( x ) = + \infty$ otherwise. For a vector or matrix $A$ we note $\| A \|$ the Frobenius (or Euclidean) norm, and $\| A \| _ { \mathsf { o p } }$ the operator norm. + +# 2 COMBINING IMPLICIT DIFFERENTIATION AND AUTODIFF + +# 2.1 GENERAL PRINCIPLES + +Overview. Contrary to autodiff through unrolled algorithm iterations, implicit differentiation typically involves a manual, sometimes complicated, mathematical derivation. For instance, numerous works (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae & Martins, 2020) use Karush–Kuhn–Tucker (KKT) conditions in order to relate a constrained optimization problem’s solution to its inputs, and to manually derive a formula for its derivatives. The derivation and implementation in these works are always case-by-case. + +![](images/fadbf82543eb3718bc2bb5820799eaf0ad81197f72ee1ae17d032dbb61832239.jpg) +Figure 1: Adding implicit differentiation on top of a ridge regression solver. The function $f ( x , \theta )$ defines the objective function and the mapping $F$ , here simply equation (4), captures the optimality conditions. Our decorator @custom_root automatically adds implicit differentiation to the solver for the user, overriding JAX’s default behavior. The last line evaluates the Jacobian at $\theta = 1 0$ . + +In this work, we propose a generic way to easily add implicit differentiation on top of existing solvers. In our approach, the user defines directly in Python a mapping function $F$ capturing the optimality conditions of the problem solved by the algorithm. We provide reusable building blocks to easily express such $F$ . The provided $F$ is then plugged into our Python decorator @custom_root, which we append on top of the solver declaration we wish to differentiate. Under the hood, we combine implicit differentiation and autodiff of $F$ to automatically differentiate the optimization problem solution. A simple illustrative example is given in Figure 1. + +Differentiating a root. Let $F \colon { \mathbb { R } } ^ { d } \times { \mathbb { R } } ^ { n } \to { \mathbb { R } } ^ { d }$ be a user-provided mapping, capturing the optimality conditions of a problem. An optimal solution, denoted $x ^ { \star } ( \theta )$ , should be a root of $F$ : + +$$ +{ \cal F } ( x ^ { \star } ( \theta ) , \theta ) = 0 . +$$ + +We can see $x ^ { \star } ( \theta )$ as an implicitly defined function of $\boldsymbol \theta \in \mathbb { R } ^ { n }$ , i.e., $x ^ { \star } \colon { \mathbb { R } } ^ { n } \to { \mathbb { R } } ^ { d }$ . More precisely, from the implicit function theorem (Griewank $\&$ Walther, 2008; Krantz & Parks, 2012), we know that for $( x _ { 0 } , \theta _ { 0 } )$ satisfying $F ( x _ { 0 } , \theta _ { 0 } ) = 0$ with a continuously differentiable $F$ , if the Jacobian $\partial _ { 1 } F$ evaluated at $( x _ { 0 } , \theta _ { 0 } )$ is a square invertible matrix, then there exists a function $x ^ { \star } ( \cdot )$ defined on a neighborhood of $\theta _ { 0 }$ such that $x ^ { \star } ( \theta _ { 0 } ) = x _ { 0 }$ . Furthermore, for all $\theta$ in this neighborhood, we have that $F ( x ^ { \star } ( \theta ) , \theta ) = 0$ and ${ \partial x ^ { \star } ( \theta ) }$ exists. Using the chain rule, the Jacobian ${ \partial x ^ { \star } ( \theta ) }$ satisfies + +$$ +\partial _ { 1 } F ( x ^ { \star } ( \theta ) , \theta ) \partial x ^ { \star } ( \theta ) + \partial _ { 2 } F ( x ^ { \star } ( \theta ) , \theta ) = 0 . +$$ + +Computing ${ \partial x ^ { \star } ( \theta ) }$ therefore boils down to the resolution of the linear system of equations + +$$ +\underbrace { - \partial _ { 1 } F ( x ^ { \star } ( \theta ) , \theta ) } _ { A \in \mathbb { R } ^ { d \times d } } \underbrace { \partial x ^ { \star } ( \theta ) } _ { J \in \mathbb { R } ^ { d \times n } } = \underbrace { \partial _ { 2 } F ( x ^ { \star } ( \theta ) , \theta ) } _ { B \in \mathbb { R } ^ { d \times n } } . +$$ + +When (1) is a one-dimensional root finding problem $\langle d = 1 \rangle$ ), (2) becomes particularly simple since we then have $\nabla x ^ { \star } ( \theta ) = B ^ { \top } / A$ , where $A$ is a scalar value. + +We will show that existing and new implicit differentiation methods all reduce to this simple principle. We call our approach hybrid, since it combines implicit differentiation (and as such requires solving a linear system) with autodiff of the optimality conditions $F$ . Our approach is efficient as it can be added on top of any state-of-the-art solver and modular as the optimality condition specification is decoupled from the implicit differentiation mechanism. This contrasts with existing works, where the mathematical derivation and implementation are specific to each optimality condition. + +Differentiating a fixed point. We will encounter numerous applications where $x ^ { \star } ( \theta )$ is implicitly defined through a fixed point: + +$$ +x ^ { \star } ( \theta ) = T ( x ^ { \star } ( \theta ) , \theta ) , +$$ + +where $T \colon { \mathbb { R } } ^ { d } \times { \mathbb { R } } ^ { n } \to { \mathbb { R } } ^ { d }$ . This can be seen as a particular case of (1) by defining the residual + +$$ +F ( x , { \boldsymbol { \theta } } ) = T ( x , { \boldsymbol { \theta } } ) - x . +$$ + +In this case, using the chain rule, we have + +$$ +A = - \partial _ { 1 } F ( x ^ { \star } ( \theta ) , \theta ) = I - \partial _ { 1 } T ( x ^ { \star } ( \theta ) , \theta ) \quad \mathrm { a n d } \quad B = \partial _ { 2 } F ( x ^ { \star } ( \theta ) , \theta ) = \partial _ { 2 } T ( x ^ { \star } ( \theta ) , \theta ) . +$$ + +Computing JVPs and VJPs. In most practical scenarios, it is not necessary to explicitly form the Jacobian matrix, and instead it is sufficient to left-multiply or right-multiply by $\partial _ { 1 } F$ and $\partial _ { 2 } F$ . These are called vector-Jacobian product (VJP) and Jacobian-vector product (JVP), and are useful for integrating $x ^ { \star } ( \theta )$ with reverse-mode and forward-mode autodiff, respectively. Oftentimes, $F$ will be explicitly defined. In this case, computing the VJP or JVP can be done via autodiff. In some cases, $F$ may itself be implicitly defined, for instance when $F$ involves the solution of a variational problem. In this case, computing the VJP or JVP will itself involve implicit differentiation. + +The right-multiplication (JVP) between $J = \partial x ^ { \star } ( \theta )$ and a vector $v$ , $J v _ { : }$ , can be computed efficiently by solving $A ( \bar { J v } ) = B v$ . The left-multiplication (VJP) of $v ^ { \top }$ with J , $v ^ { \top } J$ , can be computed by first solving $A ^ { \top } u = v$ . Then, we can obtain $\mathbf { \Delta } _ { v } \top { } _ { J }$ by $\boldsymbol { v } ^ { \top } \boldsymbol { J } = \boldsymbol { u } ^ { \top } \boldsymbol { A } \boldsymbol { J } = \boldsymbol { u } ^ { \top } \boldsymbol { B }$ . Note that when $B$ changes but $A$ and $v$ remain the same, we do not need to solve $A ^ { \top } u = v$ once again. This allows to compute the VJP w.r.t. different variables while solving only one linear system. + +To solve these linear systems, we can use the conjugate gradient method (Hestenes et al., 1952) when $A$ is symmetric positive semi-definite and GMRES (Saad & Schultz, 1986) or BiCGSTAB (Vorst & van der Vorst, 1992) otherwise. These algorithms are all matrix-free: they only require matrix-vector products. Thus, all we need from $F$ is its JVPs or VJPs. An alternative to GMRES/BiCGSTAB is to solve the normal equation $A A ^ { \top } u = A v$ using conjugate gradient. We implement this using JAX’s automatic transpose routine jax.linear_transpose (Frostig et al., 2021). + +Pre-processing and post-processing mappings. Oftentimes, the goal is not to differentiate $\theta$ per se, but the parameters of a function producing $\theta$ . One example of such pre-processing is to convert the parameters to be differentiated from one form to another canonical form, such as a quadratic program (Amos & Kolter, 2017) or a conic program (Agrawal et al., 2019a). Another example is when $x ^ { \star } ( \theta )$ is used as the output of a neural network layer, in which case $\theta$ is produced by the previous layer. Likewise, $x ^ { \star } ( \theta )$ will often not be the final output we want to differentiate. One example of such post-processing is when $x ^ { \star } ( \theta )$ is the solution of a dual program and we apply the dual-primal mapping to recover the solution of the primal program. Another example is the application of a loss function, in order to reduce $x ^ { \star } ( \theta )$ to a scalar value. We leave the differentiation of such pre/post-processing mappings to the autodiff system, allowing to compose functions in complex ways. + +Implementation details. When a solver function is decorated with @custom_root, we use jax.custom_jvp and jax.custom_vjp to automatically add custom JVP and VJP rules to the function, overriding JAX’s default behavior. As mentioned above, we use linear system solvers based on matrix-vector products and therefore we only need access to $F$ through the JVP or VJP with $\partial _ { 1 } F$ and $\partial _ { 2 } F$ . This is done by using $\ j \mathtt { a x } . \mathtt { j v p }$ and ${ \dot { \mathsf { J } } } \mathsf { a x } . \mathsf { v } { \dot { \mathsf { J } } } { \mathsf { p } }$ , respectively. Note that, as in Figure 1, the definition of $F$ will often include a gradient mapping $\bar { \nabla } _ { 1 } f ( x , \bar { \theta } )$ . Thankfully, JAX supports second-order derivatives transparently. For convenience, our library also provides a @custom_fixed_point decorator, for adding implicit differentiation on top of a solver, given a fixed point iteration $T$ ; see code examples in Appendix A. + +# 2.2 EXAMPLES + +We now give various examples of mapping $F$ or fixed point iteration $T$ , recovering existing implicit differentiation methods and creating new ones. Each choice of $F$ or $T$ implies different trade-offs in terms of computational oracles; see Table 1. Source code examples are given in Appendix A. + +Stationary point condition. The simplest example is to differentiate through the implicit function + +$$ +x ^ { \star } ( \theta ) = \underset { x \in \mathbb { R } ^ { d } } { \operatorname { a r g m i n } } f ( x , \theta ) , +$$ + +where $f \colon { \mathbb { R } ^ { d } } \times { \mathbb { R } ^ { n } } \to { \mathbb { R } }$ is twice differentiable. In this case, $F$ is simply the gradient mapping + +$$ +F ( x , \theta ) = \nabla _ { 1 } f ( x , \theta ) . +$$ + +Table 1: Summary of optimality condition mappings. Oracles are accessed through their JVP or VJP. + +
NameEquationSolution neededOracles needed
Stationary(4), (5)PrimalV1f
KKT(6)Primal and dualVif,H,G,aH,0G
Proximal gradient(7)PrimalVif,proxng
Projected gradient(9)PrimalVif,projc
Mirror descent(11)PrimalVif,proje,
Newton(14)Primal[V²f(x,0)]-1,Vif(x,0)
Block proximal gradient(15)Primal[Vif]j,[proXnglj
Conic programming(18)Residual map rootprojRP × K* XR+
+ +We then have $\partial _ { 1 } F ( x , \theta ) = \nabla _ { 1 } ^ { 2 } f ( x , \theta )$ and $\partial _ { 2 } F ( x , \theta ) = \partial _ { 2 } \nabla _ { 1 } f ( x , \theta )$ , the Hessian of $f$ in its first argument and the Jacobian in the second argument of $\nabla _ { 1 } f ( x , \theta )$ . In practice, we use autodiff to compute Jacobian products automatically. Equivalently, we can use the gradient descent fixed point + +$$ +T ( x , \theta ) = x - \eta \nabla _ { 1 } f ( x , \theta ) , +$$ + +for all $\eta > 0$ . Using (3), it is easy to check that we obtain the same linear system since $\eta$ cancels out. + +KKT conditions. We now show that the KKT conditions, manually differentiated in several works (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae $\&$ Martins, 2020), fit our framework. As we will see, the key will be to group the optimal primal and dual variables as our $x ^ { \star } ( \theta )$ . Let us consider the general problem + +$$ +\operatorname { a r g m i n } _ { z \in \mathbb { R } ^ { p } } f ( z , \theta ) \quad { \mathrm { s u b j e c t ~ t o } } \quad G ( z , \theta ) \leq 0 , \ H ( z , \theta ) = 0 , +$$ + +where $z \in \mathbb { R } ^ { p }$ is the primal variable, $f \colon { \mathbb { R } } ^ { p } \times { \mathbb { R } } ^ { n } \to { \mathbb { R } }$ , $G \colon \mathbb { R } ^ { p } \times \mathbb { R } ^ { n } \to \mathbb { R } ^ { r }$ and $H \colon \mathbb { R } ^ { p } \times \mathbb { R } ^ { n } \to \mathbb { R } ^ { q }$ The stationarity, primal feasibility and complementary slackness conditions give + +$$ +\begin{array} { r } { \nabla _ { 1 } f ( z , \theta ) + \left[ \partial _ { 1 } G ( z , \theta ) \right] ^ { \top } \boldsymbol { \lambda } + \left[ \partial _ { 1 } H ( z , \theta ) \right] ^ { \top } \boldsymbol { \nu } = 0 } \\ { H ( z , \theta ) = 0 } \\ { \boldsymbol { \lambda } \circ G ( z , \theta ) = 0 , } \end{array} +$$ + +where $\nu \in \mathbb { R } ^ { q }$ and $\lambda \in \mathbb { R } _ { + } ^ { r }$ are the dual variables, also known as KKT multipliers. The primal and dual feasibility conditions can be ignored almost everywhere. The system of (potentially nonlinear) equations (6) fits our framework, as we can group the primal and dual solutions as $x ^ { \star } ( \theta ) = ( z ^ { \star } ( { \bar { \theta } } ) , \nu ^ { \star } ( \theta ) , \lambda ^ { \star } ( \theta ) )$ to form the root of a function $F ( x ^ { \star } ( \theta ) , \theta )$ , where $F \colon { \mathbb { R } } ^ { d } \times { \mathbb { R } } ^ { n } \to { \mathbb { R } } ^ { d }$ and $d = p + q + r$ . The primal and dual solutions can be obtained from a generic solver, such as an interior point method. In practice, the above mapping $F$ will be defined directly in Python (see Figure 7 in Appendix A) and $F$ will be differentiated automatically via autodiff. + +Proximal gradient fixed point. Unfortunately, not all algorithms return both primal and dual solutions. Moreover, if the objective contains non-smooth terms, proximal gradient descent may be more efficient. We now discuss its fixed point (Niculae & Blondel, 2017; Bertrand et al., 2020; 2021). Let $x ^ { \star } ( \theta )$ be implicitly defined as + +$$ +x ^ { \star } ( \theta ) : = \operatornamewithlimits { a r g m i n } _ { x \in \mathbb { R } ^ { d } } f ( x , \theta ) + g ( x , \theta ) , +$$ + +where $f \colon { \mathbb { R } ^ { d } } \times { \mathbb { R } ^ { n } } \to { \mathbb { R } }$ is twice-differentiable convex and $g \colon { \mathbb { R } } ^ { d } \times { \mathbb { R } } ^ { n } \to { \mathbb { R } }$ is convex but possibly non-smooth. Let us define the proximity operator associated with $g$ by + +$$ +\operatorname { p r o x } _ { g } ( y , \theta ) : = \underset { x \in \mathbb { R } ^ { d } } { \operatorname { a r g m i n } } \frac { 1 } { 2 } \| x - y \| _ { 2 } ^ { 2 } + g ( x , \theta ) . +$$ + +To implicitly differentiate $x ^ { \star } ( \theta )$ , we use the fixed point mapping (Parikh & Boyd, 2014, p.150) + +$$ +T ( x , \theta ) = \mathrm { p r o x } _ { \eta g } ( x - \eta \nabla _ { 1 } f ( x , \theta ) , \theta ) , +$$ + +for any step size $\eta > 0$ . The proximity operator is 1-Lipschitz continuous (Moreau, 1965). By Rademacher’s theorem, it is differentiable almost everywhere. Many proximity operators enjoy a closed form and can easily be differentiated, as discussed in Appendix $\mathbf { B }$ . + +Projected gradient fixed point. As a special case, when $g ( x , \theta )$ is the indicator function $I _ { C ( \theta ) } ( x )$ where $\mathcal { C } ( \boldsymbol { \theta } )$ is a convex set depending on $\theta$ , we obtain + +$$ +x ^ { \star } ( \theta ) = \operatorname * { a r g m i n } _ { x \in { \mathcal { C } } ( \theta ) } f ( x , \theta ) . +$$ + +The proximity operator $\operatorname { p r o x } _ { g }$ becomes the Euclidean projection onto $\mathcal { C } ( \boldsymbol { \theta } )$ + +$$ +{ \mathrm { p r o x } } _ { g } ( y , \theta ) = { \mathrm { p r o j } } _ { \mathcal { C } } ( y , \theta ) : = { \underset { x \in { \mathcal { C } } ( \theta ) } { \operatorname { a r g m i n } } } \| x - y \| _ { 2 } ^ { 2 } +$$ + +and (7) becomes the projected gradient fixed point + +$$ +T ( x , \theta ) = \mathrm { p r o j } _ { \mathcal { C } } ( x - \eta \nabla _ { 1 } f ( x , \theta ) , \theta ) . +$$ + +Compared to the KKT conditions, this fixed point is particularly suitable when the projection enjoys a closed form. We discuss how to compute the JVP / VJP for a wealth of convex sets in Appendix B. + +Mirror descent fixed point. We again consider the case when $x ^ { \star } ( \theta )$ is implicitly defined as the solution of (8). We now generalize the projected gradient fixed point beyond Euclidean geometry. Let the Bregman divergence $D _ { \varphi }$ : $\operatorname { d o m } ( \bar { \varphi } ) \cdot \operatorname { r e l i n t } ( \operatorname { d o m } ( \varphi ) ) \to \bar { \mathbb { R } } _ { + }$ generated by $\varphi$ be defined by + +$$ +D _ { \varphi } ( x , y ) : = \varphi ( x ) - \varphi ( y ) - \langle \nabla \varphi ( y ) , x - y \rangle . +$$ + +We define the Bregman projection of $y$ onto ${ \mathcal { C } } ( \theta ) \subseteq \mathrm { d o m } ( \varphi )$ by + +$$ +\begin{array} { r } { \mathrm { p r o j } _ { \mathcal { C } } ^ { \varphi } ( y , \theta ) : = \underset { x \in \mathcal { C } ( \theta ) } { \mathrm { a r g m i n } } D _ { \varphi } ( x , \nabla \varphi ^ { \ast } ( y ) ) . } \end{array} +$$ + +Definition (10) includes the mirror map $\nabla \varphi ^ { * } ( y )$ for convenience. It can be seen as a mapping from $\mathbb { R } ^ { d }$ to $\mathrm { d o m } ( \varphi )$ , ensuring that (10) is well-defined. The mirror descent fixed point mapping is then + +$$ +\begin{array} { c } { \hat { x } = \nabla \varphi ( x ) } \\ { y = \hat { x } - \eta \nabla _ { 1 } f ( x , \theta ) } \\ { T ( x , \theta ) = \mathrm { p r o j } _ { \mathcal { C } } ^ { \varphi } ( y , \theta ) . } \end{array} +$$ + +Because $T$ involves the composition of several functions, manually deriving its JVP/VJP is error prone. This shows that our approach leveraging autodiff allows to handle more advanced fixed point mappings. A common example of $\varphi$ is $\varphi ( x ) = \langle x , \log x - 1 \rangle$ , where $\mathrm { d o m } ( \varphi ) = \mathbb { R } _ { + } ^ { d }$ . In this case, $D _ { \varphi }$ is the Kullback-Leibler divergence. An advantage of the Kullback-Leibler projection is that it sometimes easier to compute than the Euclidean projection, as we detail in Appendix B. + +Other fixed points. More fixed points are described in Appendix C. + +# 3 JACOBIAN PRECISION GUARANTEES + +In practice, either by the limitations of finite precision arithmetic or because we perform a finite number of iterations, we rarely reach the exact solution $x ^ { \star } ( \theta )$ , but instead only reach an approximate solution $\hat { x }$ and apply the implicit differentiation equation (2) at this approximate solution. This motivates the need for precision guarantees of this approach. We introduce the following formalism. + +Definition 1. Let $F : \mathbb { R } ^ { d } \times \mathbb { R } ^ { n } \mathbb { R } ^ { d }$ be an optimality criterion mapping. Let $A : = - \partial _ { 1 } F$ and $B : = \partial _ { 2 } F$ . We define the Jacobian estimate at $( x , \dot { \theta } )$ as the solution to the linear equation $A ( x , \theta ) J ( x , \theta ) = B ( x , \theta )$ . It is a function $J : \mathbb { R } ^ { d } \times \mathbb { R } ^ { \bar { n } } \mathbb { R } ^ { d \times n }$ . + +It holds by construction that $J ( x ^ { \star } ( \theta ) , \theta ) = \partial x ^ { \star } ( \theta )$ . Computing $J ( \hat { x } , \theta )$ for an approximate solution $\hat { x }$ of $x ^ { \star } ( { \boldsymbol { \theta } } )$ therefore allows to approximate the true Jacobian ${ \partial x ^ { \star } ( \theta ) }$ . In practice, an algorithm used to solve (1) depends on $\theta$ . Note however that, what we compute is not the Jacobian of ${ \hat { x } } ( \theta )$ , unlike works differentiating through unrolled algorithm iterations, but an estimate of ${ \partial x ^ { \star } ( \theta ) }$ . We therefore use the notation $\hat { x }$ , leaving the dependence on $\theta$ implicit. + +We develop bounds of the form $\| J ( \hat { x } , \theta ) - \partial x ^ { \star } ( \theta ) \| < C \| \hat { x } - x ^ { \star } ( \theta ) \|$ , hence showing that the error on the estimated Jacobian is at most of the same order as that of $\hat { x }$ as an approximation of $x ^ { \star } ( \theta )$ These bounds are based on the following main theorem, whose proof is included in Appendix D. + +Theorem 1 (Jacobian estimate). Let $F : \mathbb { R } ^ { d } \times \mathbb { R } ^ { n } \mathbb { R } ^ { d }$ . Assume that there exist $\alpha , \beta , \gamma , \varepsilon , R > 0$ such that $A = - \partial _ { 1 } F$ and $B = \partial _ { 2 } F$ satisfy, for all $v \in \mathbb { R } ^ { d }$ , $\boldsymbol \theta \in \mathbb { R } ^ { n }$ and $x$ such that $\| x - x ^ { \star } ( \theta ) \| \leq \varepsilon$ : + +$A$ is well-conditioned, Lipschitz: $\| A ( x , \theta ) v \| \geq \alpha \| v \| , \| A ( x , \theta ) - A ( x ^ { \star } ( \theta ) , \theta ) \| _ { \mathrm { o p } } \leq \gamma \| x - x ^ { \star } ( \theta ) \| .$ + +$B$ is bounded and Lipschitz: $\| B ( x ^ { \star } ( \theta ) , \theta ) \| \leq R$ , $\| B ( x , \theta ) - B ( x ^ { \star } ( \theta ) , \theta ) \| \leq \beta \| x - x ^ { \star } ( \theta ) \| .$ . + +Under these conditions, when $\| { \hat { x } } - x ^ { \star } ( \theta ) \| \leq \varepsilon$ , we have + +$$ +\begin{array} { r } { \| J ( \hat { x } , \theta ) - \partial x ^ { \star } ( \theta ) \| \leq \left( \beta \alpha ^ { - 1 } + \gamma R \alpha ^ { - 2 } \right) \| \hat { x } - x ^ { \star } ( \theta ) \| . } \end{array} +$$ + +This result is inspired by (Higham, 2002, Theorem 7.2), that is concerned with the stability of solutions to inverse problems. Here we consider that $A ( \cdot , \theta )$ is uniformly well-conditioned, rather than only at $x ^ { \star } ( \theta )$ . This does not affect the first order in $\varepsilon$ of this bound, and makes it valid for all $\hat { x }$ . It is also more tailored to applications to equation-specific cases. + +Indeed, Theorem 1 can be applied to specific cases. In particular, for the gradient descent fixed point, where $T ( x , \theta ) = x - \eta \nabla _ { 1 } \bar { f } ( x , \theta )$ and $F ( x , { \boldsymbol { \theta } } ) = T ( x , { \overline { { \theta } } } ) - x$ , this yields + +$$ +A ( x , \theta ) = \eta \nabla _ { 1 } ^ { 2 } f ( x , \theta ) \mathrm { a n d } B ( x , \theta ) = - \eta \partial _ { 2 } \nabla _ { 1 } f ( x , \theta ) . +$$ + +The guarantees on Jacobian precision under regularity conditions rely on $f$ directly; see Corollary 1 in Appendix D. This reveals in particular that Jacobian estimation by implicit differentiation gains a factor of t compared to automatic differentiation, after $t$ iterations of gradient descent in the strongly-convex setting (Ablin et al., 2020, Proposition 3.2). While our guarantees concern the Jacobian of $x ^ { \star } ( \theta )$ , we note that other studies (Grazzi et al., 2020; Ji et al., 2021; Bertrand et al., 2021) give guarantees on hypergradients (i.e., the gradient of an outer objective). + +We illustrate these results on ridge regression, where $\begin{array} { r } { x ^ { \star } ( \theta ) = \operatorname * { a r g m i n } _ { x } \| \Phi x - y \| _ { 2 } ^ { 2 } + \sum _ { i } \theta _ { i } x _ { i } ^ { \beth } } \end{array}$ . This problem has the merit that the solution $x ^ { \star } ( \theta )$ and its Jacobian ${ \partial x ^ { \star } ( \theta ) }$ are available in closed form. By running gradient descent for $t$ iterations, we obtain an estimate $\hat { x }$ of $x ^ { \star } ( \theta )$ and an estimate $J ( \hat { x } , \theta )$ of ${ \partial x ^ { \star } ( \theta ) }$ ; cf. Definition 1. By doing so for different numbers of iterations $t$ , we can graph the relation between the error $\lVert x ^ { \star } ( \theta ) - { \hat { x } } \rVert _ { 2 }$ and the error $\lVert \partial x ^ { \star } ( \theta ) - J ( \hat { x } , \theta ) \rVert _ { 2 }$ , as shown in Figure 2, empirically validating Theorem 1. The results in Figure 2 were obtained using the diabetes dataset from Efron et al. (2004), with other datasets yielding a qualitatively similar behavior. We derive similar guarantees in Corollary 2 in Appendix $\mathrm { D }$ for proximal gradient descent. + +![](images/0b12fb95ca636955ebebf8146e28306042b855cbc3a9ba0861efbd81eb53cb3a.jpg) +Figure 2: Jacobian estimate errors. Empirical error of implicit differentiation follows closely the theoretical upper bound. Unrolling achieves a much worse error for comparable iterate error. + +# 4 EXPERIMENTS + +To conclude this work, we demonstrate the ease of solving bi-level optimization problems with our framework. We also present an application to the sensitivity analysis of molecular dynamics. + +# 4.1 HYPERPARAMETER OPTIMIZATION OF MULTICLASS SVMS + +In this example, we consider the hyperparameter optimization of multiclass SVMs (Crammer & Singer, 2001) trained in the dual. Here, $x ^ { \star } ( \theta )$ is the optimal dual solution, a matrix of shape $m \times k$ , where $m$ is the number of training examples and $k$ is the number of classes, and $\theta \in \mathbb { R } _ { + }$ is the regularization parameter. The challenge in differentiating $x ^ { \star } ( \theta )$ is that each row of $x ^ { \star } ( \theta )$ is constrained to belong to the probability simplex $\triangle ^ { k }$ . More formally, let $X _ { \mathsf { t r } } \in \mathbb { R } ^ { m \times p }$ be the training feature matrix and $\breve { Y } _ { \mathrm { t r } } \in \bar { \{ 0 , 1 \} } ^ { m \times k }$ be the training labels (in row-wise one-hot encoding). Let $W ( x , \theta ) : = X _ { \mathrm { t r } } ^ { \top } ( Y _ { \mathrm { t r } } - x ) / \overset { } \theta \in \overset { } \mathbb { R } ^ { p \times k }$ be the dual-primal mapping. Then, we consider the following bi-level optimization problem + +$$ +\operatorname* { m i n } _ { = \exp ( \lambda ) } \frac { 1 } { 2 } \| X _ { \mathrm { v a l } } W ( x ^ { \star } ( \theta ) , \theta ) - Y _ { \mathrm { v a l } } \| _ { F } ^ { 2 } \quad \mathrm { s u b j e c t ~ t o } \quad x ^ { \star } ( \theta ) = \operatorname* { a r g m i n } _ { x \in C } f ( x , \theta ) : = \frac { \theta } { 2 } \| W ( x , \theta ) \| _ { F } ^ { 2 } + \langle x , Y _ { \mathrm { t r } } \rangle , +$$ + +![](images/fbcf6c03f02081b8dee912f3ff702f8674d162cde80a9f1faf84a0c101cfd9e1.jpg) +Figure 3: CPU runtime comparison of implicit differentiation and unrolling for hyperparameter optimization of multiclass SVMs for multiple problem sizes. Error bars represent $90 \%$ confidence intervals. (a) Mirror descent solver, with mirror descent fixed point for implicit differentiation. $\mathbf { ( b ) }$ Proximal gradient solver, with proximal gradient fixed point for implicit differentiation. (c) Block coordinate descent solver; for implicit differentiation we obtain $x ^ { \star } ( \theta )$ by BCD but perform differentiation with the mirror descent and proximal gradient fixed points. This showcases that the solver and fixed point can be independently chosen. + +where $\mathcal { C } = \triangle ^ { k } \times \cdot \cdot \cdot \times \triangle ^ { k }$ is the Cartesian product of $m$ probability simplices. We apply the change of variable $\theta = \exp ( \lambda )$ in order to guarantee that the hyper-parameter $\theta$ is positive. The matrix $W ( x ^ { \star } ( \theta ) , \theta ) \in \mathbb { R } ^ { p \times k }$ contains the optimal primal solution, the feature weights for each class. The outer loss is computed against validation data $X _ { \mathtt { v a l } }$ and $Y _ { \tt v a l }$ . + +While KKT conditions can be used to differentiate $x ^ { \star } ( \theta )$ , a more direct way is to use the projected gradient fixed point (9). The projection onto $\mathcal { C }$ can be easily computed by row-wise projections on the simplex. The projection’s Jacobian enjoys a closed form (Appendix B). Another way to differentiate $x ^ { \star } ( \theta )$ is using the mirror descent fixed point (11). Under the KL geometry, $\mathrm { p r o j } _ { \mathcal { C } } ^ { \varphi } ( \dot { y } , \theta )$ corresponds to a row-wise softmax. It is therefore easy to compute and differentiate. Figure 3 compares the runtime performance of implicit differentiation vs. unrolling for the latter two fixed points. + +# 4.2 DATASET DISTILLATION + +Dataset distillation (Wang et al., 2018; Lorraine et al., 2020) aims to learn a small synthetic training dataset such that a model trained on this learned data set achieves a small loss on the original training set. Formally, let $X _ { \mathsf { t r } } \in \mathbb { R } ^ { m \times p }$ and $y _ { \mathrm { t r } } \in [ k ] ^ { m }$ denote the original training set. The distilled dataset will contain one prototype example for each class and therefore $\boldsymbol { \theta } \in \mathbb { R } ^ { k \times \bar { p } }$ . The dataset distillation problem can then naturally be cast as a bi-level problem, where in the inner problem we estimate a logistic regression model $\bar { { \boldsymbol { x } } } ^ { \star } ( \theta ) \in \mathbb { R } ^ { p \times k }$ trained on the distilled images $\theta \in \bar { \mathbb { R } ^ { k \times p } }$ , while in the outer problem we want to minimize the loss achieved by $x ^ { \star } ( \theta )$ over the training set: + +$$ +\underbrace { \operatorname* { m i n } _ { \theta \in \mathbb { R } ^ { k \times p } } f ( x ^ { \star } ( \theta ) , X _ { \mathrm { t r } } ; y _ { \mathrm { t r } } ) } _ { \mathrm { o u t e r ~ p r o b l e m } } \mathrm { ~ s u b j e c t ~ t o ~ } x ^ { \star } ( \theta ) \in \underbrace { x \mathrm { g m i n } \ f ( x , \theta ; [ k ] ) + \varepsilon \| x \| ^ { 2 } } _ { \mathrm { i n n e r ~ p r o b l e m } } , +$$ + +where $f ( W , X ; y ) : = \ell ( y , X W )$ , $\ell$ denotes the multiclass logistic regression loss, and $\varepsilon = 1 0 ^ { - 3 }$ is a regularization parameter that we found had a very positive effect on convergence. + +In this problem, and unlike in the general hyperparameter optimization setup, both the inner and outer problems are high-dimensional, making it an ideal test-bed for gradient-based bi-level optimization methods. For this experiment, we use the MNIST dataset. The number of parameters in the inner problem is $p = 2 8 ^ { 2 } = 7 8 4$ while the number of parameters of the outer loss is $k \times p = 7 8 4 0$ . We solve this problem using gradient descent on both the inner and outer problem, with the gradient of the outer loss computed using implicit differentiation, as described in $\ S 2$ . This is fundamentally different from the approach used in the original paper, where they used differentiation of the unrolled iterates instead. For the same solver, we found that the implicit differentiation approach was 4 times faster than the original one. The obtained distilled images $\theta$ are visualized in Figure 4. + +![](images/ec19d71aaf0cd167c63d17522b3599bea2432677009a02da79e1979a06bf5883.jpg) +Figure 4: Distilled dataset $\theta \in$ $\mathbb { R } ^ { k \times p }$ obtained by solving (12). + +Table 2: Mean AUC (and $9 5 \%$ confidence interval) for the cancer survival prediction problem. + +
MethodL1 logregL2 logregDictL + L2 logregTask-driven DictL
AUC (%)71.6± 2.072.4 ± 2.868.3 ± 2.373.2 ± 2.1
+ +# 4.3 TASK-DRIVEN DICTIONARY LEARNING + +Task-driven dictionary learning was proposed to learn sparse codes for input data in such a way that the codes solve an outer learning problem (Mairal et al., 2012; Sprechmann et al., 2014; Zarka et al., 2019). Formally, given a data matrix $X _ { \mathsf { t r } } \in \mathbb { R } ^ { m \times p }$ and a dictionary of $k$ atoms $\boldsymbol { \theta } \in \mathbb { R } ^ { k \times p }$ , a sparse code is defined as a matrix $\boldsymbol { x } ^ { \star } ( \boldsymbol { \theta } ) \in \mathbb { R } ^ { m \times k }$ that minimizes in $x$ a reconstruction loss $f ( x , \theta ) : \bar { = } \ell ( X _ { \mathrm { t r } } , x \theta )$ regularized by a sparsity-inducing penalty $g ( x )$ . Instead of optimizing the dictionary $\theta$ to minimize the reconstruction loss, Mairal et al. (2012) proposed to optimize an outer problem that depends on the code. Given a set of labels $Y _ { \mathbf { t r } } \in \{ 0 , 1 \} ^ { m }$ , we consider a logistic regression problem which results in the bilevel optimization problem: + +![](images/d9f9382c208a39ab5a6bb575cc80df48830967a3317245d468e891117cd8e512.jpg) + +When $\ell$ is the squared Frobenius distance between matrices, and $g$ the elastic net penalty, Mairal et al. (2012, Eq. 21) derive manually, using optimality conditions (notably the support of the codes selected at the optimum), an explicit re-parameterization of $x ^ { \star } ( \theta )$ as a linear system involving $\theta$ . This closed-form allows for a direct computation of the Jacobian of $x ^ { \star }$ w.r.t. $\theta$ . Similarly, (Sprechmann et al., 2014) derive first order conditions in the case where $\ell$ is a $\beta$ -divergence, while (Zarka et al., 2019) propose to use unrolling of ISTA iterations. Our approach bypasses all of these manual derivations, giving the user more leisure to focus directly on modeling (loss, regularizer) aspects. + +We illustrate this on breast cancer survival prediction from gene expression data. We frame it as a binary classification problem to discriminate patients who survive longer than 5 years $m _ { 1 } = 2 0 0$ ) vs patients who die within 5 years of diagnosis $\mathrm { \Delta } m _ { 0 } = 9 9 \mathrm { \Delta }$ ), from $p = 1 , 0 0 0$ gene expression values. As shown in Table 2, solving (13) (Task-driven DictL) reaches a classification performance competitive with state-of-the-art $L _ { 1 }$ or $L _ { 2 }$ regularized logistic regression with 100 times fewer variables. + +# 4.4 SENSITIVITY ANALYSIS OF MOLECULAR DYNAMICS + +Many physical simulations require solving optimization problems, such as energy minimization in molecular (Schoenholz & Cubuk, 2020) and continuum (Beatson et al., 2020) mechanics, structural optimization (Hoyer et al., 2019) and data assimilation (Frerix et al., 2021). In this experiment, we revisit an example from JAXMD (Schoenholz & Cubuk, 2020), the problem of finding energy minimizing configurations to a system of $k$ packed particles in a 2-dimensional box of size $\ell$ + +$$ +x ^ { \star } ( \theta ) = \underset { x \in \mathbb { R } ^ { k \times 2 } } { \mathrm { a r g m i n } } f ( x , \theta ) : = \sum _ { i , j } U ( x _ { i , j } , \theta ) , +$$ + +where $\boldsymbol { x } ^ { \star } ( \boldsymbol { \theta } ) \in \mathbb { R } ^ { k \times 2 }$ are the optimal coordinates of the $k$ particles, $U ( x _ { i , j } , \theta )$ is the pairwise potential energy function, with half the particles at diameter 1 and half at diameter $\theta = 0 . 6$ , which we optimize with a domain-specific optimizer (Bitzek et al., 2006). Here we consider sensitivity of particle position with respect to diameter $\partial x ^ { \star } ( \theta )$ , rather than sensitivity of the total energy from the original experiment. Figure 5 shows results calculated via forward-mode implicit differentiation (JVP). Whereas differentiating the unrolled optimizer happens to work for total energy, here it typically does not even converge (see Appendix Figure 17), due the discontinuous optimization method. + +![](images/fa7e597376db214ab7df98a98a47417b874c9d7a9d57737587d398873a0acc37.jpg) +Figure 5: Particle positions and position sensitivity vectors, with respect to increasing the diameter of the blue particles. + +# REFERENCES + +Pierre Ablin, Gabriel Peyré, and Thomas Moreau. Super-efficiency of automatic differentiation for functions defined as a minimum. In Proc. of ICML, pp. 32–41, 2020. + +Akshay Agrawal, Brandon Amos, Shane Barratt, Stephen Boyd, Steven Diamond, and Zico Kolter. Differentiable convex optimization layers. arXiv preprint arXiv:1910.12430, 2019a. + +Akshay Agrawal, Shane Barratt, Stephen Boyd, Enzo Busseti, and Walaa M Moursi. 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Deep network classification by scattering and homotopy dictionary learning. arXiv preprint arXiv:1910.03561, 2019. + +# Appendix + +# A CODE EXAMPLES + +# A.1 CODE EXAMPLES FOR OPTIMALITY CONDITIONS + +Our library provides several reusable optimality condition mappings $F$ or fixed points $T$ . We nevertheless demonstrate the ease of writing some of them from scratch. + +Proximal gradient fixed point. The proximal gradient fixed point (7) with step size $\eta = 1$ is $T ( x , \theta ) = \bar { \mathrm { p r o x } } _ { g } ( x - \nabla _ { 1 } f \bar { ( } x , \theta _ { f } ) , \theta _ { g } )$ . It can be implemented as follows. + +![](images/8504037740291238ad275d7b34129068991e565646f5e8b5b135835ccf85acaf.jpg) +Figure 6: Proximal gradient fixed point $T ( x , \theta )$ + +We recall that when the proximity operator is a projection, we recover the projected gradient fixed point as a special case. Therefore, this fixed point can also be used for constrained optimization. We provide numerous proximal and projection operators in the library. + +KKT conditions. As a more advanced example, we now describe how to implement the KKT conditions (6). The stationarity, primal feasibility and complementary slackness conditions read + +$$ +\begin{array} { r } { \nabla _ { 1 } f ( z , \theta _ { f } ) + [ \partial _ { 1 } G ( z , \theta _ { G } ) ] ^ { \top } \lambda + [ \partial _ { 1 } H ( z , \theta _ { H } ) ] ^ { \top } \nu = 0 } \\ { H ( z , \theta _ { H } ) = 0 } \\ { \lambda \circ G ( z , \theta _ { G } ) = 0 . } \end{array} +$$ + +Using ${ \dot { \mathsf { J } } } \mathsf { a x } . \mathsf { v } { \dot { \mathsf { J } } } { \mathsf { p } }$ to compute vector-Jacobian products, this can be implemented as + +![](images/4c5afa2d3ce2393d2ed5ba74f895571c86cc2a907b4316ffe454af22a9a5c182.jpg) +Figure 7: KKT conditions $F ( x , \theta )$ + +Similar mappings $F$ can be written if the optimization problem contains only equality constraints or only inequality constraints. + +Mirror descent fixed point. Letting $\eta = 1$ and denoting $\theta = ( \theta _ { f } , \theta _ { \mathrm { p r o j } } )$ , the fixed point (11) is + +$$ +\begin{array} { r } { \hat { x } = \nabla \varphi ( x ) \qquad } \\ { y = \hat { x } - \nabla _ { 1 } f ( x , \theta _ { f } ) } \\ { T ( x , \theta ) = \mathrm { p r o j } _ { \mathcal { C } } ^ { \varphi } ( y , \theta _ { \mathrm { p r o j } } ) . \qquad } \end{array} +$$ + +We can then implement it as follows. + +![](images/f3cf628dfe80ab0845f2de767e7bc5f5ef762539deeda01084e875a1841612ab.jpg) +Figure 8: Mirror descent fixed point $T ( x , \theta )$ + +Although not considered in this example, the mapping $\boldsymbol { \nabla } \varphi$ could also depend on $\theta$ if necessary. + +# A.2 CODE EXAMPLES FOR EXPERIMENTS + +We now sketch how to implement our experiments using our framework. In the following, jnp is short for jax.numpy. In all experiments, we only show how to compute gradients with the outer objective. We can then use these gradients with gradient-based solvers to solve the outer objective. + +# Multiclass SVM experiment. + +![](images/cb93a45ac538a2eab5c343e4bdbe73d224d387f48173e100e6b47378bc0280ae.jpg) +Figure 9: Code example for the multiclass SVM experiment. + +Task-driven dictionary learning experiment. + +![](images/1754717219ba10a7a67584e04ef622ac4a9b7d097d3f919bda7074fe500f3cfb.jpg) +Figure 10: Code example for the task-driven dictionary learning experiment. + +# Dataset distillation experiment. + +![](images/53013e99aec5bda0635d97693c9df2f0e9dfe646d9c366131a781676d67fc39a.jpg) +Figure 11: Code example for the dataset distillation experiment. + +# Molecular dynamics experiment. + +![](images/d4d8f94f679454fe2e353a1d2e8064efa68ec64d956f4fadbd26fa10bf22a967.jpg) +Figure 12: Code for the molecular dynamics experiment. + +# B JACOBIAN PRODUCTS + +Our library provides numerous reusable building blocks. We describe in this section how to compute their Jacobian products. As a general guideline, whenever a projection enjoys a closed form, we leave the Jacobian product to the autodiff system. + +# B.1 JACOBIAN PRODUCTS OF PROJECTIONS + +We describe in this section how to compute the Jacobian products of the projections (in the Euclidean and KL senses) onto various convex sets. When the convex set does not depend on any variable, we simply denote it $\mathcal { C }$ instead of $\mathcal { C } ( \boldsymbol { \theta } )$ . + +Non-negative orthant. When $\mathcal { C }$ is the non-negative orthant, $\mathcal { C } = \mathbb { R } _ { + } ^ { d }$ , we obtain $\mathrm { p r o j } _ { \mathcal { C } } ( y ) =$ $\operatorname* { m a x } ( y , 0 )$ , where the maximum is evaluated element-wise. This is also known as the ReLu function. The projection in the KL sense reduces to the exponential function, $\mathrm { p r o j } _ { \mathcal { C } } ^ { \varphi } ( y ) = \exp ( y )$ . + +Box constraints. When $\mathcal { C } ( \boldsymbol { \theta } )$ is the box constraints $\mathcal { C } ( \theta ) = [ \theta _ { 1 } , \theta _ { 2 } ] ^ { d }$ with $\theta \in \mathbb { R } ^ { 2 }$ , we obtain + +$$ +\begin{array} { r } { \mathrm { p r o j } _ { \mathcal { C } } ( y , \theta ) = \mathrm { c l i p } ( y , \theta _ { 1 } , \theta _ { 2 } ) : = \operatorname* { m a x } ( \operatorname* { m i n } ( y , \theta _ { 2 } ) , \theta _ { 1 } ) . } \end{array} +$$ + +This is trivially extended to support different boxes for each coordinate, in which case $\theta \in \mathbb { R } ^ { d \times 2 }$ + +Probability simplex. When $\mathcal { C }$ is the standard probability simplex, ${ \mathcal { C } } = \triangle ^ { d }$ , there is no analytical solution for $\operatorname { p r o j } _ { \mathcal { C } } ( y )$ . Nevertheless, the projection can be computed exactly in $O ( d )$ expected time or $O ( d \log d )$ worst-case time (Brucker, 1984; Michelot, 1986; Duchi et al., 2008; Condat, 2016). The Jacobian is given by $\mathrm { d i a g } ( s ) - s s ^ { \top } / \| s \| _ { 1 }$ , where $s \in \{ 0 , 1 \} ^ { d }$ is a vector indicating the support of $\operatorname { p r o j } _ { \mathcal { C } } ( y )$ (Martins $\&$ Astudillo, 2016). The projection in the KL sense, on the other hand, enjoys a closed form: it reduces to the usual softmax $\begin{array} { r } { \mathrm { p r o j } _ { \mathcal { C } } ^ { \varphi } ( y ) = \exp ( y ) / \sum _ { j = 1 } ^ { d } \exp ( y _ { j } ) , } \end{array}$ . + +Box sections. Consider now the Euclidean projection $z ^ { \star } ( \theta ) = \mathrm { p r o j } _ { \mathcal { C } } ( y , \theta )$ onto the set ${ \mathcal { C } } ( \theta ) =$ $\{ z \in \mathbb { R } ^ { d } \colon \alpha _ { i } \leq z _ { i } \leq \beta _ { i } , i \in [ d ] ; w ^ { \top } z = c \}$ , where $\theta = ( \alpha , \beta , w , c )$ . This projection is a singlyconstrained bounded quadratic program. It is easy to check (see, e.g., (Niculae & Martins, 2020)) that an optimal solution satisfies for all $i \in [ d ]$ + +$$ +z _ { i } ^ { \star } ( \theta ) = [ L ( x ^ { \star } ( \theta ) , \theta ) ] _ { i } : = \mathrm { c l i p } ( w _ { i } x ^ { \star } ( \theta ) + y _ { i } , \alpha _ { i } , \beta _ { i } ) +$$ + +where $L \colon \mathbb { R } \times \mathbb { R } ^ { n } \to \mathbb { R } ^ { d }$ is the dual-primal mapping and $x ^ { \star } ( \theta ) \in \mathbb { R }$ is the optimal dual variable of the linear constraint, which should be the root of + +$$ +F ( x ^ { \star } ( \theta ) , \theta ) = L ( x ^ { \star } ( \theta ) , \theta ) ^ { \top } w - c . +$$ + +The root can be found, e.g., by bisection. The gradient $\nabla \boldsymbol { x } ^ { \star } ( \boldsymbol { \theta } )$ is given by $\nabla x ^ { \star } ( \theta ) = B ^ { \top } / A$ and the Jacobian $\partial z ^ { \star } ( \theta )$ is obtained by application of the chain rule on $L$ . + +Norm balls. When $\mathcal { C } ( \boldsymbol { \theta } ) = \{ x \in \mathbb { R } ^ { d } \colon \| x \| \leq \theta \}$ , where $\| \cdot \|$ is a norm and $\theta \in \mathbb { R } _ { + }$ , $\mathrm { p r o j } _ { \mathcal { C } } ( y , \theta )$ becomes the projection onto a norm ball. The projection onto the $\ell _ { 1 }$ -ball reduces to a projection onto the simplex, see, e.g., (Duchi et al., 2008). The projections onto the $\ell _ { 2 }$ and $\ell _ { \infty }$ balls enjoy a closed-form, see, e.g., (Parikh & Boyd, 2014, $\ S 6 . 5 )$ . Since they rely on simple composition of functions, all three projections can therefore be automatically differentiated. + +Affine sets. When ${ \mathcal { C } } ( \theta ) = \{ x \in \mathbb { R } ^ { d } \colon A x = b \}$ , where $A \in \mathbb { R } ^ { p \times d }$ , $b \in \mathbb { R } ^ { p }$ and $\theta = ( A , b )$ , we get + +$$ +\operatorname { p r o j } _ { \mathcal { C } } ( y , \theta ) = y - A ^ { \dagger } ( A y - b ) = y - A ^ { \top } ( A A ^ { \top } ) ^ { - 1 } ( A y - b ) +$$ + +where $A ^ { \dagger }$ is the Moore-Penrose pseudoinverse of $A$ . The second equality holds if $p < d$ and $A$ is full rank. A practical implementation can pre-compute a factorization of the Gram matrix $A A ^ { \top }$ . Alternatively, we can also use the KKT conditions. + +Hyperplanes and half spaces. When $\mathcal { C } ( \boldsymbol { \theta } ) = \{ x \in \mathbb { R } ^ { d } \colon a ^ { \top } x = b \}$ , where $a \in \mathbb { R } ^ { d }$ and $b \in \mathbb { R }$ and $\theta = ( a , b )$ , we get + +$$ +\mathrm { p r o j } _ { \mathcal { C } } ( y , \theta ) = y - \frac { { a ^ { \top } y - b } } { \Vert a \Vert _ { 2 } ^ { 2 } } a . +$$ + +When $\mathcal { C } ( \boldsymbol { \theta } ) = \{ x \in \mathbb { R } ^ { d } \colon a ^ { \top } x \leq b \}$ , we simply replace $a ^ { \top } y - b$ in the numerator by $\operatorname* { m a x } ( a ^ { \top } y - b , 0 )$ + +Transportation and Birkhoff polytopes. When $\mathcal { C } ( \boldsymbol { \theta } ) = \{ X \in \mathbb { R } ^ { p \times d } \colon X \mathbf { 1 } _ { d } = \boldsymbol { \theta } _ { 1 } , X ^ { \top } \mathbf { 1 } _ { p } =$ $\theta _ { 2 } , X \geq 0 \}$ , the so-called transportation polytope, where $\theta _ { 1 } \in \triangle ^ { p }$ and $\theta _ { 2 } \in \triangle ^ { d }$ are marginals, we can compute approximately the projections, both in the Euclidean and KL senses, by switching to the dual or semi-dual (Blondel et al., 2018). Since both are unconstrained optimization problems, we can compute their Jacobian product by implicit differentiation using the gradient descent fixed point. An advantage of the KL geometry here is that we can use Sinkhorn (Cuturi, 2013), which is a GPU-friendly algorithm. The Birkhoff polytope, the set of doubly stochastic matrices, is obtained by fixing ${ \theta } _ { 1 } = \bar { \theta } _ { 2 } \stackrel { - } { = } { \mathbf { 1 } } _ { d } / d$ . + +Order simplex. When ${ \mathcal { C } } ( \theta ) = \{ x \in \mathbb { R } ^ { d } \colon \theta _ { 1 } \geq x _ { 1 } \geq x _ { 2 } \geq \cdots \geq x _ { d } \geq \theta _ { 2 } \}$ , a so-called order simplex (Grotzinger & Witzgall, 1984; Blondel, 2019), the projection operations, both in the Euclidean and KL sense, reduce to isotonic optimization (Lim & Wright, 2016) and can be solved exactly in $O ( d \log d )$ time using the Pool Adjacent Violators algorithm (Best et al., 2000). The Jacobian of the projections and efficient product with it are derived in (Djolonga & Krause, 2017; Blondel et al., 2020). + +Polyhedra. More generally, we can consider polyhedra, i.e., sets of the form ${ \mathcal { C } } ( \theta ) ~ = ~ \{ x ~ \in ~$ $\mathbb { R } ^ { d } \colon A x = b , C x \leq \mathbf { \bar { d } } \}$ , where $A \in \mathbb { R } ^ { p \times d }$ , $b \in \bar { \mathbb { R } } ^ { p }$ , $C \in \mathbb { R } ^ { m \times d }$ , and $d \in \mathbb { R } ^ { m }$ . There are several ways to differentiate this projection. The first is to use the KKT conditions as detailed in $\ S 2 . 2$ . A second way is consider the dual of the projection instead, which is the maximization of a quadratic function subject to non-negative constraints (Parikh & Boyd, 2014, $\ S 6 . 2 )$ . That is, we can reduce the projection on a polyhedron to a problem of the form (8) with non-negative constraints, which we can in turn implicitly differentiate easily using the projected gradient fixed point, combined with the projection on the non-negative orthant. Finally, we apply the dual-primal mapping , which enjoys a closed form and is therefore amenable to autodiff, to obtain the primal projection. + +# B.2 JACOBIAN PRODUCTS OF PROXIMITY OPERATORS + +We provide several proximity operators, including for the lasso (soft thresholding), elastic net and group lasso (block soft thresholding). All satisfy closed form expressions and can be differentiated automatically via autodiff. For more advanced proximity operators, which do not enjoy a closed form, recent works have derived their Jacobians. The Jacobians of fused lasso and OSCAR were derived in (Niculae & Blondel, 2017). For general total variation, the Jacobians were derived in (Vaiter et al., 2013; Cherkaoui et al., 2020). + +# C MORE EXAMPLES OF OPTIMALITY CRITERIA AND FIXED POINTS + +To demonstrate the generality of our approach, we describe in this section more optimality mapping $F$ or fixed point iteration $T$ . + +Newton fixed point. Let $x$ be a root of $G ( \cdot , \theta )$ , i.e., $G ( x , \theta ) = 0$ . The fixed point iteration of Newton’s method for root-finding is + +$$ +T ( x , \theta ) = x - \eta [ \partial _ { 1 } G ( x , \theta ) ] ^ { - 1 } G ( x , \theta ) . +$$ + +By the chain and product rules, we have + +$$ +\partial _ { 1 } T ( x , \theta ) = I - \eta ( . . . ) G ( x , \theta ) - \eta [ \partial _ { 1 } G ( x , \theta ) ] ^ { - 1 } \partial _ { 1 } G ( x , \theta ) = ( 1 - \eta ) I . +$$ + +Using (3), we get $A = - \partial _ { 1 } F ( x , \theta ) = \eta I$ . Similarly, + +$$ +B = \partial _ { 2 } T ( x , \theta ) = \partial _ { 2 } F ( x , \theta ) = - \eta [ \partial _ { 1 } G ( x , \theta ) ] ^ { - 1 } \partial _ { 2 } G ( x , \theta ) . +$$ + +Newton’s method for optimization is obtained by choosing $G ( x , \theta ) = \nabla _ { 1 } f ( x , \theta )$ , which gives + +$$ +T ( x , \theta ) = x - \eta [ \nabla _ { 1 } ^ { 2 } f ( x , \theta ) ] ^ { - 1 } \nabla _ { 1 } f ( x , \theta ) . +$$ + +It is easy to check that we recover the same linear system as for the gradient descent fixed point (5). A practical implementation can pre-compute an LU decomposition of $\partial _ { 1 } G ( x , \theta )$ , or a Cholesky decomposition if $\partial _ { 1 } G ( x , \theta )$ is positive semi-definite. + +Proximal block coordinate descent fixed point. We now consider the case when $x ^ { \star } ( \theta )$ is implicitly defined as the solution + +$$ +x ^ { \star } ( \theta ) : = \operatorname * { a r g m i n } _ { x \in \mathbb { R } ^ { d } } f ( x , \theta ) + \sum _ { i = 1 } ^ { m } g _ { i } ( x _ { i } , \theta ) , +$$ + +where $g _ { 1 } , \ldots , g _ { m }$ are possibly non-smooth functions operating on subvectors (blocks) $x _ { 1 } , \ldots , x _ { m }$ of $x$ . In this case, we can use for $i \in [ m ]$ the fixed point + +$$ +\begin{array} { r } { x _ { i } = [ T ( \boldsymbol { x } , \theta ) ] _ { i } = \mathrm { p r o x } _ { \eta _ { i } g _ { i } } ( x _ { i } - \eta _ { i } [ \nabla _ { 1 } f ( \boldsymbol { x } , \theta ) ] _ { i } , \theta ) , } \end{array} +$$ + +where $\eta _ { 1 } , \dots , \eta _ { m }$ are block-wise step sizes. Clearly, when the step sizes are shared, i.e., $\eta _ { 1 } =$ $\cdots = \eta _ { m } = \eta$ this fixed point is equivalent to the proximal gradient fixed point (7) with . $g ( x , \theta ) =$ $\textstyle \sum _ { i = 1 } ^ { n } g _ { i } ( x _ { i } , \theta )$ + +Quadratic programming. We now show how to use the KKT conditions discussed in $\ S 2 . 2$ to differentiate quadratic programs, recovering Optnet (Amos & Kolter, 2017) as a special case. To give some intuition, let us start with a simple equality-constrained quadratic program (QP) + +$$ +\operatorname * { a r g m i n } _ { z \in \mathbb { R } ^ { p } } f ( z , \theta ) = \frac { 1 } { 2 } z ^ { \top } Q z + c ^ { \top } z \quad \mathrm { s u b j e c t t o } \quad H ( z , \theta ) = E z - d = 0 , +$$ + +where $Q \in \mathbb { R } ^ { p \times p }$ , $\ b { E } \in \mathbb { R } ^ { q \times p }$ , $d \in \mathbb { R } ^ { q }$ . We gather the differentiable parameters as $\theta = ( Q , E , c , d )$ . The stationarity and primal feasibility conditions give + +$$ +\begin{array} { r } { \nabla _ { 1 } f ( z , \theta ) + [ \partial _ { 1 } H ( z , \theta ) ] ^ { \top } \boldsymbol { \nu } = Q z + c + E ^ { \top } \boldsymbol { \nu } = 0 } \\ { H ( z , \theta ) = E z - d = 0 . } \end{array} +$$ + +In matrix notation, this can be rewritten as + +$$ +\left[ { Q \quad E ^ { \top } } \right] \left[ { z } \right] = \left[ { - c } \right] . +$$ + +We can write the solution of the linear system (16) as the root $\boldsymbol { x } = \left( z , \nu \right)$ of a function $F ( x , \theta )$ . More generally, the QP can also include inequality constraints + +$$ +\operatorname * { a r g m i n } _ { z \in \mathbb { R } ^ { p } } f ( z , \theta ) = \frac { 1 } { 2 } z ^ { \top } Q z + c ^ { \top } z \quad \mathrm { s u b j e c t ~ t o } \quad H ( z , \theta ) = E z - d = 0 , G ( z , \theta ) = M z - h \leq 0 . +$$ + +where $M \in \mathbb { R } ^ { r \times p }$ and $h \in \mathbb { R } ^ { r }$ . We gather the differentiable parameters as $\theta = ( Q , E , M , c , d , h )$ . The stationarity, primal feasibility and complementary slackness conditions give + +$$ +\begin{array} { r } { \nabla _ { 1 } f ( z , \theta ) + [ \partial _ { 1 } H ( z , \theta ) ] ^ { \top } \boldsymbol { \nu } + [ \partial _ { 1 } G ( z , \theta ) ] ^ { \top } \boldsymbol { \lambda } = Q z + c + E ^ { \top } \boldsymbol { \nu } + M ^ { \top } \boldsymbol { \lambda } = 0 } \\ { H ( z , \theta ) = E z - d = 0 } \\ { \boldsymbol { \lambda } \circ G ( z , \theta ) = \mathrm { d i a g } ( \boldsymbol { \lambda } ) ( M z - h ) = 0 } \end{array} +$$ + +In matrix notation, this can be written as + +$$ +{ \left[ \begin{array} { l l l } { \quad Q } & { E ^ { \top } } & { M ^ { \top } } \\ { E } & { 0 } & { 0 } \\ { \operatorname { d i a g } ( \lambda ) M } & { 0 } & { \quad 0 } \end{array} \right] } \left[ { \begin{array} { l } { z } \\ { \nu } \\ { \lambda } \end{array} } \right] = { \left[ \begin{array} { l } { - c } \\ { \ d } \\ { \lambda \circ h } \end{array} \right] } +$$ + +While $\boldsymbol { x } = ( z , \nu , \lambda )$ is no longer the solution of a linear system, it is the root of a function $F ( x , \theta )$ and therefore fits our framework. With our framework, no derivation is needed. We simply define $f$ , $H$ and $G$ directly in Python. + +Conic programming. We now show that the differentiation of conic linear programs (Agrawal et al., 2019b; Amos, 2019), at the heart of differentiating through cvxpy layers (Agrawal et al., 2019a), easily fits our framework. Consider the problem + +$$ +z ^ { \star } ( \lambda ) , s ^ { \star } ( \lambda ) = \operatorname * { a r g m i n } _ { z \in \mathbb { R } ^ { p } , s \in \mathbb { R } ^ { m } } c ^ { \top } z \quad \mathrm { s u b j e c t } \mathrm { t o } \quad E z + s = d , s \in \mathcal { K } , +$$ + +where $\lambda = ( c , E , d )$ , $\ b { E } \in \mathbb { R } ^ { m \times p }$ , $d \in \mathbb { R } ^ { m }$ , $c \in \mathbb { R } ^ { p }$ and ${ \mathcal { K } } \subseteq \mathbb { R } ^ { m }$ is a cone; $z$ and $s$ are the primal and slack variables, respectively. Every convex optimization problem can be reduced to the form (17). Let us form the skew-symmetric matrix + +$$ +\theta ( \lambda ) = \left[ \begin{array} { c c c } { 0 } & { E ^ { \top } } & { c } \\ { - E } & { 0 } & { d } \\ { - c ^ { \top } } & { - d ^ { \top } } & { 0 } \end{array} \right] \in \mathbb { R } ^ { N \times N } , +$$ + +where $N = p { + } m { + } 1$ . Following (Agrawal et al., 2019b;a; Amos, 2019), we can use the homogeneous self-dual embedding to reduce the process of solving (17) to finding a root of the residual map + +$$ +F ( x , \theta ) = \theta \Pi x + \Pi ^ { * } x = ( ( \theta - I ) \Pi + I ) x , +$$ + +where $\Pi = \mathrm { p r o j } _ { \mathbb { R } ^ { p } \times K ^ { * } \times \mathbb { R } _ { + } }$ and ${ K } ^ { * } \subseteq \mathbb { R } ^ { m }$ is the dual cone. The splitting conic solver (O’Donoghue et al., 2016), which is based on ADMM, outputs a solution $F ( x ^ { \star } ( \theta ) , \theta ) = 0$ which is decomposed as $x ^ { \star } ( \theta ) = ( u ^ { \star } ( \theta ) , v ^ { \star } ( \theta ) , w ^ { \star } ( \theta ) )$ . We can then recover the optimal solution of (17) using + +$$ +z ^ { \star } ( \lambda ) = u ^ { \star } ( \theta ( \lambda ) ) \quad \mathrm { a n d } \quad s ^ { \star } ( \lambda ) = \mathrm { p r o j } _ { \mathcal { K } ^ { * } } ( v ^ { \star } ( \theta ( \lambda ) ) ) - v ^ { \star } ( \theta ( \lambda ) ) . +$$ + +The key oracle whose JVP/VJP we need is therefore $\Pi$ , which is studied in (Ali et al., 2017). The projection onto a few cones is available in our library and can be used to express $F$ . + +Frank-Wolfe. We now consider + +$$ +x ^ { \star } ( \theta ) = \operatorname * { a r g m i n } _ { x \in \mathcal { C } ( \theta ) \subset \mathbb { R } ^ { d } } f ( x , \theta ) , +$$ + +where $\mathcal { C } ( \boldsymbol { \theta } )$ is a convex polytope, i.e., it is the convex hull of vertices $v _ { 1 } ( \theta ) , \dots , v _ { m } ( \theta )$ . The FrankWolfe algorithm requires a linear minimization oracle (LMO) + +$$ +s \mapsto \operatorname * { a r g m i n } _ { x \in { \mathcal { C } } ( \theta ) } ~ \langle s , x \rangle +$$ + +and is a popular algorithm when this LMO is easier to compute than the projection onto $\mathcal { C } ( \boldsymbol { \theta } )$ . However, since this LMO is piecewise constant, its Jacobian is null almost everywhere. Inspired by SparseMAP (Niculae et al., 2018), which corresponds to the case when $f$ is a quadratic, we rewrite (19) as + +$$ +p ^ { \star } ( \theta ) = \underset { p \in \triangle ^ { m } } { \operatorname { a r g m i n } } g ( p , \theta ) : = f ( V ( \theta ) p , \theta ) , +$$ + +where $V ( \theta )$ is a $d \times m$ matrix gathering the vertices $v _ { 1 } ( \theta ) , \dots , v _ { m } ( \theta )$ . We then have $x ^ { \star } ( \theta ) =$ $V ( \theta ) p ^ { \star } ( \theta )$ . Since we have reduced (19) to minimization over the simplex, we can use the projected gradient fixed point to obtain + +$$ +T ( p ^ { \star } ( \boldsymbol { \theta } ) , \boldsymbol { \theta } ) \stackrel { } { = } \mathrm { p r o j } _ { \triangle ^ { m } } \big ( p ^ { \star } ( \boldsymbol { \theta } ) - \nabla _ { 1 } g ( p ^ { \ * } ( \boldsymbol { \theta } ) , \boldsymbol { \theta } ) \big ) . +$$ + +We can therefore compute the derivatives of $p ^ { \star } ( \theta )$ by implicit differentiation and the derivatives of $x ^ { \star } ( \theta )$ by product rule. Frank-Wolfe implementations typically maintain the convex weights of the vertices, which we use to get an approximation of $p ^ { \star } ( \theta )$ . Moreover, it is well-known that after $t$ iterations, at most $t$ vertices are visited. We can leverage this sparsity to solve a smaller linear system. Moreover, in practice, we only need to compute VJPs of $x ^ { \star } ( \theta )$ . + +# D JACOBIAN PRECISION PROOFS + +Proof of Theorem $^ { l }$ . To simplify notations, we note $A _ { \star } : = A ( x ^ { \star } , \theta )$ and $\hat { A } : = A ( \hat { x } , \theta )$ , and similarly for $B$ and $J$ . We have by definition of the Jacobian estimate function $A _ { \star } J _ { \star } = B _ { \star }$ and $\hat { A } \hat { J } = \hat { B }$ . Therefore we have + +$$ +\begin{array} { r l } & { J ( \hat { x } , \theta ) - \partial x ^ { \star } ( \theta ) = \hat { A } ^ { - 1 } \hat { B } - A _ { \star } ^ { - 1 } B _ { \star } } \\ & { \qquad = \hat { A } ^ { - 1 } \hat { B } - \hat { A } ^ { - 1 } B _ { \star } + \hat { A } ^ { - 1 } B _ { \star } - A _ { \star } ^ { - 1 } B _ { \star } } \\ & { \qquad = \hat { A } ^ { - 1 } ( \hat { B } - B _ { \star } ) + ( \hat { A } ^ { - 1 } - A _ { \star } ^ { - 1 } ) B _ { \star } . } \end{array} +$$ + +For any invertible matrices $M _ { 1 } , M _ { 2 }$ , it holds that $M _ { 1 } ^ { - 1 } - M _ { 2 } ^ { - 1 } = M _ { 1 } ^ { - 1 } ( M _ { 2 } - M _ { 1 } ) M _ { 2 } ^ { - 1 }$ , so + +$$ +\begin{array} { r } { \| M _ { 2 } ^ { - 1 } - M _ { 2 } ^ { - 1 } \| _ { \mathrm { o p } } \leq \| M _ { 1 } ^ { - 1 } \| _ { \mathrm { o p } } \| M _ { 2 } - M _ { 1 } \| _ { \mathrm { o p } } \| M _ { 2 } ^ { - 1 } \| _ { \mathrm { o p } } . } \end{array} +$$ + +Therefore, + +$$ +\| \hat { A } ^ { - 1 } - A _ { \star } ^ { - 1 } \| _ { \mathrm { o p } } \leq \frac { 1 } { \alpha ^ { 2 } } \| \hat { A } - A _ { \star } \| _ { \mathrm { o p } } \leq \frac { \gamma } { \alpha ^ { 2 } } \| \hat { x } - x ^ { \star } ( \theta ) \| . +$$ + +As a consequence, the second term in $J ( \hat { x } , \theta ) - \partial x ^ { \star } ( \theta )$ can be upper bounded and we obtain + +$$ +\begin{array} { r l } & { \| J ( \hat { \boldsymbol { x } } , \theta ) - \partial \boldsymbol { x } ^ { \star } ( \theta ) \| \leq \| \hat { A } ^ { - 1 } ( \hat { B } - B _ { \star } ) \| + \| ( \hat { A } ^ { - 1 } - A _ { \star } ^ { - 1 } ) B _ { \star } \| } \\ & { \qquad \leq \| \hat { A } ^ { - 1 } \| _ { \mathsf { o p } } \| \hat { B } - B _ { \star } \| + \frac { \gamma } { \alpha ^ { 2 } } \| \hat { \boldsymbol { x } } - \boldsymbol { x } ^ { \star } ( \theta ) \| \ \| B _ { \star } \| , } \end{array} +$$ + +which yields the desired result. + +Corollary 1 (Jacobian precision for gradient descent fixed point). Let $f$ be such that $f ( \cdot , \theta )$ is twice differentiable and $\alpha$ -strongly convex and $\nabla _ { 1 } ^ { 2 } f ( \cdot , \theta )$ is $\gamma$ -Lipschitz (in the operator norm) and $\partial _ { 2 } \nabla _ { 1 } f ( x , \theta )$ is $\beta$ -Lipschitz and bounded in norm by $R$ . The estimated Jacobian evaluated at $\hat { x }$ is then given by + +$$ +\boldsymbol { J } ( \boldsymbol { \hat { x } } , \theta ) = - ( \nabla _ { 1 } ^ { 2 } f ( \boldsymbol { \hat { x } } , \theta ) ) ^ { - 1 } \partial _ { 2 } \nabla _ { 1 } f ( \boldsymbol { \hat { x } } , \theta ) . +$$ + +For all $\boldsymbol \theta \in \mathbb { R } ^ { n }$ , and any $\hat { x }$ estimating $x ^ { \star } ( \theta )$ , we have the following bound for the approximation error of the estimated Jacobian + +$$ +\| J ( { \hat { x } } , \theta ) - \partial x ^ { \star } ( \theta ) \| \leq \left( { \frac { \beta } { \alpha } } + { \frac { \gamma R } { \alpha ^ { 2 } } } \right) \| { \hat { x } } - x ^ { \star } ( \theta ) \| . +$$ + +Proof of Corollary $^ { l }$ . This follows from Theorem 1, applied to this specific $A ( x , \theta )$ and $B ( x , \theta )$ . + +For proximal gradient descent, where $T ( x , \theta ) = \mathrm { p r o x } _ { \eta g } ( x - \eta \nabla _ { 1 } f ( x , \theta ) , \theta )$ , this yields + +$$ +\begin{array} { r l } & { A ( x , \theta ) = I - \partial _ { 1 } T ( x , \theta ) = I - \partial _ { 1 } \mathrm { p r o x } _ { \eta g } ( x - \eta \nabla _ { 1 } f ( x , \theta ) , \theta ) ( I - \eta \nabla _ { 1 } ^ { 2 } f ( x , \theta ) ) } \\ & { B ( x , \theta ) = \partial _ { 2 } \mathrm { p r o x } _ { \eta g } ( x - \eta \nabla _ { 1 } f ( x , \theta ) , \theta ) - \eta \partial _ { 1 } \mathrm { p r o x } _ { \eta g } ( x - \eta \nabla _ { 1 } f ( x , \theta ) , \theta ) \partial _ { 2 } \nabla _ { 1 } f ( x , \theta ) . } \end{array} +$$ + +We now focus in the case of proximal gradient descent on an objective $f ( x , \theta ) + g ( x )$ , where $g$ is smooth and does not depend on $\theta$ . This is the case in our experiments in $\ S 4 . 3$ . Recent work also exploits local smoothness of solutions to derive similar bounds (Bertrand et al., 2021, Theorem 13) + +Corollary 2 (Jacobian precision for proximal gradient descent fixed point). Let $f$ be such that $f ( \cdot , \theta )$ is twice differentiable and $\alpha$ -strongly convex and $\nabla _ { 1 } ^ { 2 } f ( \cdot , \theta )$ is $\gamma$ -Lipschitz (in the operator norm) and $\partial _ { 2 } \nabla _ { 1 } f ( x , \theta )$ is $\beta$ -Lipschitz and bounded in norm by $R$ . Let $\bar { g } : \mathbb { R } ^ { d } \to \mathbb { R }$ be a twicedifferentiable $\mu$ -strongly convex (with special case $\mu = 0$ being only convex), for which the function $\Gamma _ { \eta } ( x , \theta ) = \nabla ^ { 2 } g ( p r o x _ { \eta g } ( x - \eta \nabla _ { 1 } f ( x , \theta ) )$ is $\kappa _ { \eta }$ -Lipschitz in it first argument. The estimated Jacobian evaluated at $\hat { x }$ is then given by + +$$ +\boldsymbol { J } ( \boldsymbol { \hat { x } } , \theta ) = - ( \nabla _ { 1 } ^ { 2 } f ( \boldsymbol { \hat { x } } , \theta ) + \Gamma _ { \eta } ( \boldsymbol { \hat { x } } , \theta ) ) ^ { - 1 } \partial _ { 2 } \nabla _ { 1 } f ( \boldsymbol { \hat { x } } , \theta ) . +$$ + +For all $\boldsymbol \theta \in \mathbb { R } ^ { n }$ , and any $\hat { x }$ estimating $x ^ { \star } ( \theta )$ , we have the following bound for the approximation error of the estimated Jacobian + +$$ +\| J ( \hat { x } , \theta ) - \partial x ^ { \star } ( \theta ) \| \leq \left( \frac { \beta + \kappa _ { \eta } } { \alpha + \mu } + \frac { \gamma R } { ( \alpha + \mu ) ^ { 2 } } \right) \| \hat { x } - x ^ { \star } ( \theta ) \| . +$$ + +Proof of Corollary 2. First, let us note that $\mathrm { p r o x } _ { \eta g } ( y , \theta )$ does not depend on $\theta$ , since $g$ itself does not depend on $\theta$ , and is therefore equal to classical proximity operator of $\eta g$ which, with a slight overload of notations, we denote as $\operatorname { p r o x } _ { \eta g } ( y )$ (with a single argument). In other words, + +$$ +\left\{ \begin{array} { l l } { \mathrm { p r o x } _ { \eta g } ( y , \theta ) } & { = \mathrm { p r o x } _ { \eta g } ( y ) , } \\ { \partial _ { 1 } \mathrm { p r o x } _ { \eta g } ( y , \theta ) } & { = \partial \mathrm { p r o x } _ { \eta g } ( y ) , } \\ { \partial _ { 2 } \mathrm { p r o x } _ { \eta g } ( y , \theta ) } & { = 0 . } \end{array} \right. +$$ + +Regarding the first claim (expression of the estimated Jacobian evaluated at $\hat { x }$ ), we first have that $\mathrm { p r o x } _ { \eta g } ( y )$ is the solution to $( \overline { { { x } } } ^ { \prime } - y ) + \eta \nabla g ( x ^ { \prime } ) = 0$ in $x ^ { \prime }$ - by first-order condition for a smooth convex function. We therefore have that + +$$ +\begin{array} { r l } & { \quad \mathrm { p r o x } _ { \eta g } ( y ) = ( I + \eta \nabla g ) ^ { - 1 } ( y ) } \\ & { \quad \partial \mathrm { p r o x } _ { \eta g } ( y ) = ( I _ { d } + \eta \nabla ^ { 2 } g ( \mathrm { p r o x } _ { \eta g } ( y ) ) ) ^ { - 1 } , } \end{array} +$$ + +the first $I$ and inverse being functional identity and inverse, and the second $I _ { d }$ and inverse being in the matrix sense, by inverse rule for Jacobians $\partial h ( z ) = [ \partial h ^ { - 1 } ( h ( z ) ) ] ^ { - 1 }$ (applied to the prox). + +As a consequence, we have, for $\Gamma _ { \eta } ( x , \theta ) = \nabla ^ { 2 } g ( \mathrm { p r o x } _ { \eta g } ( x - \eta \nabla _ { 1 } f ( x , \theta ) )$ that + +$$ +\begin{array} { r l } & { A ( x , \theta ) = I _ { d } - ( I _ { d } + \eta \Gamma _ { \eta } ( x , \theta ) ) ^ { - 1 } ( I _ { d } - \eta \nabla _ { 1 } ^ { 2 } f ( x , \theta ) ) } \\ & { \quad \quad \quad = ( I _ { d } + \eta \Gamma _ { \eta } ( x , \theta ) ) ^ { - 1 } [ I _ { d } + \eta \Gamma _ { \eta } ( x , \theta ) - ( I _ { d } - \eta \nabla _ { 1 } ^ { 2 } f ( x , \theta ) ) ] } \\ & { \quad \quad \quad = \eta ( I _ { d } + \eta \Gamma _ { \eta } ( x , \theta ) ) ^ { - 1 } ( \nabla _ { 1 } ^ { 2 } f ( x , \theta ) + \Gamma _ { \eta } ( x , \theta ) ) } \\ & { B ( x , \theta ) = - \eta ( I _ { d } + \eta \Gamma _ { \eta } ( x , \theta ) ) ^ { - 1 } \partial _ { 2 } \nabla _ { 1 } f ( x , \theta ) . } \end{array} +$$ + +As a consequence, for all $\boldsymbol { x } \in \mathbb { R } ^ { d }$ , we have that + +$$ +J ( x , \theta ) = - ( \nabla _ { 1 } ^ { 2 } f ( x , \theta ) + \Gamma _ { \eta } ( x , \theta ) ) ^ { - 1 } \partial _ { 2 } \nabla _ { 1 } f ( x , \theta ) . +$$ + +In the following, we modify slightly the notation of both $A$ and $B$ , writing + +$$ +\begin{array} { r l } & { \tilde { A } ( x , \theta ) = \nabla _ { 1 } ^ { 2 } f ( x , \theta ) + \Gamma _ { \eta } ( x , \theta ) } \\ & { \tilde { B } ( x , \theta ) = - \partial _ { 2 } \nabla _ { 1 } f ( x , \theta ) . } \end{array} +$$ + +With the current hypotheses, following along the proof of Theorem 1, we have that $\tilde { A }$ is $( \alpha + \mu )$ well-conditioned, and $( \gamma + \kappa _ { \eta } )$ -Lipschitz in its first argument, and $\tilde { B }$ is $\beta$ -Lipschitz in its first argument and bounded in norm by $R$ . The same reasoning yields + +$$ +\| J ( \hat { x } , \theta ) - \partial x ^ { \star } ( \theta ) \| \leq \left( \frac { \beta + \kappa _ { \eta } } { \alpha + \mu } + \frac { \gamma R } { ( \alpha + \mu ) ^ { 2 } } \right) \| \hat { x } - x ^ { \star } ( \theta ) \| . +$$ + +![](images/1e0d895f525a6e7cd14c80c0f0950c35d1838bab6b770597789ae19c3cda6d85.jpg) +Figure 13: GPU runtime comparison of implicit differentiation and unrolling for hyperparameter optimization of multiclass SVMs for multiple problem sizes (same setting as Figure 3). Error bars represent $90 \%$ confidence intervals. Absent data points were due to out-of-memory errors $( 1 6 \mathrm { G B }$ maximum). + +![](images/12b2f0934006530051b146228d2e6f05d996a17fe28c5c55485e3b5a8ff6d9f2.jpg) +Figure 14: Value of the outer problem objective function (validation loss) for hyperparameter optimization of multiclass SVMs for multiple problem sizes (same setting as Figure 3). As can be seen, all methods performed similarly in terms of validation loss. This confirms that the faster runtimes for implicit differentiation compared to unrolling shown in Figure 3 (CPU) and Figure 13 (GPU) are not at the cost of worse validation loss. + +![](images/fcc07750db1e2507aa9944019a2a73ab3473e0b2d476fcac6e196174a0b4ddb6.jpg) +Figure 15: Jacobian error $\lVert \partial x ^ { \star } ( \theta ) - J ( \hat { x } , \theta ) \rVert _ { 2 }$ (see also Definition 1) evaluated with a regularization parameter of $\theta = 1$ , as a function of solution error $\lVert \boldsymbol { x } ^ { \star } ( \theta ) - \boldsymbol { \hat { x } } \rVert _ { 2 }$ when varying the number of features, on the multiclass SVM task (see Appendix E.1 for a detailed description of the experimental setup). The ground-truth solution $x ^ { \star } ( \theta )$ is computed using the liblinear solver (Fan et al., 2008) available in scikit-learn (Pedregosa et al., 2011) with a very low tolerance of $1 0 ^ { - 9 }$ . Unlike in Figure 2, which was on ridge regression, the ground-truth Jacobian ${ \partial x ^ { \star } ( \theta ) }$ cannot be computed in closed form, in the more difficult setting of multiclass SVMs. We therefore use a finite difference to approximately compute ${ \partial x ^ { \star } ( \theta ) }$ . Our results nevertheless confirm similar trends as in Figure 2. + +E EXPERIMENTAL SETUP AND ADDITIONAL RESULTS + +Our experiments use JAX (Bradbury et al., 2018), which is Apache2-licensed and scikit-learn (Pedregosa et al., 2011), which is BSD-licensed. + +# E.1 HYPERPARAMETER OPTIMIZATION OF MULTICLASS SVMS + +Experimental setup. Synthetic datasets were generated using scikit-learn’s sklearn.datasets.make_classification (Pedregosa et al., 2011), following a model adapted from (Guyon, 2003). All datasets consist of $m = 7 0 0$ training samples belonging to $k = 5$ distinct classes. To simulate problems of different sizes, the number of features is varied as $p \in \{ 1 0 0 , 2 5 0 , 5 0 0 , 7 5 0 , 1 0 0 0 , 2 0 0 0 , 3 0 0 0 , 4 0 0 0 , 5 0 0 0 , 7 5 0 0 , 1 0 0 0 0 \}$ , with $10 \%$ of features being informative and the rest random noise. In all cases, an additional $m _ { \mathrm { v a l } } = 2 0 0$ validation samples were generated from the same model to define the outer problem. + +For the inner problem, we employed three different solvers: (i) mirror descent, (ii) (accelerated) proximal gradient descent and (iii) block coordinate descent. Hyperparameters for all solvers were individually tuned manually to ensure convergence across the range of problem sizes. For mirror descent, a stepsize of 1.0 was used for the first 100 steps, following a inverse square root decay afterwards up to a total of 2500 steps. For proximal gradient descent, a stepsize of $5 \cdot 1 0 ^ { - 4 } $ was used for 2500 steps. The block coordinate descent solver was run for 500 iterations. All solvers used the same initialization, namely, $\begin{array} { r } { x _ { \mathrm { i n i t } } = \frac { 1 } { k } 1 _ { m \times k } } \end{array}$ , which satisfies the dual constraints. + +For the outer problem, gradient descent was used with a stepsize of $5 \cdot 1 0 ^ { - 3 } ~ $ for the first 100 steps, following a inverse square root decay afterwards up to a total of 150 steps. + +Conjugate gradient was used to solve the linear systems in implicit differentiation for at most 2500 iterations. + +All results reported pertaining CPU runtimes were obtained using an internal compute cluster. GPU results were obtained using a single NVIDIA P100 GPU with 16GB of memory per dataset. For each dataset size, we report the average runtime of an individual iteration in the outer problem, alongside a $90 \%$ confidence interval estimated from the corresponding 150 runtime values. + +Additional results Figure 13 compares the runtime of implicit differentiation and unrolling on GPU. These results highlight a fundamental limitation of the unrolling approach in memory-limited systems such as accelerators, as the inner solver suffered from out-of-memory errors for most problem sizes ${ \it p } \geq 2 0 0 0$ for mirror descent, $p \geq 7 5 0$ for proximal gradient and block coordinate descent). While it might be possible to ameliorate this limitation by reducing the maximum number of iterations in the inner solver, doing so might lead to additional challenges (Wu et al., 2018) and require careful tuning. + +Figure 14 depicts the validation loss (value of the outer problem objective function) at convergence. It shows that all approaches were able to solve the outer problem, with solutions produced by different approaches being qualitatively indistinguishable from each other across the range of problem sizes considered. + +Figure 15 shows the Jacobian error achieved as a function of the solution error, when varying the number of features. + +# E.2 TASK-DRIVEN DICTIONARY LEARNING + +We downloaded from http://acgt.cs.tau.ac.il/multi_omic_benchmark/ download.html a set of breast cancer gene expression data together with survival information generated by the TCGA Research Network (https://www.cancer.gov/tcga) and processed as explained by (Rappoport & Shamir, 2018). The gene expression matrix contains the expression value for $_ { \mathrm { p = 2 0 } , 5 3 1 }$ genes in $_ \mathrm { m = 1 } , 2 1 2$ samples, from which we keep only the primary tumors $\mathrm { ( m { = } 1 , 0 9 3 }$ ). From the survival information, we select the patients who survived at least five years after diagnosis $( m _ { 1 } = 2 0 0 )$ ), and the patients who died before five years $\mathbf { \zeta } _ { m _ { 0 } } = 9 9 \mathbf { \zeta }$ ), resulting in a cohort of $m = 2 9 9$ patients with gene expression and binary label. Note that non-selected patients are those who are marked as alive but were not followed for 5 years. + +To evaluate different binary classification methods on this cohort, we repeated 10 times a random split of the full cohort into a training $( 6 0 \% )$ , validation $( 2 0 \% )$ and test $( 2 0 \% )$ sets. For each split and each method, 1) the method is trained with different parameters on the training set, 2) the parameter that maximizes the classification AUC on the validation set is selected, 3) the method is then re-trained on the union of the training and validation sets with the selected parameter, and 4) we measure the AUC of that model on the test set. We then report, for each method, the mean test AUC over the 10 repeats, together with a $9 5 \%$ confidence interval defined a mean $\pm \ : 1 . 9 6 \times$ standard error of the mean. + +We used Scikit Learn’s implementation of logistic regression regularized by $\ell _ { 1 }$ (lasso) and $\ell _ { 2 }$ (ridge) penalty from sklearn.linear_model.LogisticRegression, and varied the C regularization parameter over a grid of 10 values: $\{ 1 0 ^ { - 5 } , 1 0 ^ { - 3 } , \dots , 1 0 ^ { 4 } \}$ . For the unsupervised dictionary learning experiment method, we estimated a dictionary from the gene expression data in the training and validation sets, using sklearn.decomposition.DictionaryLearning(n_component $s { = } 1 0$ , $\mathtt { a l p h a } { = } 2 . 0 $ ), which produces sparse codes in $\begin{array} { r l r } { k } & { { } = } & { 1 0 } \end{array}$ dimensions with roughly $5 0 \%$ nonzero coefficients by minimizing the squared Frobenius reconstruction distance with lasso regularization on the code. We then use sklearn.linear_model.LogisticRegression to train a logistic regression on the codes, varying the ridge regularization parameter C over a grid of 10 values $\{ \breve { 1 } 0 ^ { - 1 } , 1 \breve { 0 ^ { 0 } } , \dots , 1 0 ^ { 8 } \}$ . + +Finally, we implemented the task-driven dictionary learning model (13) with our toolbox, following the pseudo-code in Figure 10. Like for the unsupervised dictionary learning experiment, we set the dimension of the codes to $k = 1 0$ , and a fixed elastic net regularization on the inner optimization problem to ensure that the codes have roughly $5 0 \%$ sparsity. For the outer optimization problem, we solve an $\ell _ { 2 }$ regularized ridge regression problem, varying again the ridge regularization parameter $\mathrm { C }$ over a grid of 10 values $\{ 1 0 ^ { - 1 } , 1 0 ^ { 0 } , \dot { ~ } . ~ . ~ . ~ , 1 0 ^ { 8 } \}$ . Because the outer problem is non-convex, we minimize it using the Adam optimizer (Kingma & Ba, 2014) with default parameters. + +# E.3 DATASET DISTILLATION + +Experimental setup. For the inner problem, we used gradient descent with backtracking linesearch, while for the outer problem we used gradient descent with momentum and a fixed step-size. The momentum parameter was set to 0.9 while the step-size was set to 1. + +Figure 4 was produced after 4000 iterations of the outer loop on CPU (Intel(R) Xeon(R) Platinum P-8136 CPU $ @ ~ 2 . 0 0 \mathrm { G H z }$ ), which took 1h55. Unrolled differentiation took instead 8h:05 (4 times more) to run the same number of iterations. As can be seen in Figure 16, the output is the same in both approaches. + +![](images/d14a67dc9aa1e9ca11d55deb01b59e8b2000ea8205e7371b8fdd576c7bf9e062.jpg) +Figure 16: Distilled MNIST dataset $\boldsymbol { \theta } \in \mathbb { R } ^ { k \times p }$ obtained by solving (12) through unrolled differentiation. Although there is no qualitative difference, the implicit differentiation approach is 4 times faster. + +# E.4 MOLECULAR DYNAMICS + +Our experimental setup is adapted from the JAX-MD example notebook available at https://github.com/google/jax-md/blob/master/notebooks/meta_ optimization.ipynb. + +We emphasize that calculating the gradient of the total energy objective, $\begin{array} { r } { f ( x , \theta ) = \sum _ { i j } U ( x _ { i , j } , \theta ) } \end{array}$ , with respect to the diameter $\theta$ of the smaller particles, $\nabla _ { 1 } f ( x , \theta )$ , does not require implicit differentiation or unrolling. This is because $\nabla _ { 1 } f ( x , \theta ) = 0$ at $x = x ^ { \star } ( \theta )$ : + +$$ +\nabla _ { \boldsymbol { \theta } } f ( { \boldsymbol x } ^ { \star } ( { \boldsymbol { \theta } } ) , { \boldsymbol { \theta } } ) = \partial { \boldsymbol x } ^ { \star } ( { \boldsymbol { \theta } } ) ^ { \top } \nabla _ { 1 } f ( { \boldsymbol x } ^ { \star } ( { \boldsymbol { \theta } } ) , { \boldsymbol { \theta } } ) + \nabla _ { 2 } f ( { \boldsymbol x } ^ { \star } ( { \boldsymbol { \theta } } ) , { \boldsymbol { \theta } } ) = \nabla _ { 2 } f ( { \boldsymbol { x } } ^ { \star } ( { \boldsymbol { \theta } } ) , { \boldsymbol { \theta } } ) . +$$ + +This is known as Danskin’s theorem or envelope theorem. Thus instead, we consider sensitivities of position ${ \partial x ^ { \star } ( \theta ) }$ directly, which does require implicit differentiation or unrolling. + +Our results comparing implicit and unrolled differentiation for calculating the sensitivity of position are shown in Figure 17. We use BiCGSTAB (Vorst & van der Vorst, 1992) to perform the tangent linear solve. Like in the original JAX-MD experiment, we use $k = 1 2 8$ particles in $m = 2$ dimensions. + +![](images/ceb73cf3c881abac0e260e868a99195719436064bfa46aa686abce2ad7f0ee01.jpg) +Figure 17: L1 norm of position sensitivities in the molecular dynamics simulations, for 40 different random initial conditions (different colored lines). Gradients through the unrolled FIRE optimizer (Bitzek et al., 2006) for many initial conditions do not converge, in contrast to implicit differentiation. \ No newline at end of file diff --git a/parse/dev/TQ75Md-FqQp/TQ75Md-FqQp_content_list.json b/parse/dev/TQ75Md-FqQp/TQ75Md-FqQp_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..cc6ba141690746e3a5f1790c1b9c9ad77969caba --- /dev/null +++ b/parse/dev/TQ75Md-FqQp/TQ75Md-FqQp_content_list.json @@ -0,0 +1,3988 @@ +[ + { + "type": "text", + "text": "EFFICIENT AND MODULAR IMPLICIT DIFFERENTIATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 728, + 118 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors ", + "text_level": 1, + "bbox": [ + 184, + 143, + 326, + 156 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Paper under double-blind review ", + "bbox": [ + 184, + 157, + 398, + 170 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 207, + 544, + 222 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Automatic differentiation (autodiff) has revolutionized machine learning. It allows expressing complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently, differentiation of optimization problem solutions has attracted widespread attention with applications such as optimization layers, and in bi-level problems such as hyper-parameter optimization and meta-learning. However, so far, implicit differentiation remained difficult to use for practitioners, as it often required case-by-case tedious mathematical derivations and implementations. In this paper, we propose an efficient and modular approach for implicit differentiation of optimization problems. In our approach, the user defines directly in Python a function $F$ capturing the optimality conditions of the problem to be differentiated. Once this is done, we leverage autodiff of $F$ and implicit differentiation to automatically differentiate the optimization problem. Our approach thus combines the benefits of implicit differentiation and autodiff. It is efficient as it can be added on top of any state-of-the-art solver and modular as the optimality condition specification is decoupled from the implicit differentiation mechanism. We show that seemingly simple principles allow to recover many existing implicit differentiation methods and create new ones easily. We demonstrate the ease of formulating and solving bi-level optimization problems using our framework. We also showcase an application to the sensitivity analysis of molecular dynamics. ", + "bbox": [ + 233, + 237, + 766, + 515 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 525, + 336, + 540 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Automatic differentiation (autodiff) is now an inherent part of machine learning software. It allows to express complex computations by composing elementary ones in creative ways and removes the tedious burden of computing their derivatives by hand. In parallel, the differentiation of optimization problem solutions has found many applications. A classical example is bi-level optimization, which typically involves computing the derivatives of a nested optimization problem in order to solve an outer one. Examples of applications in machine learning include hyper-parameter optimization (Chapelle et al., 2002; Seeger, 2008; Pedregosa, 2016; Franceschi et al., 2017; Bertrand et al., 2020; 2021), neural networks (Lorraine et al., 2020), and meta-learning (Franceschi et al., 2018; Rajeswaran et al., 2019). Another line of active research involving differentiation of optimization problem solutions are optimization layers (Kim et al., 2017; Amos & Kolter, 2017; Niculae & Blondel, 2017; Djolonga & Krause, 2017; Gould et al., 2019), which can be used to encourage structured outputs, and implicit deep networks (Bai et al., 2019; El Ghaoui et al., 2019), which have a smaller memory footprint than backprop-trained networks. ", + "bbox": [ + 174, + 556, + 825, + 736 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since optimization problem solutions typically do not enjoy an explicit formula in terms of their inputs, autodiff cannot be used directly to differentiate these functions. In recent years, two main approaches have been developed to circumvent this problem. The first one consists of unrolling the iterations of an optimization algorithm and using the final iteration as a proxy for the optimization problem solution (Wengert, 1964; Domke, 2012; Deledalle et al., 2014; Franceschi et al., 2018; Ablin et al., 2020). This allows to explicitly construct a computational graph relating the algorithm output to the inputs, on which autodiff can then be used transparently. However, this requires a reimplementation of the algorithm using the autodiff system, and not all algorithms are necessarily autodiff friendly. Moreover, forward-mode autodiff has time complexity that scales linearly with the number of variables and reverse-mode autodiff has memory complexity that scales linearly with the number of algorithm iterations. In contrast, a second approach consists in implicitly relating an optimization problem solution to its inputs using optimality conditions. In a machine learning context, such implicit differentiation has been used for stationarity conditions (Bengio, 2000; ", + "bbox": [ + 174, + 743, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Lorraine et al., 2020), KKT conditions (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae & Martins, 2020) and the proximal gradient fixed point (Niculae & Blondel, 2017; Bertrand et al., 2020; 2021). An advantage of implicit differentiation is that a solver reimplementation is not needed, allowing to build upon decades of state-of-the-art software. Although implicit differentiation has a long history in numerical analysis (Griewank & Walther, 2008; Bell & Burke, 2008; Krantz & Parks, 2012; Bonnans & Shapiro, 2013), so far, it remained difficult to use for practitioners, as it required a case-by-case tedious mathematical derivation and implementation. CasADi (Andersson et al., 2019) allows to differentiate various optimization and root finding problem algorithms provided by the library. However, it does not allow to easily add implicit differentiation on top of existing solvers from optimality conditions expressed by the user, as we do. A recent tutorial explains how to implement implicit differentiation in JAX (Duvenaud et al., 2020). However, the tutorial requires the user to take care of low-level technical details and does not cover a large catalog of optimality condition mappings as we do. Other work (Agrawal et al., 2019a) attempts to address this issue by adding implicit differentiation on top of cvxpy (Diamond & Boyd, 2016). This works by reducing all convex optimization problems to a conic program and using conic programming’s optimality conditions to derive an implicit differentiation formula. While this approach is very generic, solving a convex optimization problem using a conic programming solver—an ADMM-based splitting conic solver (O’Donoghue et al., 2016) in the case of cvxpy—is rarely the state-of-the-art approach for each particular problem instance. ", + "bbox": [ + 173, + 103, + 826, + 367 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we adopt a different strategy that makes it easy to add implicit differentiation on top of any existing solver. In our approach, the user defines directly in Python a mapping function $F$ capturing the optimality conditions of the problem solved by the algorithm. Once this is done, we leverage autodiff of $F$ combined with implicit differentiation to automatically differentiate the optimization problem solution. In this way, our approach is generic, yet it can exploit the efficiency of state-of-the-art solvers. It therefore combines the benefits of implicit differentiation and autodiff. To summarize, we make the following contributions. ", + "bbox": [ + 173, + 375, + 825, + 472 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We describe our framework and its JAX implementation (provided in the supplementary material). Our framework significantly lowers the barrier to use implicit differentiation thanks to the use of autodiff of the optimality conditions and the seamless integration in JAX. Our framework significantly extends JAX for numerical optimization, with low-level details all abstracted away. • We instantiate our framework on a large catalog of optimality conditions (Table 1), recovering existing schemes and obtaining new ones, such as the mirror descent fixed point based one. • On the theoretical side, we provide new bounds on the Jacobian error when the optimization problem is only solved approximately, and empirically validate them. • We implement four illustrative applications, demonstrating our framework’s ease of use. ", + "bbox": [ + 176, + 478, + 826, + 619 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Beyond our software implementation, we hope this paper provides a self-contained blueprint for creating an efficient and modular implementation of implicit differentiation. ", + "bbox": [ + 171, + 626, + 821, + 654 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Notation. We denote the gradient and Hessian of $f \\colon { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }$ evaluated at $x \\in \\mathbb { R } ^ { d }$ by $\\nabla f ( x ) \\in \\mathbb { R } ^ { d }$ and $\\nabla ^ { 2 } f ( x ) \\in \\mathbb { R } ^ { d \\times d }$ . We denote the Jacobian of $F \\colon { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } ^ { p } }$ evaluated at $x \\in \\mathbb { R } ^ { d }$ by $\\partial F ( x ) \\in$ $\\mathbb { R } ^ { p \\times d }$ . When $f$ or $F$ have several arguments, we denote the gradient, Hessian and Jacobian in the $i ^ { \\mathrm { { t h } } }$ argument by $\\nabla _ { i }$ , $\\nabla _ { i } ^ { 2 }$ and $\\partial _ { i }$ , respectively. The standard probability simplex is denoted by $\\triangle ^ { d } : = \\{ x \\in \\mathbb { R } ^ { d } \\colon \\| x \\| _ { 1 } = 1 , x \\geq 0 \\}$ . For any set $\\mathcal { C } \\subset \\mathbb { R } ^ { d }$ , we denote the indicator function $I _ { \\mathcal { C } } : \\mathbb { R } ^ { d } \\overset { \\cdot } { } \\mathbb { R } \\cup \\{ + \\infty \\}$ where $I _ { \\mathit { C } } ( x ) = 0$ if $x \\in { \\mathcal { C } }$ , $I _ { C } ( x ) = + \\infty$ otherwise. For a vector or matrix $A$ we note $\\| A \\|$ the Frobenius (or Euclidean) norm, and $\\| A \\| _ { \\mathsf { o p } }$ the operator norm. ", + "bbox": [ + 173, + 665, + 825, + 765 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 COMBINING IMPLICIT DIFFERENTIATION AND AUTODIFF ", + "text_level": 1, + "bbox": [ + 176, + 785, + 671, + 799 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 GENERAL PRINCIPLES ", + "text_level": 1, + "bbox": [ + 176, + 811, + 369, + 825 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Overview. Contrary to autodiff through unrolled algorithm iterations, implicit differentiation typically involves a manual, sometimes complicated, mathematical derivation. For instance, numerous works (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae & Martins, 2020) use Karush–Kuhn–Tucker (KKT) conditions in order to relate a constrained optimization problem’s solution to its inputs, and to manually derive a formula for its derivatives. The derivation and implementation in these works are always case-by-case. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/fadbf82543eb3718bc2bb5820799eaf0ad81197f72ee1ae17d032dbb61832239.jpg", + "image_caption": [ + "Figure 1: Adding implicit differentiation on top of a ridge regression solver. The function $f ( x , \\theta )$ defines the objective function and the mapping $F$ , here simply equation (4), captures the optimality conditions. Our decorator @custom_root automatically adds implicit differentiation to the solver for the user, overriding JAX’s default behavior. The last line evaluates the Jacobian at $\\theta = 1 0$ . " + ], + "image_footnote": [], + "bbox": [ + 290, + 102, + 705, + 328 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we propose a generic way to easily add implicit differentiation on top of existing solvers. In our approach, the user defines directly in Python a mapping function $F$ capturing the optimality conditions of the problem solved by the algorithm. We provide reusable building blocks to easily express such $F$ . The provided $F$ is then plugged into our Python decorator @custom_root, which we append on top of the solver declaration we wish to differentiate. Under the hood, we combine implicit differentiation and autodiff of $F$ to automatically differentiate the optimization problem solution. A simple illustrative example is given in Figure 1. ", + "bbox": [ + 173, + 421, + 825, + 520 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Differentiating a root. Let $F \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }$ be a user-provided mapping, capturing the optimality conditions of a problem. An optimal solution, denoted $x ^ { \\star } ( \\theta )$ , should be a root of $F$ : ", + "bbox": [ + 173, + 532, + 823, + 563 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/446159dd8937a24464d0474490428271a6a102724c6010dab3e32a02a36af2d6.jpg", + "text": "$$\n{ \\cal F } ( x ^ { \\star } ( \\theta ) , \\theta ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 439, + 565, + 558, + 582 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We can see $x ^ { \\star } ( \\theta )$ as an implicitly defined function of $\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }$ , i.e., $x ^ { \\star } \\colon { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }$ . More precisely, from the implicit function theorem (Griewank $\\&$ Walther, 2008; Krantz & Parks, 2012), we know that for $( x _ { 0 } , \\theta _ { 0 } )$ satisfying $F ( x _ { 0 } , \\theta _ { 0 } ) = 0$ with a continuously differentiable $F$ , if the Jacobian $\\partial _ { 1 } F$ evaluated at $( x _ { 0 } , \\theta _ { 0 } )$ is a square invertible matrix, then there exists a function $x ^ { \\star } ( \\cdot )$ defined on a neighborhood of $\\theta _ { 0 }$ such that $x ^ { \\star } ( \\theta _ { 0 } ) = x _ { 0 }$ . Furthermore, for all $\\theta$ in this neighborhood, we have that $F ( x ^ { \\star } ( \\theta ) , \\theta ) = 0$ and ${ \\partial x ^ { \\star } ( \\theta ) }$ exists. Using the chain rule, the Jacobian ${ \\partial x ^ { \\star } ( \\theta ) }$ satisfies ", + "bbox": [ + 173, + 584, + 825, + 670 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/81383413e0a1e51eeb2277102390a9059d7d5e75c0458a66ed2ef3eb6a3cc1d6.jpg", + "text": "$$\n\\partial _ { 1 } F ( x ^ { \\star } ( \\theta ) , \\theta ) \\partial x ^ { \\star } ( \\theta ) + \\partial _ { 2 } F ( x ^ { \\star } ( \\theta ) , \\theta ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 671, + 645, + 689 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Computing ${ \\partial x ^ { \\star } ( \\theta ) }$ therefore boils down to the resolution of the linear system of equations ", + "bbox": [ + 169, + 693, + 763, + 707 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/97fb8b270917089f469ee014e79f21f03b3d3bc21b899a3796964a33d5c97a7e.jpg", + "text": "$$\n\\underbrace { - \\partial _ { 1 } F ( x ^ { \\star } ( \\theta ) , \\theta ) } _ { A \\in \\mathbb { R } ^ { d \\times d } } \\underbrace { \\partial x ^ { \\star } ( \\theta ) } _ { J \\in \\mathbb { R } ^ { d \\times n } } = \\underbrace { \\partial _ { 2 } F ( x ^ { \\star } ( \\theta ) , \\theta ) } _ { B \\in \\mathbb { R } ^ { d \\times n } } .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 709, + 642, + 744 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "When (1) is a one-dimensional root finding problem $\\langle d = 1 \\rangle$ ), (2) becomes particularly simple since we then have $\\nabla x ^ { \\star } ( \\theta ) = B ^ { \\top } / A$ , where $A$ is a scalar value. ", + "bbox": [ + 173, + 748, + 825, + 776 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We will show that existing and new implicit differentiation methods all reduce to this simple principle. We call our approach hybrid, since it combines implicit differentiation (and as such requires solving a linear system) with autodiff of the optimality conditions $F$ . Our approach is efficient as it can be added on top of any state-of-the-art solver and modular as the optimality condition specification is decoupled from the implicit differentiation mechanism. This contrasts with existing works, where the mathematical derivation and implementation are specific to each optimality condition. ", + "bbox": [ + 173, + 782, + 825, + 867 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Differentiating a fixed point. We will encounter numerous applications where $x ^ { \\star } ( \\theta )$ is implicitly defined through a fixed point: ", + "bbox": [ + 169, + 881, + 823, + 909 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/609d578cf202525a4f3375bef14f39c5b10c200d2736646abfd45d4a9bc32541.jpg", + "text": "$$\nx ^ { \\star } ( \\theta ) = T ( x ^ { \\star } ( \\theta ) , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 424, + 907, + 571, + 925 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $T \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }$ . This can be seen as a particular case of (1) by defining the residual ", + "bbox": [ + 169, + 103, + 797, + 118 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0c4605ac9c9419470cbe650e2534a0d735361af273562369bccdd0b35057c5d0.jpg", + "text": "$$\nF ( x , { \\boldsymbol { \\theta } } ) = T ( x , { \\boldsymbol { \\theta } } ) - x .\n$$", + "text_format": "latex", + "bbox": [ + 418, + 122, + 576, + 138 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this case, using the chain rule, we have ", + "bbox": [ + 174, + 142, + 447, + 156 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ce8daec59d9a8a25ff95d9a7bb2cbf9a992612eca05cfdcaccc14e50d953adaa.jpg", + "text": "$$\nA = - \\partial _ { 1 } F ( x ^ { \\star } ( \\theta ) , \\theta ) = I - \\partial _ { 1 } T ( x ^ { \\star } ( \\theta ) , \\theta ) \\quad \\mathrm { a n d } \\quad B = \\partial _ { 2 } F ( x ^ { \\star } ( \\theta ) , \\theta ) = \\partial _ { 2 } T ( x ^ { \\star } ( \\theta ) , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 200, + 160, + 795, + 178 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Computing JVPs and VJPs. In most practical scenarios, it is not necessary to explicitly form the Jacobian matrix, and instead it is sufficient to left-multiply or right-multiply by $\\partial _ { 1 } F$ and $\\partial _ { 2 } F$ . These are called vector-Jacobian product (VJP) and Jacobian-vector product (JVP), and are useful for integrating $x ^ { \\star } ( \\theta )$ with reverse-mode and forward-mode autodiff, respectively. Oftentimes, $F$ will be explicitly defined. In this case, computing the VJP or JVP can be done via autodiff. In some cases, $F$ may itself be implicitly defined, for instance when $F$ involves the solution of a variational problem. In this case, computing the VJP or JVP will itself involve implicit differentiation. ", + "bbox": [ + 173, + 188, + 826, + 286 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The right-multiplication (JVP) between $J = \\partial x ^ { \\star } ( \\theta )$ and a vector $v$ , $J v _ { : }$ , can be computed efficiently by solving $A ( \\bar { J v } ) = B v$ . The left-multiplication (VJP) of $v ^ { \\top }$ with J , $v ^ { \\top } J$ , can be computed by first solving $A ^ { \\top } u = v$ . Then, we can obtain $\\mathbf { \\Delta } _ { v } \\top { } _ { J }$ by $\\boldsymbol { v } ^ { \\top } \\boldsymbol { J } = \\boldsymbol { u } ^ { \\top } \\boldsymbol { A } \\boldsymbol { J } = \\boldsymbol { u } ^ { \\top } \\boldsymbol { B }$ . Note that when $B$ changes but $A$ and $v$ remain the same, we do not need to solve $A ^ { \\top } u = v$ once again. This allows to compute the VJP w.r.t. different variables while solving only one linear system. ", + "bbox": [ + 174, + 292, + 825, + 363 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To solve these linear systems, we can use the conjugate gradient method (Hestenes et al., 1952) when $A$ is symmetric positive semi-definite and GMRES (Saad & Schultz, 1986) or BiCGSTAB (Vorst & van der Vorst, 1992) otherwise. These algorithms are all matrix-free: they only require matrix-vector products. Thus, all we need from $F$ is its JVPs or VJPs. An alternative to GMRES/BiCGSTAB is to solve the normal equation $A A ^ { \\top } u = A v$ using conjugate gradient. We implement this using JAX’s automatic transpose routine jax.linear_transpose (Frostig et al., 2021). ", + "bbox": [ + 173, + 369, + 825, + 454 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Pre-processing and post-processing mappings. Oftentimes, the goal is not to differentiate $\\theta$ per se, but the parameters of a function producing $\\theta$ . One example of such pre-processing is to convert the parameters to be differentiated from one form to another canonical form, such as a quadratic program (Amos & Kolter, 2017) or a conic program (Agrawal et al., 2019a). Another example is when $x ^ { \\star } ( \\theta )$ is used as the output of a neural network layer, in which case $\\theta$ is produced by the previous layer. Likewise, $x ^ { \\star } ( \\theta )$ will often not be the final output we want to differentiate. One example of such post-processing is when $x ^ { \\star } ( \\theta )$ is the solution of a dual program and we apply the dual-primal mapping to recover the solution of the primal program. Another example is the application of a loss function, in order to reduce $x ^ { \\star } ( \\theta )$ to a scalar value. We leave the differentiation of such pre/post-processing mappings to the autodiff system, allowing to compose functions in complex ways. ", + "bbox": [ + 173, + 468, + 825, + 608 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Implementation details. When a solver function is decorated with @custom_root, we use jax.custom_jvp and jax.custom_vjp to automatically add custom JVP and VJP rules to the function, overriding JAX’s default behavior. As mentioned above, we use linear system solvers based on matrix-vector products and therefore we only need access to $F$ through the JVP or VJP with $\\partial _ { 1 } F$ and $\\partial _ { 2 } F$ . This is done by using $\\ j \\mathtt { a x } . \\mathtt { j v p }$ and ${ \\dot { \\mathsf { J } } } \\mathsf { a x } . \\mathsf { v } { \\dot { \\mathsf { J } } } { \\mathsf { p } }$ , respectively. Note that, as in Figure 1, the definition of $F$ will often include a gradient mapping $\\bar { \\nabla } _ { 1 } f ( x , \\bar { \\theta } )$ . Thankfully, JAX supports second-order derivatives transparently. For convenience, our library also provides a @custom_fixed_point decorator, for adding implicit differentiation on top of a solver, given a fixed point iteration $T$ ; see code examples in Appendix A. ", + "bbox": [ + 173, + 621, + 825, + 747 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 EXAMPLES ", + "text_level": 1, + "bbox": [ + 174, + 756, + 292, + 771 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We now give various examples of mapping $F$ or fixed point iteration $T$ , recovering existing implicit differentiation methods and creating new ones. Each choice of $F$ or $T$ implies different trade-offs in terms of computational oracles; see Table 1. Source code examples are given in Appendix A. ", + "bbox": [ + 174, + 782, + 825, + 825 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Stationary point condition. The simplest example is to differentiate through the implicit function ", + "bbox": [ + 173, + 838, + 821, + 854 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/07e627d53dddc2d6519bd8cf9549e9a0d802100c83341d4451b04e472450ec43.jpg", + "text": "$$\nx ^ { \\star } ( \\theta ) = \\underset { x \\in \\mathbb { R } ^ { d } } { \\operatorname { a r g m i n } } f ( x , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 857, + 581, + 885 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $f \\colon { \\mathbb { R } ^ { d } } \\times { \\mathbb { R } ^ { n } } \\to { \\mathbb { R } }$ is twice differentiable. In this case, $F$ is simply the gradient mapping ", + "bbox": [ + 169, + 890, + 792, + 905 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c61270e26b9d8b722985b0b7fad79558a593c097b0ccde92612e28bb3c880f28.jpg", + "text": "$$\nF ( x , \\theta ) = \\nabla _ { 1 } f ( x , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 424, + 907, + 573, + 925 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/944efd29488c585cb380b987bceed6d0f68e26858bcf0e5a0961d828d2468403.jpg", + "table_caption": [ + "Table 1: Summary of optimality condition mappings. Oracles are accessed through their JVP or VJP. " + ], + "table_footnote": [], + "table_body": "
NameEquationSolution neededOracles needed
Stationary(4), (5)PrimalV1f
KKT(6)Primal and dualVif,H,G,aH,0G
Proximal gradient(7)PrimalVif,proxng
Projected gradient(9)PrimalVif,projc
Mirror descent(11)PrimalVif,proje,
Newton(14)Primal[V²f(x,0)]-1,Vif(x,0)
Block proximal gradient(15)Primal[Vif]j,[proXnglj
Conic programming(18)Residual map rootprojRP × K* XR+
", + "bbox": [ + 228, + 132, + 771, + 268 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We then have $\\partial _ { 1 } F ( x , \\theta ) = \\nabla _ { 1 } ^ { 2 } f ( x , \\theta )$ and $\\partial _ { 2 } F ( x , \\theta ) = \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta )$ , the Hessian of $f$ in its first argument and the Jacobian in the second argument of $\\nabla _ { 1 } f ( x , \\theta )$ . In practice, we use autodiff to compute Jacobian products automatically. Equivalently, we can use the gradient descent fixed point ", + "bbox": [ + 174, + 296, + 823, + 339 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ce04bff7c05cc5a9c1794520a6c422fb269f379b4b18d673f60c9e72e386ad49.jpg", + "text": "$$\nT ( x , \\theta ) = x - \\eta \\nabla _ { 1 } f ( x , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 405, + 343, + 589, + 359 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "for all $\\eta > 0$ . Using (3), it is easy to check that we obtain the same linear system since $\\eta$ cancels out. ", + "bbox": [ + 176, + 363, + 823, + 378 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "KKT conditions. We now show that the KKT conditions, manually differentiated in several works (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae $\\&$ Martins, 2020), fit our framework. As we will see, the key will be to group the optimal primal and dual variables as our $x ^ { \\star } ( \\theta )$ . Let us consider the general problem ", + "bbox": [ + 173, + 392, + 826, + 449 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4dde206746cf2aba2d0f1a638f31205f33983dc509217bd52a1f81149053551e.jpg", + "text": "$$\n\\operatorname { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } } f ( z , \\theta ) \\quad { \\mathrm { s u b j e c t ~ t o } } \\quad G ( z , \\theta ) \\leq 0 , \\ H ( z , \\theta ) = 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 310, + 452, + 684, + 478 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $z \\in \\mathbb { R } ^ { p }$ is the primal variable, $f \\colon { \\mathbb { R } } ^ { p } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } }$ , $G \\colon \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { r }$ and $H \\colon \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { q }$ The stationarity, primal feasibility and complementary slackness conditions give ", + "bbox": [ + 173, + 481, + 823, + 510 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/abb0c185cb5cc8c66837b68d239d550b3b91169c3424629a82110a4a9a606fcd.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta ) + \\left[ \\partial _ { 1 } G ( z , \\theta ) \\right] ^ { \\top } \\boldsymbol { \\lambda } + \\left[ \\partial _ { 1 } H ( z , \\theta ) \\right] ^ { \\top } \\boldsymbol { \\nu } = 0 } \\\\ { H ( z , \\theta ) = 0 } \\\\ { \\boldsymbol { \\lambda } \\circ G ( z , \\theta ) = 0 , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 511, + 666, + 569 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\nu \\in \\mathbb { R } ^ { q }$ and $\\lambda \\in \\mathbb { R } _ { + } ^ { r }$ are the dual variables, also known as KKT multipliers. The primal and dual feasibility conditions can be ignored almost everywhere. The system of (potentially nonlinear) equations (6) fits our framework, as we can group the primal and dual solutions as $x ^ { \\star } ( \\theta ) = ( z ^ { \\star } ( { \\bar { \\theta } } ) , \\nu ^ { \\star } ( \\theta ) , \\lambda ^ { \\star } ( \\theta ) )$ to form the root of a function $F ( x ^ { \\star } ( \\theta ) , \\theta )$ , where $F \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }$ and $d = p + q + r$ . The primal and dual solutions can be obtained from a generic solver, such as an interior point method. In practice, the above mapping $F$ will be defined directly in Python (see Figure 7 in Appendix A) and $F$ will be differentiated automatically via autodiff. ", + "bbox": [ + 173, + 569, + 825, + 669 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proximal gradient fixed point. Unfortunately, not all algorithms return both primal and dual solutions. Moreover, if the objective contains non-smooth terms, proximal gradient descent may be more efficient. We now discuss its fixed point (Niculae & Blondel, 2017; Bertrand et al., 2020; 2021). Let $x ^ { \\star } ( \\theta )$ be implicitly defined as ", + "bbox": [ + 173, + 681, + 826, + 738 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/92f4e6a348b59884f2774aeba492d29f0d7c2a633b2cd24da89ef622723f2cd8.jpg", + "text": "$$\nx ^ { \\star } ( \\theta ) : = \\operatornamewithlimits { a r g m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x , \\theta ) + g ( x , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 741, + 616, + 768 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $f \\colon { \\mathbb { R } ^ { d } } \\times { \\mathbb { R } ^ { n } } \\to { \\mathbb { R } }$ is twice-differentiable convex and $g \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } }$ is convex but possibly non-smooth. Let us define the proximity operator associated with $g$ by ", + "bbox": [ + 173, + 773, + 825, + 803 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/9995a050e282b9d228ea0cb44887b929e24330d07d657d0521878cff67a26d1e.jpg", + "text": "$$\n\\operatorname { p r o x } _ { g } ( y , \\theta ) : = \\underset { x \\in \\mathbb { R } ^ { d } } { \\operatorname { a r g m i n } } \\frac { 1 } { 2 } \\| x - y \\| _ { 2 } ^ { 2 } + g ( x , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 804, + 648, + 839 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To implicitly differentiate $x ^ { \\star } ( \\theta )$ , we use the fixed point mapping (Parikh & Boyd, 2014, p.150) ", + "bbox": [ + 169, + 843, + 795, + 858 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/59019423b360129e273fb0fdfd2d84ee49d86749321eeffb9554ed9c230e134d.jpg", + "text": "$$\nT ( x , \\theta ) = \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 861, + 627, + 881 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "for any step size $\\eta > 0$ . The proximity operator is 1-Lipschitz continuous (Moreau, 1965). By Rademacher’s theorem, it is differentiable almost everywhere. Many proximity operators enjoy a closed form and can easily be differentiated, as discussed in Appendix $\\mathbf { B }$ . ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Projected gradient fixed point. As a special case, when $g ( x , \\theta )$ is the indicator function $I _ { C ( \\theta ) } ( x )$ where $\\mathcal { C } ( \\boldsymbol { \\theta } )$ is a convex set depending on $\\theta$ , we obtain ", + "bbox": [ + 171, + 103, + 825, + 133 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/21d0db90801565fc4db73402a11b563257f5f09cc557cb224a7c047030239af8.jpg", + "text": "$$\nx ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x \\in { \\mathcal { C } } ( \\theta ) } f ( x , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 138, + 583, + 169 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The proximity operator $\\operatorname { p r o x } _ { g }$ becomes the Euclidean projection onto $\\mathcal { C } ( \\boldsymbol { \\theta } )$ ", + "bbox": [ + 173, + 174, + 660, + 191 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/5f9e4e81f0c3d5dbce35a254d6cab821b0da78a8f8a04166872eeae9d80f5e9e.jpg", + "text": "$$\n{ \\mathrm { p r o x } } _ { g } ( y , \\theta ) = { \\mathrm { p r o j } } _ { \\mathcal { C } } ( y , \\theta ) : = { \\underset { x \\in { \\mathcal { C } } ( \\theta ) } { \\operatorname { a r g m i n } } } \\| x - y \\| _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 341, + 199, + 655, + 228 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "and (7) becomes the projected gradient fixed point ", + "bbox": [ + 173, + 234, + 506, + 250 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9d1e1bc231914c660ca72a0e10d85b08ee3097a06cd2d8c401321814985f4fc6.jpg", + "text": "$$\nT ( x , \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 374, + 256, + 622, + 273 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Compared to the KKT conditions, this fixed point is particularly suitable when the projection enjoys a closed form. We discuss how to compute the JVP / VJP for a wealth of convex sets in Appendix B. ", + "bbox": [ + 173, + 280, + 825, + 309 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Mirror descent fixed point. We again consider the case when $x ^ { \\star } ( \\theta )$ is implicitly defined as the solution of (8). We now generalize the projected gradient fixed point beyond Euclidean geometry. Let the Bregman divergence $D _ { \\varphi }$ : $\\operatorname { d o m } ( \\bar { \\varphi } ) \\cdot \\operatorname { r e l i n t } ( \\operatorname { d o m } ( \\varphi ) ) \\to \\bar { \\mathbb { R } } _ { + }$ generated by $\\varphi$ be defined by ", + "bbox": [ + 173, + 323, + 826, + 367 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/58674b69bdba2abf82e66af3aac43fbaad9e6039780737464627a63de92b9fd4.jpg", + "text": "$$\nD _ { \\varphi } ( x , y ) : = \\varphi ( x ) - \\varphi ( y ) - \\langle \\nabla \\varphi ( y ) , x - y \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 372, + 648, + 391 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We define the Bregman projection of $y$ onto ${ \\mathcal { C } } ( \\theta ) \\subseteq \\mathrm { d o m } ( \\varphi )$ by ", + "bbox": [ + 173, + 396, + 593, + 412 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/5e5a83a15232511f9ede66f58e7405d946d6294562f410a45d6f9751f1ee6f1a.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y , \\theta ) : = \\underset { x \\in \\mathcal { C } ( \\theta ) } { \\mathrm { a r g m i n } } D _ { \\varphi } ( x , \\nabla \\varphi ^ { \\ast } ( y ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 366, + 417, + 630, + 446 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Definition (10) includes the mirror map $\\nabla \\varphi ^ { * } ( y )$ for convenience. It can be seen as a mapping from $\\mathbb { R } ^ { d }$ to $\\mathrm { d o m } ( \\varphi )$ , ensuring that (10) is well-defined. The mirror descent fixed point mapping is then ", + "bbox": [ + 173, + 453, + 823, + 483 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6318c6ba9d6dcb60b25864ed6f1fd681198ea862437c2ff09c477c12ca89ce83.jpg", + "text": "$$\n\\begin{array} { c } { \\hat { x } = \\nabla \\varphi ( x ) } \\\\ { y = \\hat { x } - \\eta \\nabla _ { 1 } f ( x , \\theta ) } \\\\ { T ( x , \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y , \\theta ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 408, + 488, + 589, + 545 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Because $T$ involves the composition of several functions, manually deriving its JVP/VJP is error prone. This shows that our approach leveraging autodiff allows to handle more advanced fixed point mappings. A common example of $\\varphi$ is $\\varphi ( x ) = \\langle x , \\log x - 1 \\rangle$ , where $\\mathrm { d o m } ( \\varphi ) = \\mathbb { R } _ { + } ^ { d }$ . In this case, $D _ { \\varphi }$ is the Kullback-Leibler divergence. An advantage of the Kullback-Leibler projection is that it sometimes easier to compute than the Euclidean projection, as we detail in Appendix B. ", + "bbox": [ + 173, + 549, + 825, + 619 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Other fixed points. More fixed points are described in Appendix C. ", + "bbox": [ + 173, + 633, + 632, + 648 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 JACOBIAN PRECISION GUARANTEES ", + "text_level": 1, + "bbox": [ + 174, + 659, + 504, + 675 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In practice, either by the limitations of finite precision arithmetic or because we perform a finite number of iterations, we rarely reach the exact solution $x ^ { \\star } ( \\theta )$ , but instead only reach an approximate solution $\\hat { x }$ and apply the implicit differentiation equation (2) at this approximate solution. This motivates the need for precision guarantees of this approach. We introduce the following formalism. ", + "bbox": [ + 174, + 690, + 826, + 746 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Definition 1. Let $F : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { n } \\mathbb { R } ^ { d }$ be an optimality criterion mapping. Let $A : = - \\partial _ { 1 } F$ and $B : = \\partial _ { 2 } F$ . We define the Jacobian estimate at $( x , \\dot { \\theta } )$ as the solution to the linear equation $A ( x , \\theta ) J ( x , \\theta ) = B ( x , \\theta )$ . It is a function $J : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { \\bar { n } } \\mathbb { R } ^ { d \\times n }$ . ", + "bbox": [ + 173, + 750, + 823, + 792 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "It holds by construction that $J ( x ^ { \\star } ( \\theta ) , \\theta ) = \\partial x ^ { \\star } ( \\theta )$ . Computing $J ( \\hat { x } , \\theta )$ for an approximate solution $\\hat { x }$ of $x ^ { \\star } ( { \\boldsymbol { \\theta } } )$ therefore allows to approximate the true Jacobian ${ \\partial x ^ { \\star } ( \\theta ) }$ . In practice, an algorithm used to solve (1) depends on $\\theta$ . Note however that, what we compute is not the Jacobian of ${ \\hat { x } } ( \\theta )$ , unlike works differentiating through unrolled algorithm iterations, but an estimate of ${ \\partial x ^ { \\star } ( \\theta ) }$ . We therefore use the notation $\\hat { x }$ , leaving the dependence on $\\theta$ implicit. ", + "bbox": [ + 174, + 804, + 825, + 876 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We develop bounds of the form $\\| J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| < C \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\|$ , hence showing that the error on the estimated Jacobian is at most of the same order as that of $\\hat { x }$ as an approximation of $x ^ { \\star } ( \\theta )$ These bounds are based on the following main theorem, whose proof is included in Appendix D. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 1 (Jacobian estimate). Let $F : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { n } \\mathbb { R } ^ { d }$ . Assume that there exist $\\alpha , \\beta , \\gamma , \\varepsilon , R > 0$ such that $A = - \\partial _ { 1 } F$ and $B = \\partial _ { 2 } F$ satisfy, for all $v \\in \\mathbb { R } ^ { d }$ , $\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }$ and $x$ such that $\\| x - x ^ { \\star } ( \\theta ) \\| \\leq \\varepsilon$ : ", + "bbox": [ + 169, + 102, + 826, + 133 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "$A$ is well-conditioned, Lipschitz: $\\| A ( x , \\theta ) v \\| \\geq \\alpha \\| v \\| , \\| A ( x , \\theta ) - A ( x ^ { \\star } ( \\theta ) , \\theta ) \\| _ { \\mathrm { o p } } \\leq \\gamma \\| x - x ^ { \\star } ( \\theta ) \\| .$ ", + "bbox": [ + 173, + 137, + 825, + 155 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "$B$ is bounded and Lipschitz: $\\| B ( x ^ { \\star } ( \\theta ) , \\theta ) \\| \\leq R$ , $\\| B ( x , \\theta ) - B ( x ^ { \\star } ( \\theta ) , \\theta ) \\| \\leq \\beta \\| x - x ^ { \\star } ( \\theta ) \\| .$ . ", + "bbox": [ + 173, + 159, + 802, + 176 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Under these conditions, when $\\| { \\hat { x } } - x ^ { \\star } ( \\theta ) \\| \\leq \\varepsilon$ , we have ", + "bbox": [ + 176, + 181, + 549, + 196 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/9f46752ccd33510fcf200f3960249df0d367cb5bb2933592959ba2cd6808261a.jpg", + "text": "$$\n\\begin{array} { r } { \\| J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| \\leq \\left( \\beta \\alpha ^ { - 1 } + \\gamma R \\alpha ^ { - 2 } \\right) \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\| . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 202, + 686, + 222 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "This result is inspired by (Higham, 2002, Theorem 7.2), that is concerned with the stability of solutions to inverse problems. Here we consider that $A ( \\cdot , \\theta )$ is uniformly well-conditioned, rather than only at $x ^ { \\star } ( \\theta )$ . This does not affect the first order in $\\varepsilon$ of this bound, and makes it valid for all $\\hat { x }$ . It is also more tailored to applications to equation-specific cases. ", + "bbox": [ + 174, + 232, + 826, + 290 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Indeed, Theorem 1 can be applied to specific cases. In particular, for the gradient descent fixed point, where $T ( x , \\theta ) = x - \\eta \\nabla _ { 1 } \\bar { f } ( x , \\theta )$ and $F ( x , { \\boldsymbol { \\theta } } ) = T ( x , { \\overline { { \\theta } } } ) - x$ , this yields ", + "bbox": [ + 173, + 295, + 825, + 325 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/575276dad4e1574671a11f4d8d13f4404cf87388aceaa094861c3d08f947de79.jpg", + "text": "$$\nA ( x , \\theta ) = \\eta \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) \\mathrm { a n d } B ( x , \\theta ) = - \\eta \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 315, + 330, + 681, + 348 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The guarantees on Jacobian precision under regularity conditions rely on $f$ directly; see Corollary 1 in Appendix D. This reveals in particular that Jacobian estimation by implicit differentiation gains a factor of t compared to automatic differentiation, after $t$ iterations of gradient descent in the strongly-convex setting (Ablin et al., 2020, Proposition 3.2). While our guarantees concern the Jacobian of $x ^ { \\star } ( \\theta )$ , we note that other studies (Grazzi et al., 2020; Ji et al., 2021; Bertrand et al., 2021) give guarantees on hypergradients (i.e., the gradient of an outer objective). ", + "bbox": [ + 173, + 353, + 826, + 438 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We illustrate these results on ridge regression, where $\\begin{array} { r } { x ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x } \\| \\Phi x - y \\| _ { 2 } ^ { 2 } + \\sum _ { i } \\theta _ { i } x _ { i } ^ { \\beth } } \\end{array}$ . This problem has the merit that the solution $x ^ { \\star } ( \\theta )$ and its Jacobian ${ \\partial x ^ { \\star } ( \\theta ) }$ are available in closed form. By running gradient descent for $t$ iterations, we obtain an estimate $\\hat { x }$ of $x ^ { \\star } ( \\theta )$ and an estimate $J ( \\hat { x } , \\theta )$ of ${ \\partial x ^ { \\star } ( \\theta ) }$ ; cf. Definition 1. By doing so for different numbers of iterations $t$ , we can graph the relation between the error $\\lVert x ^ { \\star } ( \\theta ) - { \\hat { x } } \\rVert _ { 2 }$ and the error $\\lVert \\partial x ^ { \\star } ( \\theta ) - J ( \\hat { x } , \\theta ) \\rVert _ { 2 }$ , as shown in Figure 2, empirically validating Theorem 1. The results in Figure 2 were obtained using the diabetes dataset from Efron et al. (2004), with other datasets yielding a qualitatively similar behavior. We derive similar guarantees in Corollary 2 in Appendix $\\mathrm { D }$ for proximal gradient descent. ", + "bbox": [ + 174, + 444, + 544, + 638 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/0b12fb95ca636955ebebf8146e28306042b855cbc3a9ba0861efbd81eb53cb3a.jpg", + "image_caption": [ + "Figure 2: Jacobian estimate errors. Empirical error of implicit differentiation follows closely the theoretical upper bound. Unrolling achieves a much worse error for comparable iterate error. " + ], + "image_footnote": [], + "bbox": [ + 562, + 445, + 816, + 571 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 652, + 326, + 667 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To conclude this work, we demonstrate the ease of solving bi-level optimization problems with our framework. We also present an application to the sensitivity analysis of molecular dynamics. ", + "bbox": [ + 176, + 683, + 825, + 712 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 HYPERPARAMETER OPTIMIZATION OF MULTICLASS SVMS ", + "text_level": 1, + "bbox": [ + 178, + 728, + 620, + 742 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this example, we consider the hyperparameter optimization of multiclass SVMs (Crammer & Singer, 2001) trained in the dual. Here, $x ^ { \\star } ( \\theta )$ is the optimal dual solution, a matrix of shape $m \\times k$ , where $m$ is the number of training examples and $k$ is the number of classes, and $\\theta \\in \\mathbb { R } _ { + }$ is the regularization parameter. The challenge in differentiating $x ^ { \\star } ( \\theta )$ is that each row of $x ^ { \\star } ( \\theta )$ is constrained to belong to the probability simplex $\\triangle ^ { k }$ . More formally, let $X _ { \\mathsf { t r } } \\in \\mathbb { R } ^ { m \\times p }$ be the training feature matrix and $\\breve { Y } _ { \\mathrm { t r } } \\in \\bar { \\{ 0 , 1 \\} } ^ { m \\times k }$ be the training labels (in row-wise one-hot encoding). Let $W ( x , \\theta ) : = X _ { \\mathrm { t r } } ^ { \\top } ( Y _ { \\mathrm { t r } } - x ) / \\overset { } \\theta \\in \\overset { } \\mathbb { R } ^ { p \\times k }$ be the dual-primal mapping. Then, we consider the following bi-level optimization problem ", + "bbox": [ + 173, + 753, + 826, + 866 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/5b1f1a6596fbd834f549c86507ca4ef55710196d821bce8d1bfe2b734a3fd641.jpg", + "text": "$$\n\\operatorname* { m i n } _ { = \\exp ( \\lambda ) } \\frac { 1 } { 2 } \\| X _ { \\mathrm { v a l } } W ( x ^ { \\star } ( \\theta ) , \\theta ) - Y _ { \\mathrm { v a l } } \\| _ { F } ^ { 2 } \\quad \\mathrm { s u b j e c t ~ t o } \\quad x ^ { \\star } ( \\theta ) = \\operatorname* { a r g m i n } _ { x \\in C } f ( x , \\theta ) : = \\frac { \\theta } { 2 } \\| W ( x , \\theta ) \\| _ { F } ^ { 2 } + \\langle x , Y _ { \\mathrm { t r } } \\rangle ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 871, + 831, + 901 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/fbcf6c03f02081b8dee912f3ff702f8674d162cde80a9f1faf84a0c101cfd9e1.jpg", + "image_caption": [ + "Figure 3: CPU runtime comparison of implicit differentiation and unrolling for hyperparameter optimization of multiclass SVMs for multiple problem sizes. Error bars represent $90 \\%$ confidence intervals. (a) Mirror descent solver, with mirror descent fixed point for implicit differentiation. $\\mathbf { ( b ) }$ Proximal gradient solver, with proximal gradient fixed point for implicit differentiation. (c) Block coordinate descent solver; for implicit differentiation we obtain $x ^ { \\star } ( \\theta )$ by BCD but perform differentiation with the mirror descent and proximal gradient fixed points. This showcases that the solver and fixed point can be independently chosen. " + ], + "image_footnote": [], + "bbox": [ + 174, + 101, + 821, + 237 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $\\mathcal { C } = \\triangle ^ { k } \\times \\cdot \\cdot \\cdot \\times \\triangle ^ { k }$ is the Cartesian product of $m$ probability simplices. We apply the change of variable $\\theta = \\exp ( \\lambda )$ in order to guarantee that the hyper-parameter $\\theta$ is positive. The matrix $W ( x ^ { \\star } ( \\theta ) , \\theta ) \\in \\mathbb { R } ^ { p \\times k }$ contains the optimal primal solution, the feature weights for each class. The outer loss is computed against validation data $X _ { \\mathtt { v a l } }$ and $Y _ { \\tt v a l }$ . ", + "bbox": [ + 173, + 372, + 825, + 430 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "While KKT conditions can be used to differentiate $x ^ { \\star } ( \\theta )$ , a more direct way is to use the projected gradient fixed point (9). The projection onto $\\mathcal { C }$ can be easily computed by row-wise projections on the simplex. The projection’s Jacobian enjoys a closed form (Appendix B). Another way to differentiate $x ^ { \\star } ( \\theta )$ is using the mirror descent fixed point (11). Under the KL geometry, $\\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( \\dot { y } , \\theta )$ corresponds to a row-wise softmax. It is therefore easy to compute and differentiate. Figure 3 compares the runtime performance of implicit differentiation vs. unrolling for the latter two fixed points. ", + "bbox": [ + 173, + 436, + 825, + 521 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 DATASET DISTILLATION ", + "text_level": 1, + "bbox": [ + 174, + 537, + 382, + 551 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Dataset distillation (Wang et al., 2018; Lorraine et al., 2020) aims to learn a small synthetic training dataset such that a model trained on this learned data set achieves a small loss on the original training set. Formally, let $X _ { \\mathsf { t r } } \\in \\mathbb { R } ^ { m \\times p }$ and $y _ { \\mathrm { t r } } \\in [ k ] ^ { m }$ denote the original training set. The distilled dataset will contain one prototype example for each class and therefore $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { k \\times \\bar { p } }$ . The dataset distillation problem can then naturally be cast as a bi-level problem, where in the inner problem we estimate a logistic regression model $\\bar { { \\boldsymbol { x } } } ^ { \\star } ( \\theta ) \\in \\mathbb { R } ^ { p \\times k }$ trained on the distilled images $\\theta \\in \\bar { \\mathbb { R } ^ { k \\times p } }$ , while in the outer problem we want to minimize the loss achieved by $x ^ { \\star } ( \\theta )$ over the training set: ", + "bbox": [ + 173, + 563, + 825, + 661 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/5aaec9dff222bd5c452113222d7ddf7693baffc237f0684d7b184018d1131846.jpg", + "text": "$$\n\\underbrace { \\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { k \\times p } } f ( x ^ { \\star } ( \\theta ) , X _ { \\mathrm { t r } } ; y _ { \\mathrm { t r } } ) } _ { \\mathrm { o u t e r ~ p r o b l e m } } \\mathrm { ~ s u b j e c t ~ t o ~ } x ^ { \\star } ( \\theta ) \\in \\underbrace { x \\mathrm { g m i n } \\ f ( x , \\theta ; [ k ] ) + \\varepsilon \\| x \\| ^ { 2 } } _ { \\mathrm { i n n e r ~ p r o b l e m } } ,\n$$", + "text_format": "latex", + "bbox": [ + 241, + 666, + 751, + 713 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $f ( W , X ; y ) : = \\ell ( y , X W )$ , $\\ell$ denotes the multiclass logistic regression loss, and $\\varepsilon = 1 0 ^ { - 3 }$ is a regularization parameter that we found had a very positive effect on convergence. ", + "bbox": [ + 176, + 722, + 821, + 751 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this problem, and unlike in the general hyperparameter optimization setup, both the inner and outer problems are high-dimensional, making it an ideal test-bed for gradient-based bi-level optimization methods. For this experiment, we use the MNIST dataset. The number of parameters in the inner problem is $p = 2 8 ^ { 2 } = 7 8 4$ while the number of parameters of the outer loss is $k \\times p = 7 8 4 0$ . We solve this problem using gradient descent on both the inner and outer problem, with the gradient of the outer loss computed using implicit differentiation, as described in $\\ S 2$ . This is fundamentally different from the approach used in the original paper, where they used differentiation of the unrolled iterates instead. For the same solver, we found that the implicit differentiation approach was 4 times faster than the original one. The obtained distilled images $\\theta$ are visualized in Figure 4. ", + "bbox": [ + 174, + 757, + 609, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/ec19d71aaf0cd167c63d17522b3599bea2432677009a02da79e1979a06bf5883.jpg", + "image_caption": [ + "Figure 4: Distilled dataset $\\theta \\in$ $\\mathbb { R } ^ { k \\times p }$ obtained by solving (12). " + ], + "image_footnote": [], + "bbox": [ + 624, + 757, + 820, + 906 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/b7a1feba8f86acef1ff2293d6f676f80da3490c3e9f965b29334afdf2a422549.jpg", + "table_caption": [ + "Table 2: Mean AUC (and $9 5 \\%$ confidence interval) for the cancer survival prediction problem. " + ], + "table_footnote": [], + "table_body": "
MethodL1 logregL2 logregDictL + L2 logregTask-driven DictL
AUC (%)71.6± 2.072.4 ± 2.868.3 ± 2.373.2 ± 2.1
", + "bbox": [ + 246, + 133, + 748, + 176 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 203, + 607, + 232 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 TASK-DRIVEN DICTIONARY LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 247, + 482, + 262 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Task-driven dictionary learning was proposed to learn sparse codes for input data in such a way that the codes solve an outer learning problem (Mairal et al., 2012; Sprechmann et al., 2014; Zarka et al., 2019). Formally, given a data matrix $X _ { \\mathsf { t r } } \\in \\mathbb { R } ^ { m \\times p }$ and a dictionary of $k$ atoms $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { k \\times p }$ , a sparse code is defined as a matrix $\\boldsymbol { x } ^ { \\star } ( \\boldsymbol { \\theta } ) \\in \\mathbb { R } ^ { m \\times k }$ that minimizes in $x$ a reconstruction loss $f ( x , \\theta ) : \\bar { = } \\ell ( X _ { \\mathrm { t r } } , x \\theta )$ regularized by a sparsity-inducing penalty $g ( x )$ . Instead of optimizing the dictionary $\\theta$ to minimize the reconstruction loss, Mairal et al. (2012) proposed to optimize an outer problem that depends on the code. Given a set of labels $Y _ { \\mathbf { t r } } \\in \\{ 0 , 1 \\} ^ { m }$ , we consider a logistic regression problem which results in the bilevel optimization problem: ", + "bbox": [ + 173, + 273, + 826, + 385 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/d9f9382c208a39ab5a6bb575cc80df48830967a3317245d468e891117cd8e512.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 207, + 387, + 754, + 431 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "When $\\ell$ is the squared Frobenius distance between matrices, and $g$ the elastic net penalty, Mairal et al. (2012, Eq. 21) derive manually, using optimality conditions (notably the support of the codes selected at the optimum), an explicit re-parameterization of $x ^ { \\star } ( \\theta )$ as a linear system involving $\\theta$ . This closed-form allows for a direct computation of the Jacobian of $x ^ { \\star }$ w.r.t. $\\theta$ . Similarly, (Sprechmann et al., 2014) derive first order conditions in the case where $\\ell$ is a $\\beta$ -divergence, while (Zarka et al., 2019) propose to use unrolling of ISTA iterations. Our approach bypasses all of these manual derivations, giving the user more leisure to focus directly on modeling (loss, regularizer) aspects. ", + "bbox": [ + 173, + 435, + 826, + 532 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We illustrate this on breast cancer survival prediction from gene expression data. We frame it as a binary classification problem to discriminate patients who survive longer than 5 years $m _ { 1 } = 2 0 0$ ) vs patients who die within 5 years of diagnosis $\\mathrm { \\Delta } m _ { 0 } = 9 9 \\mathrm { \\Delta }$ ), from $p = 1 , 0 0 0$ gene expression values. As shown in Table 2, solving (13) (Task-driven DictL) reaches a classification performance competitive with state-of-the-art $L _ { 1 }$ or $L _ { 2 }$ regularized logistic regression with 100 times fewer variables. ", + "bbox": [ + 173, + 537, + 825, + 609 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.4 SENSITIVITY ANALYSIS OF MOLECULAR DYNAMICS ", + "text_level": 1, + "bbox": [ + 173, + 626, + 573, + 638 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Many physical simulations require solving optimization problems, such as energy minimization in molecular (Schoenholz & Cubuk, 2020) and continuum (Beatson et al., 2020) mechanics, structural optimization (Hoyer et al., 2019) and data assimilation (Frerix et al., 2021). In this experiment, we revisit an example from JAXMD (Schoenholz & Cubuk, 2020), the problem of finding energy minimizing configurations to a system of $k$ packed particles in a 2-dimensional box of size $\\ell$ ", + "bbox": [ + 173, + 650, + 609, + 761 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/b22de2644b4d8d25a31214f62ebfef4a87b30413b843fbe278ea339f9585340a.jpg", + "text": "$$\nx ^ { \\star } ( \\theta ) = \\underset { x \\in \\mathbb { R } ^ { k \\times 2 } } { \\mathrm { a r g m i n } } f ( x , \\theta ) : = \\sum _ { i , j } U ( x _ { i , j } , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 248, + 761, + 531, + 795 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where $\\boldsymbol { x } ^ { \\star } ( \\boldsymbol { \\theta } ) \\in \\mathbb { R } ^ { k \\times 2 }$ are the optimal coordinates of the $k$ particles, $U ( x _ { i , j } , \\theta )$ is the pairwise potential energy function, with half the particles at diameter 1 and half at diameter $\\theta = 0 . 6$ , which we optimize with a domain-specific optimizer (Bitzek et al., 2006). Here we consider sensitivity of particle position with respect to diameter $\\partial x ^ { \\star } ( \\theta )$ , rather than sensitivity of the total energy from the original experiment. Figure 5 shows results calculated via forward-mode implicit differentiation (JVP). Whereas differentiating the unrolled optimizer happens to work for total energy, here it typically does not even converge (see Appendix Figure 17), due the discontinuous optimization method. ", + "bbox": [ + 174, + 797, + 609, + 868 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/fa7e597376db214ab7df98a98a47417b874c9d7a9d57737587d398873a0acc37.jpg", + "image_caption": [ + "Figure 5: Particle positions and position sensitivity vectors, with respect to increasing the diameter of the blue particles. " + ], + "image_footnote": [], + "bbox": [ + 624, + 640, + 820, + 791 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 869, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 287, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Pierre Ablin, Gabriel Peyré, and Thomas Moreau. Super-efficiency of automatic differentiation for functions defined as a minimum. In Proc. of ICML, pp. 32–41, 2020. ", + "bbox": [ + 173, + 126, + 825, + 155 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Akshay Agrawal, Brandon Amos, Shane Barratt, Stephen Boyd, Steven Diamond, and Zico Kolter. Differentiable convex optimization layers. arXiv preprint arXiv:1910.12430, 2019a. ", + "bbox": [ + 171, + 162, + 825, + 194 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Akshay Agrawal, Shane Barratt, Stephen Boyd, Enzo Busseti, and Walaa M Moursi. 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URL https://openreview.net/forum?id $=$ H1MczcgR-. ", + "bbox": [ + 174, + 821, + 825, + 878 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "John Zarka, Louis Thiry, Tomás Angles, and Stéphane Mallat. Deep network classification by scattering and homotopy dictionary learning. arXiv preprint arXiv:1910.03561, 2019. ", + "bbox": [ + 174, + 887, + 825, + 916 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Appendix ", + "text_level": 1, + "bbox": [ + 429, + 98, + 570, + 127 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A CODE EXAMPLES ", + "text_level": 1, + "bbox": [ + 176, + 143, + 354, + 160 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 CODE EXAMPLES FOR OPTIMALITY CONDITIONS ", + "text_level": 1, + "bbox": [ + 174, + 174, + 555, + 189 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Our library provides several reusable optimality condition mappings $F$ or fixed points $T$ . We nevertheless demonstrate the ease of writing some of them from scratch. ", + "bbox": [ + 174, + 200, + 823, + 229 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proximal gradient fixed point. The proximal gradient fixed point (7) with step size $\\eta = 1$ is $T ( x , \\theta ) = \\bar { \\mathrm { p r o x } } _ { g } ( x - \\nabla _ { 1 } f \\bar { ( } x , \\theta _ { f } ) , \\theta _ { g } )$ . It can be implemented as follows. ", + "bbox": [ + 174, + 243, + 823, + 273 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/8504037740291238ad275d7b34129068991e565646f5e8b5b135835ccf85acaf.jpg", + "image_caption": [ + "Figure 6: Proximal gradient fixed point $T ( x , \\theta )$ " + ], + "image_footnote": [], + "bbox": [ + 207, + 287, + 790, + 371 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We recall that when the proximity operator is a projection, we recover the projected gradient fixed point as a special case. Therefore, this fixed point can also be used for constrained optimization. We provide numerous proximal and projection operators in the library. ", + "bbox": [ + 174, + 420, + 825, + 463 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "KKT conditions. As a more advanced example, we now describe how to implement the KKT conditions (6). The stationarity, primal feasibility and complementary slackness conditions read ", + "bbox": [ + 173, + 477, + 823, + 506 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/72abfbbf405af6a079f3008bbe7591c2f2d200e4f287cea7a05ce5068997ffe4.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta _ { f } ) + [ \\partial _ { 1 } G ( z , \\theta _ { G } ) ] ^ { \\top } \\lambda + [ \\partial _ { 1 } H ( z , \\theta _ { H } ) ] ^ { \\top } \\nu = 0 } \\\\ { H ( z , \\theta _ { H } ) = 0 } \\\\ { \\lambda \\circ G ( z , \\theta _ { G } ) = 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 316, + 508, + 681, + 566 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Using ${ \\dot { \\mathsf { J } } } \\mathsf { a x } . \\mathsf { v } { \\dot { \\mathsf { J } } } { \\mathsf { p } }$ to compute vector-Jacobian products, this can be implemented as ", + "bbox": [ + 174, + 569, + 717, + 584 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/4c5afa2d3ce2393d2ed5ba74f895571c86cc2a907b4316ffe454af22a9a5c182.jpg", + "image_caption": [ + "Figure 7: KKT conditions $F ( x , \\theta )$ " + ], + "image_footnote": [], + "bbox": [ + 209, + 597, + 790, + 844 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Similar mappings $F$ can be written if the optimization problem contains only equality constraints or only inequality constraints. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Mirror descent fixed point. Letting $\\eta = 1$ and denoting $\\theta = ( \\theta _ { f } , \\theta _ { \\mathrm { p r o j } } )$ , the fixed point (11) is ", + "bbox": [ + 169, + 103, + 803, + 119 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/18b2f515e83e9bbf3c35308e37e8e93484833446a3551300e1e69eb22af64fc8.jpg", + "text": "$$\n\\begin{array} { r } { \\hat { x } = \\nabla \\varphi ( x ) \\qquad } \\\\ { y = \\hat { x } - \\nabla _ { 1 } f ( x , \\theta _ { f } ) } \\\\ { T ( x , \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y , \\theta _ { \\mathrm { p r o j } } ) . \\qquad } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 406, + 125, + 589, + 183 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We can then implement it as follows. ", + "bbox": [ + 173, + 189, + 416, + 203 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/f3cf628dfe80ab0845f2de767e7bc5f5ef762539deeda01084e875a1841612ab.jpg", + "image_caption": [ + "Figure 8: Mirror descent fixed point $T ( x , \\theta )$ " + ], + "image_footnote": [], + "bbox": [ + 207, + 218, + 790, + 332 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Although not considered in this example, the mapping $\\boldsymbol { \\nabla } \\varphi$ could also depend on $\\theta$ if necessary. ", + "bbox": [ + 171, + 383, + 795, + 400 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 CODE EXAMPLES FOR EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 417, + 472, + 433 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We now sketch how to implement our experiments using our framework. In the following, jnp is short for jax.numpy. In all experiments, we only show how to compute gradients with the outer objective. We can then use these gradients with gradient-based solvers to solve the outer objective. ", + "bbox": [ + 174, + 444, + 826, + 487 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Multiclass SVM experiment. ", + "text_level": 1, + "bbox": [ + 174, + 503, + 374, + 518 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/cb93a45ac538a2eab5c343e4bdbe73d224d387f48173e100e6b47378bc0280ae.jpg", + "image_caption": [ + "Figure 9: Code example for the multiclass SVM experiment. " + ], + "image_footnote": [], + "bbox": [ + 235, + 535, + 761, + 895 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Task-driven dictionary learning experiment. ", + "bbox": [ + 174, + 103, + 483, + 118 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/1754717219ba10a7a67584e04ef622ac4a9b7d097d3f919bda7074fe500f3cfb.jpg", + "image_caption": [ + "Figure 10: Code example for the task-driven dictionary learning experiment. " + ], + "image_footnote": [], + "bbox": [ + 209, + 133, + 789, + 457 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Dataset distillation experiment. ", + "text_level": 1, + "bbox": [ + 173, + 510, + 393, + 525 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/53013e99aec5bda0635d97693c9df2f0e9dfe646d9c366131a781676d67fc39a.jpg", + "image_caption": [ + "Figure 11: Code example for the dataset distillation experiment. " + ], + "image_footnote": [], + "bbox": [ + 209, + 541, + 789, + 893 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Molecular dynamics experiment. ", + "text_level": 1, + "bbox": [ + 174, + 103, + 403, + 118 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/d4d8f94f679454fe2e353a1d2e8064efa68ec64d956f4fadbd26fa10bf22a967.jpg", + "image_caption": [ + "Figure 12: Code for the molecular dynamics experiment. " + ], + "image_footnote": [], + "bbox": [ + 207, + 133, + 789, + 458 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B JACOBIAN PRODUCTS ", + "text_level": 1, + "bbox": [ + 174, + 513, + 388, + 530 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Our library provides numerous reusable building blocks. We describe in this section how to compute their Jacobian products. As a general guideline, whenever a projection enjoys a closed form, we leave the Jacobian product to the autodiff system. ", + "bbox": [ + 174, + 545, + 825, + 588 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.1 JACOBIAN PRODUCTS OF PROJECTIONS ", + "text_level": 1, + "bbox": [ + 176, + 607, + 486, + 621 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We describe in this section how to compute the Jacobian products of the projections (in the Euclidean and KL senses) onto various convex sets. When the convex set does not depend on any variable, we simply denote it $\\mathcal { C }$ instead of $\\mathcal { C } ( \\boldsymbol { \\theta } )$ . ", + "bbox": [ + 174, + 633, + 825, + 676 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Non-negative orthant. When $\\mathcal { C }$ is the non-negative orthant, $\\mathcal { C } = \\mathbb { R } _ { + } ^ { d }$ , we obtain $\\mathrm { p r o j } _ { \\mathcal { C } } ( y ) =$ $\\operatorname* { m a x } ( y , 0 )$ , where the maximum is evaluated element-wise. This is also known as the ReLu function. The projection in the KL sense reduces to the exponential function, $\\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y ) = \\exp ( y )$ . ", + "bbox": [ + 174, + 693, + 825, + 736 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Box constraints. When $\\mathcal { C } ( \\boldsymbol { \\theta } )$ is the box constraints $\\mathcal { C } ( \\theta ) = [ \\theta _ { 1 } , \\theta _ { 2 } ] ^ { d }$ with $\\theta \\in \\mathbb { R } ^ { 2 }$ , we obtain ", + "bbox": [ + 173, + 753, + 789, + 770 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5023ae9f47f81eca27c5171abe04c3c37deb6538bfbff98d071383acd0349193.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta ) = \\mathrm { c l i p } ( y , \\theta _ { 1 } , \\theta _ { 2 } ) : = \\operatorname* { m a x } ( \\operatorname* { m i n } ( y , \\theta _ { 2 } ) , \\theta _ { 1 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 777, + 673, + 795 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "This is trivially extended to support different boxes for each coordinate, in which case $\\theta \\in \\mathbb { R } ^ { d \\times 2 }$ ", + "bbox": [ + 173, + 804, + 805, + 820 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Probability simplex. When $\\mathcal { C }$ is the standard probability simplex, ${ \\mathcal { C } } = \\triangle ^ { d }$ , there is no analytical solution for $\\operatorname { p r o j } _ { \\mathcal { C } } ( y )$ . Nevertheless, the projection can be computed exactly in $O ( d )$ expected time or $O ( d \\log d )$ worst-case time (Brucker, 1984; Michelot, 1986; Duchi et al., 2008; Condat, 2016). The Jacobian is given by $\\mathrm { d i a g } ( s ) - s s ^ { \\top } / \\| s \\| _ { 1 }$ , where $s \\in \\{ 0 , 1 \\} ^ { d }$ is a vector indicating the support of $\\operatorname { p r o j } _ { \\mathcal { C } } ( y )$ (Martins $\\&$ Astudillo, 2016). The projection in the KL sense, on the other hand, enjoys a closed form: it reduces to the usual softmax $\\begin{array} { r } { \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y ) = \\exp ( y ) / \\sum _ { j = 1 } ^ { d } \\exp ( y _ { j } ) , } \\end{array}$ . ", + "bbox": [ + 173, + 835, + 826, + 926 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Box sections. Consider now the Euclidean projection $z ^ { \\star } ( \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta )$ onto the set ${ \\mathcal { C } } ( \\theta ) =$ $\\{ z \\in \\mathbb { R } ^ { d } \\colon \\alpha _ { i } \\leq z _ { i } \\leq \\beta _ { i } , i \\in [ d ] ; w ^ { \\top } z = c \\}$ , where $\\theta = ( \\alpha , \\beta , w , c )$ . This projection is a singlyconstrained bounded quadratic program. It is easy to check (see, e.g., (Niculae & Martins, 2020)) that an optimal solution satisfies for all $i \\in [ d ]$ ", + "bbox": [ + 173, + 102, + 826, + 162 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/f785b0379b0466b943cf009cc33655fc6617a40801a0665a3c04308d86c93a17.jpg", + "text": "$$\nz _ { i } ^ { \\star } ( \\theta ) = [ L ( x ^ { \\star } ( \\theta ) , \\theta ) ] _ { i } : = \\mathrm { c l i p } ( w _ { i } x ^ { \\star } ( \\theta ) + y _ { i } , \\alpha _ { i } , \\beta _ { i } )\n$$", + "text_format": "latex", + "bbox": [ + 325, + 166, + 673, + 184 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $L \\colon \\mathbb { R } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { d }$ is the dual-primal mapping and $x ^ { \\star } ( \\theta ) \\in \\mathbb { R }$ is the optimal dual variable of the linear constraint, which should be the root of ", + "bbox": [ + 176, + 189, + 823, + 217 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/456d01b81789a3d57b2cac845912db216d847a471e5e54dd18a0eaab79ee20f5.jpg", + "text": "$$\nF ( x ^ { \\star } ( \\theta ) , \\theta ) = L ( x ^ { \\star } ( \\theta ) , \\theta ) ^ { \\top } w - c .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 220, + 616, + 239 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The root can be found, e.g., by bisection. The gradient $\\nabla \\boldsymbol { x } ^ { \\star } ( \\boldsymbol { \\theta } )$ is given by $\\nabla x ^ { \\star } ( \\theta ) = B ^ { \\top } / A$ and the Jacobian $\\partial z ^ { \\star } ( \\theta )$ is obtained by application of the chain rule on $L$ . ", + "bbox": [ + 174, + 244, + 823, + 273 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Norm balls. When $\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon \\| x \\| \\leq \\theta \\}$ , where $\\| \\cdot \\|$ is a norm and $\\theta \\in \\mathbb { R } _ { + }$ , $\\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta )$ becomes the projection onto a norm ball. The projection onto the $\\ell _ { 1 }$ -ball reduces to a projection onto the simplex, see, e.g., (Duchi et al., 2008). The projections onto the $\\ell _ { 2 }$ and $\\ell _ { \\infty }$ balls enjoy a closed-form, see, e.g., (Parikh & Boyd, 2014, $\\ S 6 . 5 )$ . Since they rely on simple composition of functions, all three projections can therefore be automatically differentiated. ", + "bbox": [ + 173, + 286, + 825, + 359 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Affine sets. When ${ \\mathcal { C } } ( \\theta ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon A x = b \\}$ , where $A \\in \\mathbb { R } ^ { p \\times d }$ , $b \\in \\mathbb { R } ^ { p }$ and $\\theta = ( A , b )$ , we get ", + "bbox": [ + 171, + 372, + 818, + 388 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/c04a70cad22aa1a789af6aab9449d621c81b2d4668f11a17fd1147384ac7f868.jpg", + "text": "$$\n\\operatorname { p r o j } _ { \\mathcal { C } } ( y , \\theta ) = y - A ^ { \\dagger } ( A y - b ) = y - A ^ { \\top } ( A A ^ { \\top } ) ^ { - 1 } ( A y - b )\n$$", + "text_format": "latex", + "bbox": [ + 299, + 393, + 699, + 412 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $A ^ { \\dagger }$ is the Moore-Penrose pseudoinverse of $A$ . The second equality holds if $p < d$ and $A$ is full rank. A practical implementation can pre-compute a factorization of the Gram matrix $A A ^ { \\top }$ . Alternatively, we can also use the KKT conditions. ", + "bbox": [ + 174, + 419, + 826, + 462 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Hyperplanes and half spaces. When $\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon a ^ { \\top } x = b \\}$ , where $a \\in \\mathbb { R } ^ { d }$ and $b \\in \\mathbb { R }$ and $\\theta = ( a , b )$ , we get ", + "bbox": [ + 173, + 474, + 825, + 505 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/117a5d69337ae35d7b97e3b7471c1416afc6aeace0cd0ef458716fda906109a0.jpg", + "text": "$$\n\\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta ) = y - \\frac { { a ^ { \\top } y - b } } { \\Vert a \\Vert _ { 2 } ^ { 2 } } a .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 502, + 598, + 539 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "When $\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon a ^ { \\top } x \\leq b \\}$ , we simply replace $a ^ { \\top } y - b$ in the numerator by $\\operatorname* { m a x } ( a ^ { \\top } y - b , 0 )$ ", + "bbox": [ + 174, + 540, + 825, + 558 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Transportation and Birkhoff polytopes. When $\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ X \\in \\mathbb { R } ^ { p \\times d } \\colon X \\mathbf { 1 } _ { d } = \\boldsymbol { \\theta } _ { 1 } , X ^ { \\top } \\mathbf { 1 } _ { p } =$ $\\theta _ { 2 } , X \\geq 0 \\}$ , the so-called transportation polytope, where $\\theta _ { 1 } \\in \\triangle ^ { p }$ and $\\theta _ { 2 } \\in \\triangle ^ { d }$ are marginals, we can compute approximately the projections, both in the Euclidean and KL senses, by switching to the dual or semi-dual (Blondel et al., 2018). Since both are unconstrained optimization problems, we can compute their Jacobian product by implicit differentiation using the gradient descent fixed point. An advantage of the KL geometry here is that we can use Sinkhorn (Cuturi, 2013), which is a GPU-friendly algorithm. The Birkhoff polytope, the set of doubly stochastic matrices, is obtained by fixing ${ \\theta } _ { 1 } = \\bar { \\theta } _ { 2 } \\stackrel { - } { = } { \\mathbf { 1 } } _ { d } / d$ . ", + "bbox": [ + 173, + 570, + 826, + 685 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Order simplex. When ${ \\mathcal { C } } ( \\theta ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon \\theta _ { 1 } \\geq x _ { 1 } \\geq x _ { 2 } \\geq \\cdots \\geq x _ { d } \\geq \\theta _ { 2 } \\}$ , a so-called order simplex (Grotzinger & Witzgall, 1984; Blondel, 2019), the projection operations, both in the Euclidean and KL sense, reduce to isotonic optimization (Lim & Wright, 2016) and can be solved exactly in $O ( d \\log d )$ time using the Pool Adjacent Violators algorithm (Best et al., 2000). The Jacobian of the projections and efficient product with it are derived in (Djolonga & Krause, 2017; Blondel et al., 2020). ", + "bbox": [ + 173, + 698, + 825, + 784 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Polyhedra. More generally, we can consider polyhedra, i.e., sets of the form ${ \\mathcal { C } } ( \\theta ) ~ = ~ \\{ x ~ \\in ~$ $\\mathbb { R } ^ { d } \\colon A x = b , C x \\leq \\mathbf { \\bar { d } } \\}$ , where $A \\in \\mathbb { R } ^ { p \\times d }$ , $b \\in \\bar { \\mathbb { R } } ^ { p }$ , $C \\in \\mathbb { R } ^ { m \\times d }$ , and $d \\in \\mathbb { R } ^ { m }$ . There are several ways to differentiate this projection. The first is to use the KKT conditions as detailed in $\\ S 2 . 2$ . A second way is consider the dual of the projection instead, which is the maximization of a quadratic function subject to non-negative constraints (Parikh & Boyd, 2014, $\\ S 6 . 2 )$ . That is, we can reduce the projection on a polyhedron to a problem of the form (8) with non-negative constraints, which we can in turn implicitly differentiate easily using the projected gradient fixed point, combined with the projection on the non-negative orthant. Finally, we apply the dual-primal mapping , which enjoys a closed form and is therefore amenable to autodiff, to obtain the primal projection. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2 JACOBIAN PRODUCTS OF PROXIMITY OPERATORS", + "text_level": 1, + "bbox": [ + 173, + 103, + 558, + 117 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We provide several proximity operators, including for the lasso (soft thresholding), elastic net and group lasso (block soft thresholding). All satisfy closed form expressions and can be differentiated automatically via autodiff. For more advanced proximity operators, which do not enjoy a closed form, recent works have derived their Jacobians. The Jacobians of fused lasso and OSCAR were derived in (Niculae & Blondel, 2017). For general total variation, the Jacobians were derived in (Vaiter et al., 2013; Cherkaoui et al., 2020). ", + "bbox": [ + 173, + 128, + 826, + 214 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C MORE EXAMPLES OF OPTIMALITY CRITERIA AND FIXED POINTS", + "text_level": 1, + "bbox": [ + 174, + 234, + 738, + 251 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To demonstrate the generality of our approach, we describe in this section more optimality mapping $F$ or fixed point iteration $T$ . ", + "bbox": [ + 173, + 265, + 825, + 295 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Newton fixed point. Let $x$ be a root of $G ( \\cdot , \\theta )$ , i.e., $G ( x , \\theta ) = 0$ . The fixed point iteration of Newton’s method for root-finding is ", + "bbox": [ + 174, + 309, + 823, + 339 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/4438f43ac03ecd4a13b305550b2d53894c16f57a38c6d94bec19af984e5cb5b3.jpg", + "text": "$$\nT ( x , \\theta ) = x - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } G ( x , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 345, + 629, + 364 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "By the chain and product rules, we have ", + "bbox": [ + 174, + 371, + 437, + 386 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/357418c817bcdf725ae771c319609759511fd44e9c79cb633cd32f955df2cca6.jpg", + "text": "$$\n\\partial _ { 1 } T ( x , \\theta ) = I - \\eta ( . . . ) G ( x , \\theta ) - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } \\partial _ { 1 } G ( x , \\theta ) = ( 1 - \\eta ) I .\n$$", + "text_format": "latex", + "bbox": [ + 261, + 392, + 736, + 410 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Using (3), we get $A = - \\partial _ { 1 } F ( x , \\theta ) = \\eta I$ . Similarly, ", + "bbox": [ + 173, + 416, + 517, + 433 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/f704a3247fc0e7fec8441bbd283a79a54d1c785071cf979cf4c52597ed198315.jpg", + "text": "$$\nB = \\partial _ { 2 } T ( x , \\theta ) = \\partial _ { 2 } F ( x , \\theta ) = - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } \\partial _ { 2 } G ( x , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 300, + 439, + 697, + 458 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Newton’s method for optimization is obtained by choosing $G ( x , \\theta ) = \\nabla _ { 1 } f ( x , \\theta )$ , which gives ", + "bbox": [ + 169, + 464, + 787, + 481 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/e7763fd63af2daa4b5024ad287cbdeaa5a94a907a09ffe43677f3a312d16f24e.jpg", + "text": "$$\nT ( x , \\theta ) = x - \\eta [ \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) ] ^ { - 1 } \\nabla _ { 1 } f ( x , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 487, + 638, + 506 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "It is easy to check that we recover the same linear system as for the gradient descent fixed point (5). A practical implementation can pre-compute an LU decomposition of $\\partial _ { 1 } G ( x , \\theta )$ , or a Cholesky decomposition if $\\partial _ { 1 } G ( x , \\theta )$ is positive semi-definite. ", + "bbox": [ + 173, + 513, + 825, + 556 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proximal block coordinate descent fixed point. We now consider the case when $x ^ { \\star } ( \\theta )$ is implicitly defined as the solution ", + "bbox": [ + 174, + 570, + 820, + 598 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/e0304c49bb0bd86b1c474ae72fe12a17971e2de1cc9f3b422a5e2645f37b585d.jpg", + "text": "$$\nx ^ { \\star } ( \\theta ) : = \\operatorname * { a r g m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x , \\theta ) + \\sum _ { i = 1 } ^ { m } g _ { i } ( x _ { i } , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 362, + 594, + 633, + 637 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $g _ { 1 } , \\ldots , g _ { m }$ are possibly non-smooth functions operating on subvectors (blocks) $x _ { 1 } , \\ldots , x _ { m }$ of $x$ . In this case, we can use for $i \\in [ m ]$ the fixed point ", + "bbox": [ + 173, + 640, + 823, + 669 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/864791e9ff2ab5e109fafb0c9bdd5b925e3e34f2d216a79b3b14e4b7259c1249.jpg", + "text": "$$\n\\begin{array} { r } { x _ { i } = [ T ( \\boldsymbol { x } , \\theta ) ] _ { i } = \\mathrm { p r o x } _ { \\eta _ { i } g _ { i } } ( x _ { i } - \\eta _ { i } [ \\nabla _ { 1 } f ( \\boldsymbol { x } , \\theta ) ] _ { i } , \\theta ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 675, + 669, + 694 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $\\eta _ { 1 } , \\dots , \\eta _ { m }$ are block-wise step sizes. Clearly, when the step sizes are shared, i.e., $\\eta _ { 1 } =$ $\\cdots = \\eta _ { m } = \\eta$ this fixed point is equivalent to the proximal gradient fixed point (7) with . $g ( x , \\theta ) =$ $\\textstyle \\sum _ { i = 1 } ^ { n } g _ { i } ( x _ { i } , \\theta )$ ", + "bbox": [ + 173, + 699, + 826, + 743 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Quadratic programming. We now show how to use the KKT conditions discussed in $\\ S 2 . 2$ to differentiate quadratic programs, recovering Optnet (Amos & Kolter, 2017) as a special case. To give some intuition, let us start with a simple equality-constrained quadratic program (QP) ", + "bbox": [ + 173, + 757, + 825, + 801 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/4df53a57f56b2ff7f07ea428fbf7530d7e372e67dd72edacf790c0890ffb5d09.jpg", + "text": "$$\n\\operatorname * { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } } f ( z , \\theta ) = \\frac { 1 } { 2 } z ^ { \\top } Q z + c ^ { \\top } z \\quad \\mathrm { s u b j e c t t o } \\quad H ( z , \\theta ) = E z - d = 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 259, + 806, + 736, + 840 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $Q \\in \\mathbb { R } ^ { p \\times p }$ , $\\ b { E } \\in \\mathbb { R } ^ { q \\times p }$ , $d \\in \\mathbb { R } ^ { q }$ . We gather the differentiable parameters as $\\theta = ( Q , E , c , d )$ . The stationarity and primal feasibility conditions give ", + "bbox": [ + 173, + 848, + 826, + 877 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/10a7a8c2bfa8548f59ed7dbf2fb7e892b123dcf70faeda1083e6446022af3e3a.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta ) + [ \\partial _ { 1 } H ( z , \\theta ) ] ^ { \\top } \\boldsymbol { \\nu } = Q z + c + E ^ { \\top } \\boldsymbol { \\nu } = 0 } \\\\ { H ( z , \\theta ) = E z - d = 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 883, + 673, + 922 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In matrix notation, this can be rewritten as ", + "bbox": [ + 174, + 103, + 452, + 118 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ca8b212b581e40473dd4dc5169335875be5456fbcda798bdd8c189bfbfb8985c.jpg", + "text": "$$\n\\left[ { Q \\quad E ^ { \\top } } \\right] \\left[ { z } \\right] = \\left[ { - c } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 122, + 583, + 157 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We can write the solution of the linear system (16) as the root $\\boldsymbol { x } = \\left( z , \\nu \\right)$ of a function $F ( x , \\theta )$ . More generally, the QP can also include inequality constraints ", + "bbox": [ + 174, + 161, + 823, + 190 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/c35a0e0d7ffcbd3d0a0fb863709e3ede718b8f9c734aa13959026b5398a7264f.jpg", + "text": "$$\n\\operatorname * { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } } f ( z , \\theta ) = \\frac { 1 } { 2 } z ^ { \\top } Q z + c ^ { \\top } z \\quad \\mathrm { s u b j e c t ~ t o } \\quad H ( z , \\theta ) = E z - d = 0 , G ( z , \\theta ) = M z - h \\leq 0 .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 195, + 813, + 226 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $M \\in \\mathbb { R } ^ { r \\times p }$ and $h \\in \\mathbb { R } ^ { r }$ . We gather the differentiable parameters as $\\theta = ( Q , E , M , c , d , h )$ . The stationarity, primal feasibility and complementary slackness conditions give ", + "bbox": [ + 173, + 234, + 825, + 263 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/fbb65e91f54b8309ea2c307445fc7907e0a5a3e89babace5166113674103c56f.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta ) + [ \\partial _ { 1 } H ( z , \\theta ) ] ^ { \\top } \\boldsymbol { \\nu } + [ \\partial _ { 1 } G ( z , \\theta ) ] ^ { \\top } \\boldsymbol { \\lambda } = Q z + c + E ^ { \\top } \\boldsymbol { \\nu } + M ^ { \\top } \\boldsymbol { \\lambda } = 0 } \\\\ { H ( z , \\theta ) = E z - d = 0 } \\\\ { \\boldsymbol { \\lambda } \\circ G ( z , \\theta ) = \\mathrm { d i a g } ( \\boldsymbol { \\lambda } ) ( M z - h ) = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 241, + 266, + 758, + 324 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In matrix notation, this can be written as ", + "bbox": [ + 174, + 327, + 439, + 342 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/d1d2a4995d1154f6f0c22d70597483f1fee03f4565de7025a05cd5d71b00b9ab.jpg", + "text": "$$\n{ \\left[ \\begin{array} { l l l } { \\quad Q } & { E ^ { \\top } } & { M ^ { \\top } } \\\\ { E } & { 0 } & { 0 } \\\\ { \\operatorname { d i a g } ( \\lambda ) M } & { 0 } & { \\quad 0 } \\end{array} \\right] } \\left[ { \\begin{array} { l } { z } \\\\ { \\nu } \\\\ { \\lambda } \\end{array} } \\right] = { \\left[ \\begin{array} { l } { - c } \\\\ { \\ d } \\\\ { \\lambda \\circ h } \\end{array} \\right] }\n$$", + "text_format": "latex", + "bbox": [ + 356, + 345, + 643, + 396 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "While $\\boldsymbol { x } = ( z , \\nu , \\lambda )$ is no longer the solution of a linear system, it is the root of a function $F ( x , \\theta )$ and therefore fits our framework. With our framework, no derivation is needed. We simply define $f$ , $H$ and $G$ directly in Python. ", + "bbox": [ + 173, + 400, + 826, + 443 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Conic programming. We now show that the differentiation of conic linear programs (Agrawal et al., 2019b; Amos, 2019), at the heart of differentiating through cvxpy layers (Agrawal et al., 2019a), easily fits our framework. Consider the problem ", + "bbox": [ + 173, + 457, + 826, + 500 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/82bea159c2902c9b51e155aef2c56c618e8802e5c479299db514216d84430c7d.jpg", + "text": "$$\nz ^ { \\star } ( \\lambda ) , s ^ { \\star } ( \\lambda ) = \\operatorname * { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } , s \\in \\mathbb { R } ^ { m } } c ^ { \\top } z \\quad \\mathrm { s u b j e c t } \\mathrm { t o } \\quad E z + s = d , s \\in \\mathcal { K } ,\n$$", + "text_format": "latex", + "bbox": [ + 281, + 505, + 714, + 532 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\lambda = ( c , E , d )$ , $\\ b { E } \\in \\mathbb { R } ^ { m \\times p }$ , $d \\in \\mathbb { R } ^ { m }$ , $c \\in \\mathbb { R } ^ { p }$ and ${ \\mathcal { K } } \\subseteq \\mathbb { R } ^ { m }$ is a cone; $z$ and $s$ are the primal and slack variables, respectively. Every convex optimization problem can be reduced to the form (17). Let us form the skew-symmetric matrix ", + "bbox": [ + 174, + 539, + 825, + 582 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/e6f11544cc1ef48ceee7670cade7f5e32f50afd42af4e8f955f08d94e946be27.jpg", + "text": "$$\n\\theta ( \\lambda ) = \\left[ \\begin{array} { c c c } { 0 } & { E ^ { \\top } } & { c } \\\\ { - E } & { 0 } & { d } \\\\ { - c ^ { \\top } } & { - d ^ { \\top } } & { 0 } \\end{array} \\right] \\in \\mathbb { R } ^ { N \\times N } ,\n$$", + "text_format": "latex", + "bbox": [ + 372, + 585, + 624, + 637 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $N = p { + } m { + } 1$ . Following (Agrawal et al., 2019b;a; Amos, 2019), we can use the homogeneous self-dual embedding to reduce the process of solving (17) to finding a root of the residual map ", + "bbox": [ + 169, + 641, + 825, + 670 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/e7370eb1ef21e658f878423743d609b8492228a2bdefd5ea26267aa20b282eac.jpg", + "text": "$$\nF ( x , \\theta ) = \\theta \\Pi x + \\Pi ^ { * } x = ( ( \\theta - I ) \\Pi + I ) x ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 674, + 643, + 693 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\Pi = \\mathrm { p r o j } _ { \\mathbb { R } ^ { p } \\times K ^ { * } \\times \\mathbb { R } _ { + } }$ and ${ K } ^ { * } \\subseteq \\mathbb { R } ^ { m }$ is the dual cone. The splitting conic solver (O’Donoghue et al., 2016), which is based on ADMM, outputs a solution $F ( x ^ { \\star } ( \\theta ) , \\theta ) = 0$ which is decomposed as $x ^ { \\star } ( \\theta ) = ( u ^ { \\star } ( \\theta ) , v ^ { \\star } ( \\theta ) , w ^ { \\star } ( \\theta ) )$ . We can then recover the optimal solution of (17) using ", + "bbox": [ + 173, + 696, + 826, + 742 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/a39afbb580d55a04c8d6fed81e605b9a50a902c08cb7b3eeb30b80e966811c90.jpg", + "text": "$$\nz ^ { \\star } ( \\lambda ) = u ^ { \\star } ( \\theta ( \\lambda ) ) \\quad \\mathrm { a n d } \\quad s ^ { \\star } ( \\lambda ) = \\mathrm { p r o j } _ { \\mathcal { K } ^ { * } } ( v ^ { \\star } ( \\theta ( \\lambda ) ) ) - v ^ { \\star } ( \\theta ( \\lambda ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 746, + 717, + 763 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The key oracle whose JVP/VJP we need is therefore $\\Pi$ , which is studied in (Ali et al., 2017). The projection onto a few cones is available in our library and can be used to express $F$ . ", + "bbox": [ + 174, + 768, + 823, + 797 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Frank-Wolfe. We now consider ", + "bbox": [ + 173, + 811, + 398, + 827 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/de83a2d74258321559a6fe2ada43cbc330f287c42f57a80b8100146a0d523fb5.jpg", + "text": "$$\nx ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x \\in \\mathcal { C } ( \\theta ) \\subset \\mathbb { R } ^ { d } } f ( x , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 406, + 830, + 589, + 861 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\mathcal { C } ( \\boldsymbol { \\theta } )$ is a convex polytope, i.e., it is the convex hull of vertices $v _ { 1 } ( \\theta ) , \\dots , v _ { m } ( \\theta )$ . The FrankWolfe algorithm requires a linear minimization oracle (LMO) ", + "bbox": [ + 171, + 866, + 825, + 895 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/b0ecff09defea20f90f6d8bffffd7944f812d68a890aa7a03f671cc0e5efdcb9.jpg", + "text": "$$\ns \\mapsto \\operatorname * { a r g m i n } _ { x \\in { \\mathcal { C } } ( \\theta ) } ~ \\langle s , x \\rangle\n$$", + "text_format": "latex", + "bbox": [ + 433, + 900, + 565, + 928 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "and is a popular algorithm when this LMO is easier to compute than the projection onto $\\mathcal { C } ( \\boldsymbol { \\theta } )$ . However, since this LMO is piecewise constant, its Jacobian is null almost everywhere. Inspired by SparseMAP (Niculae et al., 2018), which corresponds to the case when $f$ is a quadratic, we rewrite (19) as ", + "bbox": [ + 173, + 102, + 825, + 159 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/2806c56d422ef608e306f21ee869a7464829a762b80d5487090e07b932111bb9.jpg", + "text": "$$\np ^ { \\star } ( \\theta ) = \\underset { p \\in \\triangle ^ { m } } { \\operatorname { a r g m i n } } g ( p , \\theta ) : = f ( V ( \\theta ) p , \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 157, + 632, + 184 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $V ( \\theta )$ is a $d \\times m$ matrix gathering the vertices $v _ { 1 } ( \\theta ) , \\dots , v _ { m } ( \\theta )$ . We then have $x ^ { \\star } ( \\theta ) =$ $V ( \\theta ) p ^ { \\star } ( \\theta )$ . Since we have reduced (19) to minimization over the simplex, we can use the projected gradient fixed point to obtain ", + "bbox": [ + 173, + 186, + 825, + 229 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/26335ef9f5b29258f3409890b9591693786eddfffde92b77eeb7b1ce764136f2.jpg", + "text": "$$\nT ( p ^ { \\star } ( \\boldsymbol { \\theta } ) , \\boldsymbol { \\theta } ) \\stackrel { } { = } \\mathrm { p r o j } _ { \\triangle ^ { m } } \\big ( p ^ { \\star } ( \\boldsymbol { \\theta } ) - \\nabla _ { 1 } g ( p ^ { \\ * } ( \\boldsymbol { \\theta } ) , \\boldsymbol { \\theta } ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 234, + 656, + 252 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We can therefore compute the derivatives of $p ^ { \\star } ( \\theta )$ by implicit differentiation and the derivatives of $x ^ { \\star } ( \\theta )$ by product rule. Frank-Wolfe implementations typically maintain the convex weights of the vertices, which we use to get an approximation of $p ^ { \\star } ( \\theta )$ . Moreover, it is well-known that after $t$ iterations, at most $t$ vertices are visited. We can leverage this sparsity to solve a smaller linear system. Moreover, in practice, we only need to compute VJPs of $x ^ { \\star } ( \\theta )$ . ", + "bbox": [ + 173, + 257, + 826, + 329 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "D JACOBIAN PRECISION PROOFS ", + "text_level": 1, + "bbox": [ + 174, + 348, + 459, + 364 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof of Theorem $^ { l }$ . To simplify notations, we note $A _ { \\star } : = A ( x ^ { \\star } , \\theta )$ and $\\hat { A } : = A ( \\hat { x } , \\theta )$ , and similarly for $B$ and $J$ . We have by definition of the Jacobian estimate function $A _ { \\star } J _ { \\star } = B _ { \\star }$ and $\\hat { A } \\hat { J } = \\hat { B }$ . Therefore we have ", + "bbox": [ + 174, + 377, + 826, + 422 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/ea19f87c8539544562bd4b7de5e16b2fd89fdc51869d3b5430f2a2930582992e.jpg", + "text": "$$\n\\begin{array} { r l } & { J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) = \\hat { A } ^ { - 1 } \\hat { B } - A _ { \\star } ^ { - 1 } B _ { \\star } } \\\\ & { \\qquad = \\hat { A } ^ { - 1 } \\hat { B } - \\hat { A } ^ { - 1 } B _ { \\star } + \\hat { A } ^ { - 1 } B _ { \\star } - A _ { \\star } ^ { - 1 } B _ { \\star } } \\\\ & { \\qquad = \\hat { A } ^ { - 1 } ( \\hat { B } - B _ { \\star } ) + ( \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } ) B _ { \\star } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 425, + 696, + 489 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For any invertible matrices $M _ { 1 } , M _ { 2 }$ , it holds that $M _ { 1 } ^ { - 1 } - M _ { 2 } ^ { - 1 } = M _ { 1 } ^ { - 1 } ( M _ { 2 } - M _ { 1 } ) M _ { 2 } ^ { - 1 }$ , so ", + "bbox": [ + 173, + 493, + 781, + 511 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/f6fdde9603b4506641a20ca93b4ced9e9979aafe4a2e4ecfb32bdf5f13d112f5.jpg", + "text": "$$\n\\begin{array} { r } { \\| M _ { 2 } ^ { - 1 } - M _ { 2 } ^ { - 1 } \\| _ { \\mathrm { o p } } \\leq \\| M _ { 1 } ^ { - 1 } \\| _ { \\mathrm { o p } } \\| M _ { 2 } - M _ { 1 } \\| _ { \\mathrm { o p } } \\| M _ { 2 } ^ { - 1 } \\| _ { \\mathrm { o p } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 515, + 684, + 535 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 539, + 243, + 553 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/8d2dfeee4d4a36ec4a8042a257662c7a48d0017fa610f8c68a701ac7abbbe53c.jpg", + "text": "$$\n\\| \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } \\| _ { \\mathrm { o p } } \\leq \\frac { 1 } { \\alpha ^ { 2 } } \\| \\hat { A } - A _ { \\star } \\| _ { \\mathrm { o p } } \\leq \\frac { \\gamma } { \\alpha ^ { 2 } } \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\| .\n$$", + "text_format": "latex", + "bbox": [ + 313, + 550, + 683, + 580 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "As a consequence, the second term in $J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta )$ can be upper bounded and we obtain ", + "bbox": [ + 173, + 582, + 782, + 597 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/04a120c2c8d35d56824e45203ba451c01c1e663a25dc0d466c67e313514736dc.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| J ( \\hat { \\boldsymbol { x } } , \\theta ) - \\partial \\boldsymbol { x } ^ { \\star } ( \\theta ) \\| \\leq \\| \\hat { A } ^ { - 1 } ( \\hat { B } - B _ { \\star } ) \\| + \\| ( \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } ) B _ { \\star } \\| } \\\\ & { \\qquad \\leq \\| \\hat { A } ^ { - 1 } \\| _ { \\mathsf { o p } } \\| \\hat { B } - B _ { \\star } \\| + \\frac { \\gamma } { \\alpha ^ { 2 } } \\| \\hat { \\boldsymbol { x } } - \\boldsymbol { x } ^ { \\star } ( \\theta ) \\| \\ \\| B _ { \\star } \\| , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 601, + 727, + 651 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "which yields the desired result. ", + "bbox": [ + 173, + 652, + 379, + 667 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Corollary 1 (Jacobian precision for gradient descent fixed point). Let $f$ be such that $f ( \\cdot , \\theta )$ is twice differentiable and $\\alpha$ -strongly convex and $\\nabla _ { 1 } ^ { 2 } f ( \\cdot , \\theta )$ is $\\gamma$ -Lipschitz (in the operator norm) and $\\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta )$ is $\\beta$ -Lipschitz and bounded in norm by $R$ . The estimated Jacobian evaluated at $\\hat { x }$ is then given by ", + "bbox": [ + 173, + 675, + 826, + 731 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/13e69f1b0fef0bf383fb6263539535ab5dabbdb6037c601d6674f7dce9189dd2.jpg", + "text": "$$\n\\boldsymbol { J } ( \\boldsymbol { \\hat { x } } , \\theta ) = - ( \\nabla _ { 1 } ^ { 2 } f ( \\boldsymbol { \\hat { x } } , \\theta ) ) ^ { - 1 } \\partial _ { 2 } \\nabla _ { 1 } f ( \\boldsymbol { \\hat { x } } , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 728, + 635, + 747 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For all $\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }$ , and any $\\hat { x }$ estimating $x ^ { \\star } ( \\theta )$ , we have the following bound for the approximation error of the estimated Jacobian ", + "bbox": [ + 173, + 748, + 823, + 776 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/dcad6b4bd5caf54e48896a9959995a9337c2e18bd05e29aacdfd1f541fbb2309.jpg", + "text": "$$\n\\| J ( { \\hat { x } } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| \\leq \\left( { \\frac { \\beta } { \\alpha } } + { \\frac { \\gamma R } { \\alpha ^ { 2 } } } \\right) \\| { \\hat { x } } - x ^ { \\star } ( \\theta ) \\| .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 781, + 666, + 816 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof of Corollary $^ { l }$ . This follows from Theorem 1, applied to this specific $A ( x , \\theta )$ and $B ( x , \\theta )$ . ", + "bbox": [ + 171, + 829, + 795, + 847 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For proximal gradient descent, where $T ( x , \\theta ) = \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta )$ , this yields ", + "bbox": [ + 171, + 859, + 750, + 877 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/4bf082d2069c3545cb03be603d1c989ae677864cd1a72cf06dd97beee3d24bc0.jpg", + "text": "$$\n\\begin{array} { r l } & { A ( x , \\theta ) = I - \\partial _ { 1 } T ( x , \\theta ) = I - \\partial _ { 1 } \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) ( I - \\eta \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) ) } \\\\ & { B ( x , \\theta ) = \\partial _ { 2 } \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) - \\eta \\partial _ { 1 } \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 882, + 792, + 922 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We now focus in the case of proximal gradient descent on an objective $f ( x , \\theta ) + g ( x )$ , where $g$ is smooth and does not depend on $\\theta$ . This is the case in our experiments in $\\ S 4 . 3$ . Recent work also exploits local smoothness of solutions to derive similar bounds (Bertrand et al., 2021, Theorem 13) ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Corollary 2 (Jacobian precision for proximal gradient descent fixed point). Let $f$ be such that $f ( \\cdot , \\theta )$ is twice differentiable and $\\alpha$ -strongly convex and $\\nabla _ { 1 } ^ { 2 } f ( \\cdot , \\theta )$ is $\\gamma$ -Lipschitz (in the operator norm) and $\\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta )$ is $\\beta$ -Lipschitz and bounded in norm by $R$ . Let $\\bar { g } : \\mathbb { R } ^ { d } \\to \\mathbb { R }$ be a twicedifferentiable $\\mu$ -strongly convex (with special case $\\mu = 0$ being only convex), for which the function $\\Gamma _ { \\eta } ( x , \\theta ) = \\nabla ^ { 2 } g ( p r o x _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) )$ is $\\kappa _ { \\eta }$ -Lipschitz in it first argument. The estimated Jacobian evaluated at $\\hat { x }$ is then given by ", + "bbox": [ + 173, + 148, + 826, + 234 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/0adde6ed30ea227e3fc2d9eb352c3f8316d1333ca5d1cdbacc441fdf6281b5bd.jpg", + "text": "$$\n\\boldsymbol { J } ( \\boldsymbol { \\hat { x } } , \\theta ) = - ( \\nabla _ { 1 } ^ { 2 } f ( \\boldsymbol { \\hat { x } } , \\theta ) + \\Gamma _ { \\eta } ( \\boldsymbol { \\hat { x } } , \\theta ) ) ^ { - 1 } \\partial _ { 2 } \\nabla _ { 1 } f ( \\boldsymbol { \\hat { x } } , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 238, + 673, + 257 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For all $\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }$ , and any $\\hat { x }$ estimating $x ^ { \\star } ( \\theta )$ , we have the following bound for the approximation error of the estimated Jacobian ", + "bbox": [ + 173, + 262, + 823, + 291 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/e2839a6046c9f24535d7bd27cdc01f1016f47f68a8a3a19f66f6bb0ad0396976.jpg", + "text": "$$\n\\| J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| \\leq \\left( \\frac { \\beta + \\kappa _ { \\eta } } { \\alpha + \\mu } + \\frac { \\gamma R } { ( \\alpha + \\mu ) ^ { 2 } } \\right) \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\| .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 296, + 702, + 332 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof of Corollary 2. First, let us note that $\\mathrm { p r o x } _ { \\eta g } ( y , \\theta )$ does not depend on $\\theta$ , since $g$ itself does not depend on $\\theta$ , and is therefore equal to classical proximity operator of $\\eta g$ which, with a slight overload of notations, we denote as $\\operatorname { p r o x } _ { \\eta g } ( y )$ (with a single argument). In other words, ", + "bbox": [ + 173, + 344, + 826, + 388 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/9e0d51648cf62bf13f46df43a00738f762257d296a199f5731f7d7ba8a76a3b5.jpg", + "text": "$$\n\\left\\{ \\begin{array} { l l } { \\mathrm { p r o x } _ { \\eta g } ( y , \\theta ) } & { = \\mathrm { p r o x } _ { \\eta g } ( y ) , } \\\\ { \\partial _ { 1 } \\mathrm { p r o x } _ { \\eta g } ( y , \\theta ) } & { = \\partial \\mathrm { p r o x } _ { \\eta g } ( y ) , } \\\\ { \\partial _ { 2 } \\mathrm { p r o x } _ { \\eta g } ( y , \\theta ) } & { = 0 . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 380, + 397, + 612, + 450 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Regarding the first claim (expression of the estimated Jacobian evaluated at $\\hat { x }$ ), we first have that $\\mathrm { p r o x } _ { \\eta g } ( y )$ is the solution to $( \\overline { { { x } } } ^ { \\prime } - y ) + \\eta \\nabla g ( x ^ { \\prime } ) = 0$ in $x ^ { \\prime }$ - by first-order condition for a smooth convex function. We therefore have that ", + "bbox": [ + 173, + 462, + 825, + 505 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/66ff28fd2cd139035577be6245ccb3f4bae0f07ccf77ded77c4a4ce4c4b8c8d6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathrm { p r o x } _ { \\eta g } ( y ) = ( I + \\eta \\nabla g ) ^ { - 1 } ( y ) } \\\\ & { \\quad \\partial \\mathrm { p r o x } _ { \\eta g } ( y ) = ( I _ { d } + \\eta \\nabla ^ { 2 } g ( \\mathrm { p r o x } _ { \\eta g } ( y ) ) ) ^ { - 1 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 508, + 643, + 553 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "the first $I$ and inverse being functional identity and inverse, and the second $I _ { d }$ and inverse being in the matrix sense, by inverse rule for Jacobians $\\partial h ( z ) = [ \\partial h ^ { - 1 } ( h ( z ) ) ] ^ { - 1 }$ (applied to the prox). ", + "bbox": [ + 173, + 555, + 825, + 584 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "As a consequence, we have, for $\\Gamma _ { \\eta } ( x , \\theta ) = \\nabla ^ { 2 } g ( \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) )$ that ", + "bbox": [ + 174, + 588, + 692, + 607 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/5bb676eed71cac0401ce4aede0a8508c2cccb3d56676cf85d6f87eb2fd201e79.jpg", + "text": "$$\n\\begin{array} { r l } & { A ( x , \\theta ) = I _ { d } - ( I _ { d } + \\eta \\Gamma _ { \\eta } ( x , \\theta ) ) ^ { - 1 } ( I _ { d } - \\eta \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) ) } \\\\ & { \\quad \\quad \\quad = ( I _ { d } + \\eta \\Gamma _ { \\eta } ( x , \\theta ) ) ^ { - 1 } [ I _ { d } + \\eta \\Gamma _ { \\eta } ( x , \\theta ) - ( I _ { d } - \\eta \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) ) ] } \\\\ & { \\quad \\quad \\quad = \\eta ( I _ { d } + \\eta \\Gamma _ { \\eta } ( x , \\theta ) ) ^ { - 1 } ( \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) + \\Gamma _ { \\eta } ( x , \\theta ) ) } \\\\ & { B ( x , \\theta ) = - \\eta ( I _ { d } + \\eta \\Gamma _ { \\eta } ( x , \\theta ) ) ^ { - 1 } \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 612, + 728, + 694 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "As a consequence, for all $\\boldsymbol { x } \\in \\mathbb { R } ^ { d }$ , we have that ", + "bbox": [ + 174, + 699, + 483, + 714 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/07c5d0690e205b365dc9ccfc915b1776537746c8686405c584d5089c5ee4ba3f.jpg", + "text": "$$\nJ ( x , \\theta ) = - ( \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) + \\Gamma _ { \\eta } ( x , \\theta ) ) ^ { - 1 } \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 718, + 674, + 738 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In the following, we modify slightly the notation of both $A$ and $B$ , writing ", + "bbox": [ + 173, + 742, + 661, + 758 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/94c7c70f86348844d6bc7ff2d9b1c0644cce79fa2f618714e0bdf2decfb37973.jpg", + "text": "$$\n\\begin{array} { r l } & { \\tilde { A } ( x , \\theta ) = \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) + \\Gamma _ { \\eta } ( x , \\theta ) } \\\\ & { \\tilde { B } ( x , \\theta ) = - \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 388, + 762, + 609, + 806 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "With the current hypotheses, following along the proof of Theorem 1, we have that $\\tilde { A }$ is $( \\alpha + \\mu )$ well-conditioned, and $( \\gamma + \\kappa _ { \\eta } )$ -Lipschitz in its first argument, and $\\tilde { B }$ is $\\beta$ -Lipschitz in its first argument and bounded in norm by $R$ . The same reasoning yields ", + "bbox": [ + 173, + 819, + 826, + 864 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/ab1ec383217960ea12c7a5a0a8d744c6926992ff497e8f88083016f61ecdc2e3.jpg", + "text": "$$\n\\| J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| \\leq \\left( \\frac { \\beta + \\kappa _ { \\eta } } { \\alpha + \\mu } + \\frac { \\gamma R } { ( \\alpha + \\mu ) ^ { 2 } } \\right) \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\| .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 869, + 702, + 905 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/1e0d895f525a6e7cd14c80c0f0950c35d1838bab6b770597789ae19c3cda6d85.jpg", + "image_caption": [ + "Figure 13: GPU runtime comparison of implicit differentiation and unrolling for hyperparameter optimization of multiclass SVMs for multiple problem sizes (same setting as Figure 3). Error bars represent $90 \\%$ confidence intervals. Absent data points were due to out-of-memory errors $( 1 6 \\mathrm { G B }$ maximum). " + ], + "image_footnote": [], + "bbox": [ + 174, + 123, + 821, + 260 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/12b2f0934006530051b146228d2e6f05d996a17fe28c5c55485e3b5a8ff6d9f2.jpg", + "image_caption": [ + "Figure 14: Value of the outer problem objective function (validation loss) for hyperparameter optimization of multiclass SVMs for multiple problem sizes (same setting as Figure 3). As can be seen, all methods performed similarly in terms of validation loss. This confirms that the faster runtimes for implicit differentiation compared to unrolling shown in Figure 3 (CPU) and Figure 13 (GPU) are not at the cost of worse validation loss. " + ], + "image_footnote": [], + "bbox": [ + 174, + 377, + 821, + 517 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/fcc07750db1e2507aa9944019a2a73ab3473e0b2d476fcac6e196174a0b4ddb6.jpg", + "image_caption": [ + "Figure 15: Jacobian error $\\lVert \\partial x ^ { \\star } ( \\theta ) - J ( \\hat { x } , \\theta ) \\rVert _ { 2 }$ (see also Definition 1) evaluated with a regularization parameter of $\\theta = 1$ , as a function of solution error $\\lVert \\boldsymbol { x } ^ { \\star } ( \\theta ) - \\boldsymbol { \\hat { x } } \\rVert _ { 2 }$ when varying the number of features, on the multiclass SVM task (see Appendix E.1 for a detailed description of the experimental setup). The ground-truth solution $x ^ { \\star } ( \\theta )$ is computed using the liblinear solver (Fan et al., 2008) available in scikit-learn (Pedregosa et al., 2011) with a very low tolerance of $1 0 ^ { - 9 }$ . Unlike in Figure 2, which was on ridge regression, the ground-truth Jacobian ${ \\partial x ^ { \\star } ( \\theta ) }$ cannot be computed in closed form, in the more difficult setting of multiclass SVMs. We therefore use a finite difference to approximately compute ${ \\partial x ^ { \\star } ( \\theta ) }$ . Our results nevertheless confirm similar trends as in Figure 2. " + ], + "image_footnote": [], + "bbox": [ + 174, + 648, + 823, + 777 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E EXPERIMENTAL SETUP AND ADDITIONAL RESULTS ", + "bbox": [ + 174, + 103, + 629, + 117 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Our experiments use JAX (Bradbury et al., 2018), which is Apache2-licensed and scikit-learn (Pedregosa et al., 2011), which is BSD-licensed. ", + "bbox": [ + 176, + 133, + 820, + 161 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "E.1 HYPERPARAMETER OPTIMIZATION OF MULTICLASS SVMS ", + "text_level": 1, + "bbox": [ + 173, + 180, + 620, + 194 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Experimental setup. Synthetic datasets were generated using scikit-learn’s sklearn.datasets.make_classification (Pedregosa et al., 2011), following a model adapted from (Guyon, 2003). All datasets consist of $m = 7 0 0$ training samples belonging to $k = 5$ distinct classes. To simulate problems of different sizes, the number of features is varied as $p \\in \\{ 1 0 0 , 2 5 0 , 5 0 0 , 7 5 0 , 1 0 0 0 , 2 0 0 0 , 3 0 0 0 , 4 0 0 0 , 5 0 0 0 , 7 5 0 0 , 1 0 0 0 0 \\}$ , with $10 \\%$ of features being informative and the rest random noise. In all cases, an additional $m _ { \\mathrm { v a l } } = 2 0 0$ validation samples were generated from the same model to define the outer problem. ", + "bbox": [ + 174, + 205, + 825, + 303 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For the inner problem, we employed three different solvers: (i) mirror descent, (ii) (accelerated) proximal gradient descent and (iii) block coordinate descent. Hyperparameters for all solvers were individually tuned manually to ensure convergence across the range of problem sizes. For mirror descent, a stepsize of 1.0 was used for the first 100 steps, following a inverse square root decay afterwards up to a total of 2500 steps. For proximal gradient descent, a stepsize of $5 \\cdot 1 0 ^ { - 4 } $ was used for 2500 steps. The block coordinate descent solver was run for 500 iterations. All solvers used the same initialization, namely, $\\begin{array} { r } { x _ { \\mathrm { i n i t } } = \\frac { 1 } { k } 1 _ { m \\times k } } \\end{array}$ , which satisfies the dual constraints. ", + "bbox": [ + 174, + 310, + 825, + 409 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For the outer problem, gradient descent was used with a stepsize of $5 \\cdot 1 0 ^ { - 3 } ~ $ for the first 100 steps, following a inverse square root decay afterwards up to a total of 150 steps. ", + "bbox": [ + 174, + 416, + 823, + 445 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Conjugate gradient was used to solve the linear systems in implicit differentiation for at most 2500 iterations. ", + "bbox": [ + 174, + 452, + 823, + 479 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "All results reported pertaining CPU runtimes were obtained using an internal compute cluster. GPU results were obtained using a single NVIDIA P100 GPU with 16GB of memory per dataset. For each dataset size, we report the average runtime of an individual iteration in the outer problem, alongside a $90 \\%$ confidence interval estimated from the corresponding 150 runtime values. ", + "bbox": [ + 174, + 487, + 825, + 542 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Additional results Figure 13 compares the runtime of implicit differentiation and unrolling on GPU. These results highlight a fundamental limitation of the unrolling approach in memory-limited systems such as accelerators, as the inner solver suffered from out-of-memory errors for most problem sizes ${ \\it p } \\geq 2 0 0 0$ for mirror descent, $p \\geq 7 5 0$ for proximal gradient and block coordinate descent). While it might be possible to ameliorate this limitation by reducing the maximum number of iterations in the inner solver, doing so might lead to additional challenges (Wu et al., 2018) and require careful tuning. ", + "bbox": [ + 174, + 559, + 825, + 656 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Figure 14 depicts the validation loss (value of the outer problem objective function) at convergence. It shows that all approaches were able to solve the outer problem, with solutions produced by different approaches being qualitatively indistinguishable from each other across the range of problem sizes considered. ", + "bbox": [ + 174, + 664, + 825, + 719 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Figure 15 shows the Jacobian error achieved as a function of the solution error, when varying the number of features. ", + "bbox": [ + 174, + 726, + 823, + 753 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "E.2 TASK-DRIVEN DICTIONARY LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 772, + 483, + 786 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We downloaded from http://acgt.cs.tau.ac.il/multi_omic_benchmark/ download.html a set of breast cancer gene expression data together with survival information generated by the TCGA Research Network (https://www.cancer.gov/tcga) and processed as explained by (Rappoport & Shamir, 2018). The gene expression matrix contains the expression value for $_ { \\mathrm { p = 2 0 } , 5 3 1 }$ genes in $_ \\mathrm { m = 1 } , 2 1 2$ samples, from which we keep only the primary tumors $\\mathrm { ( m { = } 1 , 0 9 3 }$ ). From the survival information, we select the patients who survived at least five years after diagnosis $( m _ { 1 } = 2 0 0 )$ ), and the patients who died before five years $\\mathbf { \\zeta } _ { m _ { 0 } } = 9 9 \\mathbf { \\zeta }$ ), resulting in a cohort of $m = 2 9 9$ patients with gene expression and binary label. Note that non-selected patients are those who are marked as alive but were not followed for 5 years. ", + "bbox": [ + 174, + 799, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "To evaluate different binary classification methods on this cohort, we repeated 10 times a random split of the full cohort into a training $( 6 0 \\% )$ , validation $( 2 0 \\% )$ and test $( 2 0 \\% )$ sets. For each split and each method, 1) the method is trained with different parameters on the training set, 2) the parameter that maximizes the classification AUC on the validation set is selected, 3) the method is then re-trained on the union of the training and validation sets with the selected parameter, and 4) we measure the AUC of that model on the test set. We then report, for each method, the mean test AUC over the 10 repeats, together with a $9 5 \\%$ confidence interval defined a mean $\\pm \\ : 1 . 9 6 \\times$ standard error of the mean. ", + "bbox": [ + 173, + 103, + 825, + 200 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We used Scikit Learn’s implementation of logistic regression regularized by $\\ell _ { 1 }$ (lasso) and $\\ell _ { 2 }$ (ridge) penalty from sklearn.linear_model.LogisticRegression, and varied the C regularization parameter over a grid of 10 values: $\\{ 1 0 ^ { - 5 } , 1 0 ^ { - 3 } , \\dots , 1 0 ^ { 4 } \\}$ . For the unsupervised dictionary learning experiment method, we estimated a dictionary from the gene expression data in the training and validation sets, using sklearn.decomposition.DictionaryLearning(n_component $s { = } 1 0$ , $\\mathtt { a l p h a } { = } 2 . 0 $ ), which produces sparse codes in $\\begin{array} { r l r } { k } & { { } = } & { 1 0 } \\end{array}$ dimensions with roughly $5 0 \\%$ nonzero coefficients by minimizing the squared Frobenius reconstruction distance with lasso regularization on the code. We then use sklearn.linear_model.LogisticRegression to train a logistic regression on the codes, varying the ridge regularization parameter C over a grid of 10 values $\\{ \\breve { 1 } 0 ^ { - 1 } , 1 \\breve { 0 ^ { 0 } } , \\dots , 1 0 ^ { 8 } \\}$ . ", + "bbox": [ + 174, + 208, + 826, + 361 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Finally, we implemented the task-driven dictionary learning model (13) with our toolbox, following the pseudo-code in Figure 10. Like for the unsupervised dictionary learning experiment, we set the dimension of the codes to $k = 1 0$ , and a fixed elastic net regularization on the inner optimization problem to ensure that the codes have roughly $5 0 \\%$ sparsity. For the outer optimization problem, we solve an $\\ell _ { 2 }$ regularized ridge regression problem, varying again the ridge regularization parameter $\\mathrm { C }$ over a grid of 10 values $\\{ 1 0 ^ { - 1 } , 1 0 ^ { 0 } , \\dot { ~ } . ~ . ~ . ~ , 1 0 ^ { 8 } \\}$ . Because the outer problem is non-convex, we minimize it using the Adam optimizer (Kingma & Ba, 2014) with default parameters. ", + "bbox": [ + 174, + 367, + 825, + 465 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "E.3 DATASET DISTILLATION ", + "text_level": 1, + "bbox": [ + 174, + 486, + 385, + 501 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Experimental setup. For the inner problem, we used gradient descent with backtracking linesearch, while for the outer problem we used gradient descent with momentum and a fixed step-size. The momentum parameter was set to 0.9 while the step-size was set to 1. ", + "bbox": [ + 174, + 513, + 825, + 555 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Figure 4 was produced after 4000 iterations of the outer loop on CPU (Intel(R) Xeon(R) Platinum P-8136 CPU $ @ ~ 2 . 0 0 \\mathrm { G H z }$ ), which took 1h55. Unrolled differentiation took instead 8h:05 (4 times more) to run the same number of iterations. As can be seen in Figure 16, the output is the same in both approaches. ", + "bbox": [ + 174, + 563, + 825, + 618 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/d14a67dc9aa1e9ca11d55deb01b59e8b2000ea8205e7371b8fdd576c7bf9e062.jpg", + "image_caption": [ + "Figure 16: Distilled MNIST dataset $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { k \\times p }$ obtained by solving (12) through unrolled differentiation. Although there is no qualitative difference, the implicit differentiation approach is 4 times faster. " + ], + "image_footnote": [], + "bbox": [ + 169, + 636, + 823, + 693 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "E.4 MOLECULAR DYNAMICS ", + "text_level": 1, + "bbox": [ + 174, + 775, + 387, + 789 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Our experimental setup is adapted from the JAX-MD example notebook available at https://github.com/google/jax-md/blob/master/notebooks/meta_ optimization.ipynb. ", + "bbox": [ + 173, + 801, + 826, + 844 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We emphasize that calculating the gradient of the total energy objective, $\\begin{array} { r } { f ( x , \\theta ) = \\sum _ { i j } U ( x _ { i , j } , \\theta ) } \\end{array}$ , with respect to the diameter $\\theta$ of the smaller particles, $\\nabla _ { 1 } f ( x , \\theta )$ , does not require implicit differentiation or unrolling. This is because $\\nabla _ { 1 } f ( x , \\theta ) = 0$ at $x = x ^ { \\star } ( \\theta )$ : ", + "bbox": [ + 174, + 851, + 826, + 896 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/f072a260627c1312a19fa6e1bd114fcea22e76534d25592c79aa7c3632d8d965.jpg", + "text": "$$\n\\nabla _ { \\boldsymbol { \\theta } } f ( { \\boldsymbol x } ^ { \\star } ( { \\boldsymbol { \\theta } } ) , { \\boldsymbol { \\theta } } ) = \\partial { \\boldsymbol x } ^ { \\star } ( { \\boldsymbol { \\theta } } ) ^ { \\top } \\nabla _ { 1 } f ( { \\boldsymbol x } ^ { \\star } ( { \\boldsymbol { \\theta } } ) , { \\boldsymbol { \\theta } } ) + \\nabla _ { 2 } f ( { \\boldsymbol x } ^ { \\star } ( { \\boldsymbol { \\theta } } ) , { \\boldsymbol { \\theta } } ) = \\nabla _ { 2 } f ( { \\boldsymbol { x } } ^ { \\star } ( { \\boldsymbol { \\theta } } ) , { \\boldsymbol { \\theta } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 240, + 906, + 756, + 925 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "This is known as Danskin’s theorem or envelope theorem. Thus instead, we consider sensitivities of position ${ \\partial x ^ { \\star } ( \\theta ) }$ directly, which does require implicit differentiation or unrolling. ", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Our results comparing implicit and unrolled differentiation for calculating the sensitivity of position are shown in Figure 17. We use BiCGSTAB (Vorst & van der Vorst, 1992) to perform the tangent linear solve. Like in the original JAX-MD experiment, we use $k = 1 2 8$ particles in $m = 2$ dimensions. ", + "bbox": [ + 176, + 138, + 823, + 181 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/ceb73cf3c881abac0e260e868a99195719436064bfa46aa686abce2ad7f0ee01.jpg", + "image_caption": [ + "Figure 17: L1 norm of position sensitivities in the molecular dynamics simulations, for 40 different random initial conditions (different colored lines). Gradients through the unrolled FIRE optimizer (Bitzek et al., 2006) for many initial conditions do not converge, in contrast to implicit differentiation. 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It allows", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 200, + 470, + 213 + ], + "spans": [ + { + "bbox": [ + 141, + 200, + 470, + 213 + ], + "score": 1.0, + "content": "expressing complex computations by composing elementary ones in creative ways", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 210, + 470, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 470, + 223 + ], + "score": 1.0, + "content": "and removes the burden of computing their derivatives by hand. More recently,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 222, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 469, + 234 + ], + "score": 1.0, + "content": "differentiation of optimization problem solutions has attracted widespread attention", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 233, + 469, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 469, + 245 + ], + "score": 1.0, + "content": "with applications such as optimization layers, and in bi-level problems such as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 244, + 470, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 470, + 255 + ], + "score": 1.0, + "content": "hyper-parameter optimization and meta-learning. However, so far, implicit differen-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 254, + 469, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 469, + 267 + ], + "score": 1.0, + "content": "tiation remained difficult to use for practitioners, as it often required case-by-case", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 265, + 469, + 278 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 469, + 278 + ], + "score": 1.0, + "content": "tedious mathematical derivations and implementations. In this paper, we propose", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 277, + 470, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 470, + 289 + ], + "score": 1.0, + "content": "an efficient and modular approach for implicit differentiation of optimization prob-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 286, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 286, + 418, + 301 + ], + "score": 1.0, + "content": "lems. In our approach, the user defines directly in Python a function", + "type": "text" + }, + { + "bbox": [ + 419, + 288, + 428, + 297 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 286, + 470, + 301 + ], + "score": 1.0, + "content": "capturing", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 299, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 469, + 311 + ], + "score": 1.0, + "content": "the optimality conditions of the problem to be differentiated. Once this is done, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 310, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 221, + 322 + ], + "score": 1.0, + "content": "leverage autodiff of", + "type": "text" + }, + { + "bbox": [ + 221, + 310, + 230, + 319 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 310, + 469, + 322 + ], + "score": 1.0, + "content": "and implicit differentiation to automatically differentiate the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 321, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 333 + ], + "score": 1.0, + "content": "optimization problem. Our approach thus combines the benefits of implicit differ-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 331, + 469, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 469, + 343 + ], + "score": 1.0, + "content": "entiation and autodiff. It is efficient as it can be added on top of any state-of-the-art", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 343, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 142, + 343, + 469, + 354 + ], + "score": 1.0, + "content": "solver and modular as the optimality condition specification is decoupled from", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "score": 1.0, + "content": "the implicit differentiation mechanism. We show that seemingly simple principles", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 364, + 470, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 470, + 377 + ], + "score": 1.0, + "content": "allow to recover many existing implicit differentiation methods and create new ones", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 375, + 470, + 388 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 470, + 388 + ], + "score": 1.0, + "content": "easily. We demonstrate the ease of formulating and solving bi-level optimization", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 386, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 469, + 399 + ], + "score": 1.0, + "content": "problems using our framework. We also showcase an application to the sensitivity", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 398, + 273, + 409 + ], + "spans": [ + { + "bbox": [ + 142, + 398, + 273, + 409 + ], + "score": 1.0, + "content": "analysis of molecular dynamics.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 13.5, + "bbox_fs": [ + 141, + 189, + 470, + 409 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 416, + 206, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 208, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 208, + 431 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "Automatic differentiation (autodiff) is now an inherent part of machine learning software. It allows", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 452, + 504, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 504, + 464 + ], + "score": 1.0, + "content": "to express complex computations by composing elementary ones in creative ways and removes the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "tedious burden of computing their derivatives by hand. In parallel, the differentiation of optimization", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "score": 1.0, + "content": "problem solutions has found many applications. A classical example is bi-level optimization, which", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "typically involves computing the derivatives of a nested optimization problem in order to solve an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "outer one. Examples of applications in machine learning include hyper-parameter optimization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "(Chapelle et al., 2002; Seeger, 2008; Pedregosa, 2016; Franceschi et al., 2017; Bertrand et al., 2020;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "score": 1.0, + "content": "2021), neural networks (Lorraine et al., 2020), and meta-learning (Franceschi et al., 2018; Rajeswaran", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "et al., 2019). Another line of active research involving differentiation of optimization problem", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "solutions are optimization layers (Kim et al., 2017; Amos & Kolter, 2017; Niculae & Blondel, 2017;", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "Djolonga & Krause, 2017; Gould et al., 2019), which can be used to encourage structured outputs,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "and implicit deep networks (Bai et al., 2019; El Ghaoui et al., 2019), which have a smaller memory", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 573, + 274, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 274, + 583 + ], + "score": 1.0, + "content": "footprint than backprop-trained networks.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 441, + 506, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "Since optimization problem solutions typically do not enjoy an explicit formula in terms of their", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "inputs, autodiff cannot be used directly to differentiate these functions. In recent years, two main", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "approaches have been developed to circumvent this problem. The first one consists of unrolling the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "iterations of an optimization algorithm and using the final iteration as a proxy for the optimization", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "problem solution (Wengert, 1964; Domke, 2012; Deledalle et al., 2014; Franceschi et al., 2018;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "Ablin et al., 2020). This allows to explicitly construct a computational graph relating the algorithm", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "output to the inputs, on which autodiff can then be used transparently. However, this requires a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "reimplementation of the algorithm using the autodiff system, and not all algorithms are necessarily", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "autodiff friendly. Moreover, forward-mode autodiff has time complexity that scales linearly with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "the number of variables and reverse-mode autodiff has memory complexity that scales linearly", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "with the number of algorithm iterations. In contrast, a second approach consists in implicitly", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "relating an optimization problem solution to its inputs using optimality conditions. In a machine", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "learning context, such implicit differentiation has been used for stationarity conditions (Bengio, 2000;", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 588, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Lorraine et al., 2020), KKT conditions (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "2017; Niculae et al., 2018; Niculae & Martins, 2020) and the proximal gradient fixed point (Niculae", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "& Blondel, 2017; Bertrand et al., 2020; 2021). An advantage of implicit differentiation is that a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "solver reimplementation is not needed, allowing to build upon decades of state-of-the-art software.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 139 + ], + "score": 1.0, + "content": "Although implicit differentiation has a long history in numerical analysis (Griewank & Walther,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "2008; Bell & Burke, 2008; Krantz & Parks, 2012; Bonnans & Shapiro, 2013), so far, it remained", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "score": 1.0, + "content": "difficult to use for practitioners, as it required a case-by-case tedious mathematical derivation and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "score": 1.0, + "content": "implementation. CasADi (Andersson et al., 2019) allows to differentiate various optimization and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 182 + ], + "score": 1.0, + "content": "root finding problem algorithms provided by the library. However, it does not allow to easily add", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 192 + ], + "score": 1.0, + "content": "implicit differentiation on top of existing solvers from optimality conditions expressed by the user, as", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "score": 1.0, + "content": "we do. A recent tutorial explains how to implement implicit differentiation in JAX (Duvenaud et al.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 214 + ], + "score": 1.0, + "content": "2020). However, the tutorial requires the user to take care of low-level technical details and does", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 226 + ], + "score": 1.0, + "content": "not cover a large catalog of optimality condition mappings as we do. Other work (Agrawal et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "2019a) attempts to address this issue by adding implicit differentiation on top of cvxpy (Diamond", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "& Boyd, 2016). This works by reducing all convex optimization problems to a conic program and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "using conic programming’s optimality conditions to derive an implicit differentiation formula. While", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "this approach is very generic, solving a convex optimization problem using a conic programming", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "solver—an ADMM-based splitting conic solver (O’Donoghue et al., 2016) in the case of cvxpy—is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 397, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 397, + 293 + ], + "score": 1.0, + "content": "rarely the state-of-the-art approach for each particular problem instance.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 311 + ], + "score": 1.0, + "content": "In this work, we adopt a different strategy that makes it easy to add implicit differentiation on top", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "of any existing solver. In our approach, the user defines directly in Python a mapping function", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 318, + 507, + 332 + ], + "spans": [ + { + "bbox": [ + 107, + 319, + 116, + 329 + ], + "score": 0.77, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 318, + 507, + 332 + ], + "score": 1.0, + "content": "capturing the optimality conditions of the problem solved by the algorithm. Once this is done,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 202, + 343 + ], + "score": 1.0, + "content": "we leverage autodiff of", + "type": "text" + }, + { + "bbox": [ + 203, + 330, + 212, + 339 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "combined with implicit differentiation to automatically differentiate the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "optimization problem solution. In this way, our approach is generic, yet it can exploit the efficiency", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 363 + ], + "score": 1.0, + "content": "of state-of-the-art solvers. It therefore combines the benefits of implicit differentiation and autodiff.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 319, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 319, + 375 + ], + "score": 1.0, + "content": "To summarize, we make the following contributions.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 506, + 491 + ], + "lines": [ + { + "bbox": [ + 107, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 107, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "• We describe our framework and its JAX implementation (provided in the supplementary material).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 116, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 116, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "Our framework significantly lowers the barrier to use implicit differentiation thanks to the use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 115, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "of autodiff of the optimality conditions and the seamless integration in JAX. Our framework", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 412, + 502, + 425 + ], + "spans": [ + { + "bbox": [ + 115, + 412, + 502, + 425 + ], + "score": 1.0, + "content": "significantly extends JAX for numerical optimization, with low-level details all abstracted away.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 107, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "• We instantiate our framework on a large catalog of optimality conditions (Table 1), recovering", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 115, + 438, + 482, + 451 + ], + "spans": [ + { + "bbox": [ + 115, + 438, + 482, + 451 + ], + "score": 1.0, + "content": "existing schemes and obtaining new ones, such as the mirror descent fixed point based one.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "• On the theoretical side, we provide new bounds on the Jacobian error when the optimization", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 115, + 464, + 395, + 477 + ], + "spans": [ + { + "bbox": [ + 115, + 464, + 395, + 477 + ], + "score": 1.0, + "content": "problem is only solved approximately, and empirically validate them.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 478, + 477, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 477, + 492 + ], + "score": 1.0, + "content": "• We implement four illustrative applications, demonstrating our framework’s ease of use.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 105, + 496, + 503, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Beyond our software implementation, we hope this paper provides a self-contained blueprint for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 507, + 411, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 411, + 519 + ], + "score": 1.0, + "content": "creating an efficient and modular implementation of implicit differentiation.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 104, + 525, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 525, + 308, + 538 + ], + "score": 1.0, + "content": "Notation. We denote the gradient and Hessian of", + "type": "text" + }, + { + "bbox": [ + 308, + 526, + 357, + 538 + ], + "score": 0.9, + "content": "f \\colon { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 525, + 406, + 538 + ], + "score": 1.0, + "content": "evaluated at", + "type": "text" + }, + { + "bbox": [ + 407, + 526, + 437, + 537 + ], + "score": 0.91, + "content": "x \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 525, + 451, + 538 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 451, + 526, + 504, + 538 + ], + "score": 0.9, + "content": "\\nabla f ( x ) \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 534, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 124, + 553 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 537, + 193, + 550 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } f ( x ) \\in \\mathbb { R } ^ { d \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 534, + 309, + 553 + ], + "score": 1.0, + "content": ". 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When", + "type": "text" + }, + { + "bbox": [ + 165, + 550, + 172, + 560 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 545, + 186, + 563 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 187, + 550, + 196, + 559 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 545, + 506, + 563 + ], + "score": 1.0, + "content": "have several arguments, we denote the gradient, Hessian and Jacobian in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 557, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 557, + 122, + 574 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 560, + 133, + 570 + ], + "score": 0.85, + "content": "i ^ { \\mathrm { { t h } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 557, + 188, + 574 + ], + "score": 1.0, + "content": "argument by", + "type": "text" + }, + { + "bbox": [ + 189, + 560, + 201, + 571 + ], + "score": 0.81, + "content": "\\nabla _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 557, + 205, + 574 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 559, + 220, + 572 + ], + "score": 0.85, + "content": "\\nabla _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 557, + 239, + 574 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 560, + 249, + 571 + ], + "score": 0.87, + "content": "\\partial _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 557, + 506, + 574 + ], + "score": 1.0, + "content": ", respectively. The standard probability simplex is denoted by", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 569, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 267, + 584 + ], + "score": 0.88, + "content": "\\triangle ^ { d } : = \\{ x \\in \\mathbb { R } ^ { d } \\colon \\| x \\| _ { 1 } = 1 , x \\geq 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 569, + 325, + 586 + ], + "score": 1.0, + "content": ". For any set", + "type": "text" + }, + { + "bbox": [ + 325, + 571, + 361, + 582 + ], + "score": 0.9, + "content": "\\mathcal { C } \\subset \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 569, + 506, + 586 + ], + "score": 1.0, + "content": ", we denote the indicator function", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 582, + 502, + 596 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 198, + 595 + ], + "score": 0.87, + "content": "I _ { \\mathcal { C } } : \\mathbb { R } ^ { d } \\overset { \\cdot } { } \\mathbb { R } \\cup \\{ + \\infty \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 582, + 225, + 596 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 225, + 583, + 268, + 595 + ], + "score": 0.91, + "content": "I _ { \\mathit { C } } ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 582, + 277, + 596 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 277, + 583, + 302, + 594 + ], + "score": 0.85, + "content": "x \\in { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 582, + 306, + 596 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 306, + 583, + 361, + 595 + ], + "score": 0.86, + "content": "I _ { C } ( x ) = + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 582, + 494, + 596 + ], + "score": 1.0, + "content": "otherwise. For a vector or matrix", + "type": "text" + }, + { + "bbox": [ + 494, + 584, + 502, + 593 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 594, + 424, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 140, + 608 + ], + "score": 1.0, + "content": "we note", + "type": "text" + }, + { + "bbox": [ + 141, + 595, + 158, + 606 + ], + "score": 0.9, + "content": "\\| A \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 594, + 320, + 608 + ], + "score": 1.0, + "content": "the Frobenius (or Euclidean) norm, and", + "type": "text" + }, + { + "bbox": [ + 320, + 595, + 346, + 606 + ], + "score": 0.92, + "content": "\\| A \\| _ { \\mathsf { o p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 594, + 424, + 608 + ], + "score": 1.0, + "content": "the operator norm.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 108, + 622, + 411, + 633 + ], + "lines": [ + { + "bbox": [ + 104, + 619, + 413, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 413, + 636 + ], + "score": 1.0, + "content": "2 COMBINING IMPLICIT DIFFERENTIATION AND AUTODIFF", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "title", + "bbox": [ + 108, + 643, + 226, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 227, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 227, + 655 + ], + "score": 1.0, + "content": "2.1 GENERAL PRINCIPLES", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Overview. Contrary to autodiff through unrolled algorithm iterations, implicit differentiation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "typically involves a manual, sometimes complicated, mathematical derivation. For instance, numerous", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "works (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "& Martins, 2020) use Karush–Kuhn–Tucker (KKT) conditions in order to relate a constrained", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "optimization problem’s solution to its inputs, and to manually derive a formula for its derivatives.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 719, + 409, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 409, + 733 + ], + "score": 1.0, + "content": "The derivation and implementation in these works are always case-by-case.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Lorraine et al., 2020), KKT conditions (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "2017; Niculae et al., 2018; Niculae & Martins, 2020) and the proximal gradient fixed point (Niculae", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "& Blondel, 2017; Bertrand et al., 2020; 2021). An advantage of implicit differentiation is that a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "solver reimplementation is not needed, allowing to build upon decades of state-of-the-art software.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 139 + ], + "score": 1.0, + "content": "Although implicit differentiation has a long history in numerical analysis (Griewank & Walther,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "2008; Bell & Burke, 2008; Krantz & Parks, 2012; Bonnans & Shapiro, 2013), so far, it remained", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "score": 1.0, + "content": "difficult to use for practitioners, as it required a case-by-case tedious mathematical derivation and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "score": 1.0, + "content": "implementation. CasADi (Andersson et al., 2019) allows to differentiate various optimization and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 182 + ], + "score": 1.0, + "content": "root finding problem algorithms provided by the library. However, it does not allow to easily add", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 192 + ], + "score": 1.0, + "content": "implicit differentiation on top of existing solvers from optimality conditions expressed by the user, as", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "score": 1.0, + "content": "we do. A recent tutorial explains how to implement implicit differentiation in JAX (Duvenaud et al.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 214 + ], + "score": 1.0, + "content": "2020). However, the tutorial requires the user to take care of low-level technical details and does", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 226 + ], + "score": 1.0, + "content": "not cover a large catalog of optimality condition mappings as we do. Other work (Agrawal et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "2019a) attempts to address this issue by adding implicit differentiation on top of cvxpy (Diamond", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "& Boyd, 2016). This works by reducing all convex optimization problems to a conic program and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "using conic programming’s optimality conditions to derive an implicit differentiation formula. While", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "this approach is very generic, solving a convex optimization problem using a conic programming", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "solver—an ADMM-based splitting conic solver (O’Donoghue et al., 2016) in the case of cvxpy—is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 397, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 397, + 293 + ], + "score": 1.0, + "content": "rarely the state-of-the-art approach for each particular problem instance.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 82, + 506, + 293 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 311 + ], + "score": 1.0, + "content": "In this work, we adopt a different strategy that makes it easy to add implicit differentiation on top", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "of any existing solver. In our approach, the user defines directly in Python a mapping function", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 318, + 507, + 332 + ], + "spans": [ + { + "bbox": [ + 107, + 319, + 116, + 329 + ], + "score": 0.77, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 318, + 507, + 332 + ], + "score": 1.0, + "content": "capturing the optimality conditions of the problem solved by the algorithm. Once this is done,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 202, + 343 + ], + "score": 1.0, + "content": "we leverage autodiff of", + "type": "text" + }, + { + "bbox": [ + 203, + 330, + 212, + 339 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "combined with implicit differentiation to automatically differentiate the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "optimization problem solution. In this way, our approach is generic, yet it can exploit the efficiency", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 363 + ], + "score": 1.0, + "content": "of state-of-the-art solvers. It therefore combines the benefits of implicit differentiation and autodiff.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 319, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 319, + 375 + ], + "score": 1.0, + "content": "To summarize, we make the following contributions.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 295, + 507, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 506, + 491 + ], + "lines": [ + { + "bbox": [ + 107, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 107, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "• We describe our framework and its JAX implementation (provided in the supplementary material).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 116, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 116, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "Our framework significantly lowers the barrier to use implicit differentiation thanks to the use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 115, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "of autodiff of the optimality conditions and the seamless integration in JAX. Our framework", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 412, + 502, + 425 + ], + "spans": [ + { + "bbox": [ + 115, + 412, + 502, + 425 + ], + "score": 1.0, + "content": "significantly extends JAX for numerical optimization, with low-level details all abstracted away.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 107, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "• We instantiate our framework on a large catalog of optimality conditions (Table 1), recovering", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 115, + 438, + 482, + 451 + ], + "spans": [ + { + "bbox": [ + 115, + 438, + 482, + 451 + ], + "score": 1.0, + "content": "existing schemes and obtaining new ones, such as the mirror descent fixed point based one.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "• On the theoretical side, we provide new bounds on the Jacobian error when the optimization", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 115, + 464, + 395, + 477 + ], + "spans": [ + { + "bbox": [ + 115, + 464, + 395, + 477 + ], + "score": 1.0, + "content": "problem is only solved approximately, and empirically validate them.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 478, + 477, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 477, + 492 + ], + "score": 1.0, + "content": "• We implement four illustrative applications, demonstrating our framework’s ease of use.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 378, + 506, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 496, + 503, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Beyond our software implementation, we hope this paper provides a self-contained blueprint for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 507, + 411, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 411, + 519 + ], + "score": 1.0, + "content": "creating an efficient and modular implementation of implicit differentiation.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 496, + 505, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 104, + 525, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 525, + 308, + 538 + ], + "score": 1.0, + "content": "Notation. We denote the gradient and Hessian of", + "type": "text" + }, + { + "bbox": [ + 308, + 526, + 357, + 538 + ], + "score": 0.9, + "content": "f \\colon { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 525, + 406, + 538 + ], + "score": 1.0, + "content": "evaluated at", + "type": "text" + }, + { + "bbox": [ + 407, + 526, + 437, + 537 + ], + "score": 0.91, + "content": "x \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 525, + 451, + 538 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 451, + 526, + 504, + 538 + ], + "score": 0.9, + "content": "\\nabla f ( x ) \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 534, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 124, + 553 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 537, + 193, + 550 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } f ( x ) \\in \\mathbb { R } ^ { d \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 534, + 309, + 553 + ], + "score": 1.0, + "content": ". 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For any set", + "type": "text" + }, + { + "bbox": [ + 325, + 571, + 361, + 582 + ], + "score": 0.9, + "content": "\\mathcal { C } \\subset \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 569, + 506, + 586 + ], + "score": 1.0, + "content": ", we denote the indicator function", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 582, + 502, + 596 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 198, + 595 + ], + "score": 0.87, + "content": "I _ { \\mathcal { C } } : \\mathbb { R } ^ { d } \\overset { \\cdot } { } \\mathbb { R } \\cup \\{ + \\infty \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 582, + 225, + 596 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 225, + 583, + 268, + 595 + ], + "score": 0.91, + "content": "I _ { \\mathit { C } } ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 582, + 277, + 596 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 277, + 583, + 302, + 594 + ], + "score": 0.85, + "content": "x \\in { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 582, + 306, + 596 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 306, + 583, + 361, + 595 + ], + "score": 0.86, + "content": "I _ { C } ( x ) = + \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 582, + 494, + 596 + ], + "score": 1.0, + "content": "otherwise. For a vector or matrix", + "type": "text" + }, + { + "bbox": [ + 494, + 584, + 502, + 593 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 594, + 424, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 140, + 608 + ], + "score": 1.0, + "content": "we note", + "type": "text" + }, + { + "bbox": [ + 141, + 595, + 158, + 606 + ], + "score": 0.9, + "content": "\\| A \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 594, + 320, + 608 + ], + "score": 1.0, + "content": "the Frobenius (or Euclidean) norm, and", + "type": "text" + }, + { + "bbox": [ + 320, + 595, + 346, + 606 + ], + "score": 0.92, + "content": "\\| A \\| _ { \\mathsf { o p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 594, + 424, + 608 + ], + "score": 1.0, + "content": "the operator norm.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 525, + 506, + 608 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 622, + 411, + 633 + ], + "lines": [ + { + "bbox": [ + 104, + 619, + 413, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 413, + 636 + ], + "score": 1.0, + "content": "2 COMBINING IMPLICIT DIFFERENTIATION AND AUTODIFF", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "title", + "bbox": [ + 108, + 643, + 226, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 227, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 227, + 655 + ], + "score": 1.0, + "content": "2.1 GENERAL PRINCIPLES", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Overview. Contrary to autodiff through unrolled algorithm iterations, implicit differentiation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "typically involves a manual, sometimes complicated, mathematical derivation. For instance, numerous", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "works (Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "& Martins, 2020) use Karush–Kuhn–Tucker (KKT) conditions in order to relate a constrained", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "optimization problem’s solution to its inputs, and to manually derive a formula for its derivatives.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 719, + 409, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 409, + 733 + ], + "score": 1.0, + "content": "The derivation and implementation in these works are always case-by-case.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 666, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 178, + 81, + 432, + 260 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 178, + 81, + 432, + 260 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 178, + 81, + 432, + 260 + ], + "spans": [ + { + "bbox": [ + 178, + 81, + 432, + 260 + ], + "score": 0.97, + "type": "image", + "image_path": "fadbf82543eb3718bc2bb5820799eaf0ad81197f72ee1ae17d032dbb61832239.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 178, + 81, + 432, + 140.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 178, + 140.66666666666666, + 432, + 200.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 178, + 200.33333333333331, + 432, + 260.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 271, + 506, + 315 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 475, + 283 + ], + "score": 1.0, + "content": "Figure 1: Adding implicit differentiation on top of a ridge regression solver. The function", + "type": "text" + }, + { + "bbox": [ + 475, + 271, + 505, + 283 + ], + "score": 0.92, + "content": "f ( x , \\theta )", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 295, + 294 + ], + "score": 1.0, + "content": "defines the objective function and the mapping", + "type": "text" + }, + { + "bbox": [ + 296, + 282, + 304, + 292 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 281, + 505, + 294 + ], + "score": 1.0, + "content": ", here simply equation (4), captures the optimality", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "conditions. Our decorator @custom_root automatically adds implicit differentiation to the solver", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 304, + 482, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 449, + 315 + ], + "score": 1.0, + "content": "for the user, overriding JAX’s default behavior. The last line evaluates the Jacobian at", + "type": "text" + }, + { + "bbox": [ + 449, + 304, + 479, + 314 + ], + "score": 0.9, + "content": "\\theta = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 304, + 482, + 315 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "In this work, we propose a generic way to easily add implicit differentiation on top of existing solvers.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 396, + 358 + ], + "score": 1.0, + "content": "In our approach, the user defines directly in Python a mapping function", + "type": "text" + }, + { + "bbox": [ + 396, + 346, + 405, + 355 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "capturing the optimality", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "conditions of the problem solved by the algorithm. We provide reusable building blocks to easily", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 158, + 380 + ], + "score": 1.0, + "content": "express such", + "type": "text" + }, + { + "bbox": [ + 159, + 368, + 167, + 377 + ], + "score": 0.78, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 367, + 226, + 380 + ], + "score": 1.0, + "content": ". The provided", + "type": "text" + }, + { + "bbox": [ + 227, + 368, + 236, + 377 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "is then plugged into our Python decorator @custom_root, which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "we append on top of the solver declaration we wish to differentiate. Under the hood, we combine", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 266, + 402 + ], + "score": 1.0, + "content": "implicit differentiation and autodiff of", + "type": "text" + }, + { + "bbox": [ + 266, + 390, + 276, + 399 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "to automatically differentiate the optimization problem", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 400, + 345, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 345, + 413 + ], + "score": 1.0, + "content": "solution. A simple illustrative example is given in Figure 1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 422, + 504, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 221, + 436 + ], + "score": 1.0, + "content": "Differentiating a root. Let", + "type": "text" + }, + { + "bbox": [ + 222, + 422, + 297, + 434 + ], + "score": 0.92, + "content": "F \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 420, + 505, + 436 + ], + "score": 1.0, + "content": "be a user-provided mapping, capturing the optimality", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 434, + 446, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 326, + 447 + ], + "score": 1.0, + "content": "conditions of a problem. An optimal solution, denoted", + "type": "text" + }, + { + "bbox": [ + 326, + 434, + 349, + 446 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 434, + 432, + 447 + ], + "score": 1.0, + "content": ", should be a root of", + "type": "text" + }, + { + "bbox": [ + 433, + 435, + 441, + 444 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 434, + 446, + 447 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 269, + 448, + 342, + 461 + ], + "lines": [ + { + "bbox": [ + 269, + 448, + 342, + 461 + ], + "spans": [ + { + "bbox": [ + 269, + 448, + 342, + 461 + ], + "score": 0.91, + "content": "{ \\cal F } ( x ^ { \\star } ( \\theta ) , \\theta ) = 0 .", + "type": "interline_equation", + "image_path": "446159dd8937a24464d0474490428271a6a102724c6010dab3e32a02a36af2d6.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 269, + 448, + 342, + 461 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 463, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 507, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 154, + 477 + ], + "score": 1.0, + "content": "We can see", + "type": "text" + }, + { + "bbox": [ + 154, + 464, + 178, + 476 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 461, + 325, + 477 + ], + "score": 1.0, + "content": "as an implicitly defined function of", + "type": "text" + }, + { + "bbox": [ + 325, + 464, + 356, + 474 + ], + "score": 0.91, + "content": "\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 461, + 377, + 477 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 378, + 463, + 436, + 474 + ], + "score": 0.91, + "content": "x ^ { \\star } \\colon { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 461, + 507, + 477 + ], + "score": 1.0, + "content": ". More precisely,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 299, + 487 + ], + "score": 1.0, + "content": "from the implicit function theorem (Griewank", + "type": "text" + }, + { + "bbox": [ + 299, + 475, + 309, + 485 + ], + "score": 0.26, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "Walther, 2008; Krantz & Parks, 2012), we know", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 485, + 504, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 138, + 498 + ], + "score": 1.0, + "content": "that for", + "type": "text" + }, + { + "bbox": [ + 139, + 486, + 171, + 498 + ], + "score": 0.92, + "content": "( x _ { 0 } , \\theta _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 485, + 213, + 498 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 214, + 486, + 272, + 498 + ], + "score": 0.93, + "content": "F ( x _ { 0 } , \\theta _ { 0 } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 485, + 411, + 498 + ], + "score": 1.0, + "content": "with a continuously differentiable", + "type": "text" + }, + { + "bbox": [ + 411, + 487, + 420, + 496 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 485, + 485, + 498 + ], + "score": 1.0, + "content": ", if the Jacobian", + "type": "text" + }, + { + "bbox": [ + 485, + 486, + 504, + 497 + ], + "score": 0.88, + "content": "\\partial _ { 1 } F", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 159, + 509 + ], + "score": 1.0, + "content": "evaluated at", + "type": "text" + }, + { + "bbox": [ + 159, + 497, + 191, + 509 + ], + "score": 0.91, + "content": "( x _ { 0 } , \\theta _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 496, + 428, + 509 + ], + "score": 1.0, + "content": "is a square invertible matrix, then there exists a function", + "type": "text" + }, + { + "bbox": [ + 428, + 497, + 450, + 509 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "defined on a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 174, + 520 + ], + "score": 1.0, + "content": "neighborhood of", + "type": "text" + }, + { + "bbox": [ + 174, + 509, + 184, + 519 + ], + "score": 0.87, + "content": "\\theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 507, + 223, + 520 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 224, + 507, + 275, + 520 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta _ { 0 } ) = x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 507, + 360, + 520 + ], + "score": 1.0, + "content": ". 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\\partial _ { 1 } F ( x ^ { \\star } ( \\theta ) , \\theta ) } _ { A \\in \\mathbb { R } ^ { d \\times d } } \\underbrace { \\partial x ^ { \\star } ( \\theta ) } _ { J \\in \\mathbb { R } ^ { d \\times n } } = \\underbrace { \\partial _ { 2 } F ( x ^ { \\star } ( \\theta ) , \\theta ) } _ { B \\in \\mathbb { R } ^ { d \\times n } } .", + "type": "interline_equation", + "image_path": "97fb8b270917089f469ee014e79f21f03b3d3bc21b899a3796964a33d5c97a7e.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 216, + 562, + 393, + 576.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 216, + 576.0, + 393, + 590.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 320, + 606 + ], + "score": 1.0, + "content": "When (1) is a one-dimensional root finding problem", + "type": "text" + }, + { + "bbox": [ + 320, + 593, + 346, + 604 + ], + "score": 0.86, + "content": "\\langle d = 1 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "), (2) becomes particularly simple since", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 603, + 342, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 161, + 616 + ], + "score": 1.0, + "content": "we then have", + "type": "text" + }, + { + "bbox": [ + 162, + 603, + 233, + 616 + ], + "score": 0.94, + "content": "\\nabla x ^ { \\star } ( \\theta ) = B ^ { \\top } / A", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 603, + 264, + 616 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 264, + 604, + 273, + 614 + ], + "score": 0.81, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 603, + 342, + 616 + ], + "score": 1.0, + "content": "is a scalar value.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 620, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "We will show that existing and new implicit differentiation methods all reduce to this simple principle.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 630, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 507, + 646 + ], + "score": 1.0, + "content": "We call our approach hybrid, since it combines implicit differentiation (and as such requires solving a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 338, + 655 + ], + "score": 1.0, + "content": "linear system) with autodiff of the optimality conditions", + "type": "text" + }, + { + "bbox": [ + 338, + 643, + 347, + 653 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 643, + 505, + 655 + ], + "score": 1.0, + "content": ". Our approach is efficient as it can be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "added on top of any state-of-the-art solver and modular as the optimality condition specification is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "decoupled from the implicit differentiation mechanism. This contrasts with existing works, where", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 675, + 465, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 465, + 687 + ], + "score": 1.0, + "content": "the mathematical derivation and implementation are specific to each optimality condition.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 104, + 698, + 504, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 430, + 712 + ], + "score": 1.0, + "content": "Differentiating a fixed point. 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The function", + "type": "text" + }, + { + "bbox": [ + 475, + 271, + 505, + 283 + ], + "score": 0.92, + "content": "f ( x , \\theta )", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 295, + 294 + ], + "score": 1.0, + "content": "defines the objective function and the mapping", + "type": "text" + }, + { + "bbox": [ + 296, + 282, + 304, + 292 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 281, + 505, + 294 + ], + "score": 1.0, + "content": ", here simply equation (4), captures the optimality", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "conditions. 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Under the hood, we combine", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 266, + 402 + ], + "score": 1.0, + "content": "implicit differentiation and autodiff of", + "type": "text" + }, + { + "bbox": [ + 266, + 390, + 276, + 399 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "to automatically differentiate the optimization problem", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 400, + 345, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 345, + 413 + ], + "score": 1.0, + "content": "solution. A simple illustrative example is given in Figure 1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 334, + 506, + 413 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 422, + 504, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 221, + 436 + ], + "score": 1.0, + "content": "Differentiating a root. Let", + "type": "text" + }, + { + "bbox": [ + 222, + 422, + 297, + 434 + ], + "score": 0.92, + "content": "F \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 420, + 505, + 436 + ], + "score": 1.0, + "content": "be a user-provided mapping, capturing the optimality", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 434, + 446, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 326, + 447 + ], + "score": 1.0, + "content": "conditions of a problem. An optimal solution, denoted", + "type": "text" + }, + { + "bbox": [ + 326, + 434, + 349, + 446 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 434, + 432, + 447 + ], + "score": 1.0, + "content": ", should be a root of", + "type": "text" + }, + { + "bbox": [ + 433, + 435, + 441, + 444 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 434, + 446, + 447 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 420, + 505, + 447 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 269, + 448, + 342, + 461 + ], + "lines": [ + { + "bbox": [ + 269, + 448, + 342, + 461 + ], + "spans": [ + { + "bbox": [ + 269, + 448, + 342, + 461 + ], + "score": 0.91, + "content": "{ \\cal F } ( x ^ { \\star } ( \\theta ) , \\theta ) = 0 .", + "type": "interline_equation", + "image_path": "446159dd8937a24464d0474490428271a6a102724c6010dab3e32a02a36af2d6.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 269, + 448, + 342, + 461 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 463, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 507, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 154, + 477 + ], + "score": 1.0, + "content": "We can see", + "type": "text" + }, + { + "bbox": [ + 154, + 464, + 178, + 476 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 461, + 325, + 477 + ], + "score": 1.0, + "content": "as an implicitly defined function of", + "type": "text" + }, + { + "bbox": [ + 325, + 464, + 356, + 474 + ], + "score": 0.91, + "content": "\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 461, + 377, + 477 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 378, + 463, + 436, + 474 + ], + "score": 0.91, + "content": "x ^ { \\star } \\colon { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 461, + 507, + 477 + ], + "score": 1.0, + "content": ". 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\\partial _ { 1 } F ( x ^ { \\star } ( \\theta ) , \\theta ) } _ { A \\in \\mathbb { R } ^ { d \\times d } } \\underbrace { \\partial x ^ { \\star } ( \\theta ) } _ { J \\in \\mathbb { R } ^ { d \\times n } } = \\underbrace { \\partial _ { 2 } F ( x ^ { \\star } ( \\theta ) , \\theta ) } _ { B \\in \\mathbb { R } ^ { d \\times n } } .", + "type": "interline_equation", + "image_path": "97fb8b270917089f469ee014e79f21f03b3d3bc21b899a3796964a33d5c97a7e.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 216, + 562, + 393, + 576.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 216, + 576.0, + 393, + 590.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 320, + 606 + ], + "score": 1.0, + "content": "When (1) is a one-dimensional root finding problem", + "type": "text" + }, + { + "bbox": [ + 320, + 593, + 346, + 604 + ], + "score": 0.86, + "content": "\\langle d = 1 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "), (2) becomes particularly simple since", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 603, + 342, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 161, + 616 + ], + "score": 1.0, + "content": "we then have", + "type": "text" + }, + { + "bbox": [ + 162, + 603, + 233, + 616 + ], + "score": 0.94, + "content": "\\nabla x ^ { \\star } ( \\theta ) = B ^ { \\top } / A", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 603, + 264, + 616 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 264, + 604, + 273, + 614 + ], + "score": 0.81, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 603, + 342, + 616 + ], + "score": 1.0, + "content": "is a scalar value.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 592, + 505, + 616 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 620, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "We will show that existing and new implicit differentiation methods all reduce to this simple principle.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 630, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 507, + 646 + ], + "score": 1.0, + "content": "We call our approach hybrid, since it combines implicit differentiation (and as such requires solving a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 338, + 655 + ], + "score": 1.0, + "content": "linear system) with autodiff of the optimality conditions", + "type": "text" + }, + { + "bbox": [ + 338, + 643, + 347, + 653 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 643, + 505, + 655 + ], + "score": 1.0, + "content": ". Our approach is efficient as it can be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "added on top of any state-of-the-art solver and modular as the optimality condition specification is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "decoupled from the implicit differentiation mechanism. This contrasts with existing works, where", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 675, + 465, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 465, + 687 + ], + "score": 1.0, + "content": "the mathematical derivation and implementation are specific to each optimality condition.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 620, + 507, + 687 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 698, + 504, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 430, + 712 + ], + "score": 1.0, + "content": "Differentiating a fixed point. We will encounter numerous applications where", + "type": "text" + }, + { + "bbox": [ + 430, + 699, + 454, + 711 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 697, + 505, + 712 + ], + "score": 1.0, + "content": "is implicitly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 709, + 229, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 229, + 722 + ], + "score": 1.0, + "content": "defined through a fixed point:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 697, + 505, + 722 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 719, + 350, + 733 + ], + "lines": [ + { + "bbox": [ + 260, + 719, + 350, + 733 + ], + "spans": [ + { + "bbox": [ + 260, + 719, + 350, + 733 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta ) = T ( x ^ { \\star } ( \\theta ) , \\theta ) ,", + "type": "interline_equation", + "image_path": "609d578cf202525a4f3375bef14f39c5b10c200d2736646abfd45d4a9bc32541.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 260, + 719, + 350, + 733 + ], + "spans": [], + "index": 37 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 488, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 487, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 133, + 96 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 81, + 213, + 93 + ], + "score": 0.92, + "content": "T \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 80, + 487, + 96 + ], + "score": 1.0, + "content": ". This can be seen as a particular case of (1) by defining the residual", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 97, + 353, + 110 + ], + "lines": [ + { + "bbox": [ + 256, + 97, + 353, + 110 + ], + "spans": [ + { + "bbox": [ + 256, + 97, + 353, + 110 + ], + "score": 0.92, + "content": "F ( x , { \\boldsymbol { \\theta } } ) = T ( x , { \\boldsymbol { \\theta } } ) - x .", + "type": "interline_equation", + "image_path": "0c4605ac9c9419470cbe650e2534a0d735361af273562369bccdd0b35057c5d0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 256, + 97, + 353, + 110 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 113, + 274, + 124 + ], + "lines": [ + { + "bbox": [ + 106, + 113, + 275, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 275, + 126 + ], + "score": 1.0, + "content": "In this case, using the chain rule, we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 127, + 487, + 141 + ], + "lines": [ + { + "bbox": [ + 123, + 127, + 487, + 141 + ], + "spans": [ + { + "bbox": [ + 123, + 127, + 487, + 141 + ], + "score": 0.9, + "content": "A = - \\partial _ { 1 } F ( x ^ { \\star } ( \\theta ) , \\theta ) = I - \\partial _ { 1 } T ( x ^ { \\star } ( \\theta ) , \\theta ) \\quad \\mathrm { a n d } \\quad B = \\partial _ { 2 } F ( x ^ { \\star } ( \\theta ) , \\theta ) = \\partial _ { 2 } T ( x ^ { \\star } ( \\theta ) , \\theta ) .", + "type": "interline_equation", + "image_path": "ce8daec59d9a8a25ff95d9a7bb2cbf9a992612eca05cfdcaccc14e50d953adaa.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 123, + 127, + 487, + 141 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 162 + ], + "score": 1.0, + "content": "Computing JVPs and VJPs. In most practical scenarios, it is not necessary to explicitly form", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 161, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 446, + 173 + ], + "score": 1.0, + "content": "the Jacobian matrix, and instead it is sufficient to left-multiply or right-multiply by", + "type": "text" + }, + { + "bbox": [ + 447, + 161, + 466, + 172 + ], + "score": 0.91, + "content": "\\partial _ { 1 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 161, + 484, + 173 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 484, + 161, + 503, + 172 + ], + "score": 0.9, + "content": "\\partial _ { 2 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 161, + 506, + 173 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "These are called vector-Jacobian product (VJP) and Jacobian-vector product (JVP), and are useful for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 151, + 196 + ], + "score": 1.0, + "content": "integrating", + "type": "text" + }, + { + "bbox": [ + 151, + 183, + 176, + 195 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 183, + 465, + 196 + ], + "score": 1.0, + "content": "with reverse-mode and forward-mode autodiff, respectively. Oftentimes,", + "type": "text" + }, + { + "bbox": [ + 465, + 183, + 474, + 193 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "will be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 504, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 495, + 207 + ], + "score": 1.0, + "content": "explicitly defined. In this case, computing the VJP or JVP can be done via autodiff. In some cases,", + "type": "text" + }, + { + "bbox": [ + 495, + 194, + 504, + 204 + ], + "score": 0.81, + "content": "F", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 310, + 217 + ], + "score": 1.0, + "content": "may itself be implicitly defined, for instance when", + "type": "text" + }, + { + "bbox": [ + 310, + 205, + 319, + 215 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "involves the solution of a variational problem.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 216, + 432, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 432, + 228 + ], + "score": 1.0, + "content": "In this case, computing the VJP or JVP will itself involve implicit differentiation.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 265, + 245 + ], + "score": 1.0, + "content": "The right-multiplication (JVP) between", + "type": "text" + }, + { + "bbox": [ + 265, + 232, + 315, + 244 + ], + "score": 0.92, + "content": "J = \\partial x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 232, + 367, + 245 + ], + "score": 1.0, + "content": "and a vector", + "type": "text" + }, + { + "bbox": [ + 367, + 235, + 373, + 243 + ], + "score": 0.66, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 232, + 377, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 377, + 233, + 390, + 243 + ], + "score": 0.77, + "content": "J v _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 232, + 505, + 245 + ], + "score": 1.0, + "content": ", can be computed efficiently", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 150, + 256 + ], + "score": 1.0, + "content": "by solving", + "type": "text" + }, + { + "bbox": [ + 151, + 243, + 205, + 255 + ], + "score": 0.92, + "content": "A ( \\bar { J v } ) = B v", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 243, + 336, + 256 + ], + "score": 1.0, + "content": ". The left-multiplication (VJP) of", + "type": "text" + }, + { + "bbox": [ + 337, + 243, + 349, + 253 + ], + "score": 0.88, + "content": "v ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 243, + 382, + 256 + ], + "score": 1.0, + "content": "with J ,", + "type": "text" + }, + { + "bbox": [ + 383, + 243, + 402, + 254 + ], + "score": 0.56, + "content": "v ^ { \\top } J", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 243, + 505, + 256 + ], + "score": 1.0, + "content": ", can be computed by first", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 252, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 138, + 268 + ], + "score": 1.0, + "content": "solving", + "type": "text" + }, + { + "bbox": [ + 139, + 254, + 178, + 265 + ], + "score": 0.91, + "content": "A ^ { \\top } u = v", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 252, + 265, + 268 + ], + "score": 1.0, + "content": ". Then, we can obtain", + "type": "text" + }, + { + "bbox": [ + 266, + 254, + 285, + 265 + ], + "score": 0.91, + "content": "\\mathbf { \\Delta } _ { v } \\top { } _ { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 252, + 299, + 268 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 299, + 254, + 393, + 264 + ], + "score": 0.9, + "content": "\\boldsymbol { v } ^ { \\top } \\boldsymbol { J } = \\boldsymbol { u } ^ { \\top } \\boldsymbol { A } \\boldsymbol { J } = \\boldsymbol { u } ^ { \\top } \\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 252, + 460, + 268 + ], + "score": 1.0, + "content": ". Note that when", + "type": "text" + }, + { + "bbox": [ + 460, + 255, + 469, + 264 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 252, + 506, + 268 + ], + "score": 1.0, + "content": "changes", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 121, + 279 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 122, + 266, + 130, + 275 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 264, + 147, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 148, + 267, + 155, + 275 + ], + "score": 0.74, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 264, + 321, + 279 + ], + "score": 1.0, + "content": "remain the same, we do not need to solve", + "type": "text" + }, + { + "bbox": [ + 322, + 265, + 361, + 276 + ], + "score": 0.89, + "content": "A ^ { \\top } u = v", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 264, + 505, + 279 + ], + "score": 1.0, + "content": "once again. This allows to compute", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 389, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 389, + 289 + ], + "score": 1.0, + "content": "the VJP w.r.t. different variables while solving only one linear system.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "To solve these linear systems, we can use the conjugate gradient method (Hestenes et al., 1952) when", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 107, + 305, + 115, + 314 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "is symmetric positive semi-definite and GMRES (Saad & Schultz, 1986) or BiCGSTAB (Vorst &", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "van der Vorst, 1992) otherwise. These algorithms are all matrix-free: they only require matrix-vector", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 240, + 338 + ], + "score": 1.0, + "content": "products. Thus, all we need from", + "type": "text" + }, + { + "bbox": [ + 240, + 327, + 249, + 336 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "is its JVPs or VJPs. An alternative to GMRES/BiCGSTAB is to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 214, + 351 + ], + "score": 1.0, + "content": "solve the normal equation", + "type": "text" + }, + { + "bbox": [ + 214, + 337, + 268, + 348 + ], + "score": 0.91, + "content": "A A ^ { \\top } u = A v", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 336, + 506, + 351 + ], + "score": 1.0, + "content": "using conjugate gradient. We implement this using JAX’s", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 427, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 427, + 360 + ], + "score": 1.0, + "content": "automatic transpose routine jax.linear_transpose (Frostig et al., 2021).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 482, + 384 + ], + "score": 1.0, + "content": "Pre-processing and post-processing mappings. Oftentimes, the goal is not to differentiate", + "type": "text" + }, + { + "bbox": [ + 483, + 372, + 489, + 381 + ], + "score": 0.72, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "per", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 286, + 395 + ], + "score": 1.0, + "content": "se, but the parameters of a function producing", + "type": "text" + }, + { + "bbox": [ + 287, + 383, + 293, + 392 + ], + "score": 0.61, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 382, + 505, + 395 + ], + "score": 1.0, + "content": ". One example of such pre-processing is to convert the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "parameters to be differentiated from one form to another canonical form, such as a quadratic program", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 403, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 480, + 418 + ], + "score": 1.0, + "content": "(Amos & Kolter, 2017) or a conic program (Agrawal et al., 2019a). Another example is when", + "type": "text" + }, + { + "bbox": [ + 481, + 404, + 505, + 416 + ], + "score": 0.91, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 415, + 507, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 358, + 428 + ], + "score": 1.0, + "content": "is used as the output of a neural network layer, in which case", + "type": "text" + }, + { + "bbox": [ + 358, + 416, + 365, + 425 + ], + "score": 0.76, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 415, + 507, + 428 + ], + "score": 1.0, + "content": "is produced by the previous layer.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 149, + 439 + ], + "score": 1.0, + "content": "Likewise,", + "type": "text" + }, + { + "bbox": [ + 149, + 426, + 173, + 438 + ], + "score": 0.93, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "will often not be the final output we want to differentiate. One example of such", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 436, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 436, + 201, + 452 + ], + "score": 1.0, + "content": "post-processing is when", + "type": "text" + }, + { + "bbox": [ + 202, + 437, + 226, + 449 + ], + "score": 0.93, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 436, + 506, + 452 + ], + "score": 1.0, + "content": "is the solution of a dual program and we apply the dual-primal mapping", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "to recover the solution of the primal program. Another example is the application of a loss function,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 458, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 180, + 473 + ], + "score": 1.0, + "content": "in order to reduce", + "type": "text" + }, + { + "bbox": [ + 181, + 459, + 205, + 471 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 458, + 505, + 473 + ], + "score": 1.0, + "content": "to a scalar value. We leave the differentiation of such pre/post-processing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 437, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 437, + 483 + ], + "score": 1.0, + "content": "mappings to the autodiff system, allowing to compose functions in complex ways.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "Implementation details. When a solver function is decorated with @custom_root, we use", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "jax.custom_jvp and jax.custom_vjp to automatically add custom JVP and VJP rules to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "the function, overriding JAX’s default behavior. As mentioned above, we use linear system solvers", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 394, + 538 + ], + "score": 1.0, + "content": "based on matrix-vector products and therefore we only need access to", + "type": "text" + }, + { + "bbox": [ + 395, + 527, + 404, + 536 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "through the JVP or VJP", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 536, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 127, + 551 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 537, + 146, + 548 + ], + "score": 0.9, + "content": "\\partial _ { 1 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 536, + 165, + 551 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 166, + 537, + 185, + 548 + ], + "score": 0.89, + "content": "\\partial _ { 2 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 536, + 284, + 551 + ], + "score": 1.0, + "content": ". This is done by using", + "type": "text" + }, + { + "bbox": [ + 285, + 538, + 327, + 549 + ], + "score": 0.84, + "content": "\\ j \\mathtt { a x } . \\mathtt { j v p }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 536, + 347, + 551 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 347, + 538, + 389, + 549 + ], + "score": 0.84, + "content": "{ \\dot { \\mathsf { J } } } \\mathsf { a x } . \\mathsf { v } { \\dot { \\mathsf { J } } } { \\mathsf { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 536, + 506, + 551 + ], + "score": 1.0, + "content": ", respectively. Note that, as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 547, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 231, + 562 + ], + "score": 1.0, + "content": "in Figure 1, the definition of", + "type": "text" + }, + { + "bbox": [ + 231, + 549, + 241, + 558 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 547, + 405, + 562 + ], + "score": 1.0, + "content": "will often include a gradient mapping", + "type": "text" + }, + { + "bbox": [ + 406, + 548, + 448, + 560 + ], + "score": 0.93, + "content": "\\bar { \\nabla } _ { 1 } f ( x , \\bar { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 547, + 506, + 562 + ], + "score": 1.0, + "content": ". Thankfully,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "JAX supports second-order derivatives transparently. For convenience, our library also provides a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "@custom_fixed_point decorator, for adding implicit differentiation on top of a solver, given a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 341, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 187, + 594 + ], + "score": 1.0, + "content": "fixed point iteration", + "type": "text" + }, + { + "bbox": [ + 187, + 581, + 196, + 591 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 580, + 341, + 594 + ], + "score": 1.0, + "content": "; see code examples in Appendix A.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 599, + 179, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 180, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 180, + 613 + ], + "score": 1.0, + "content": "2.2 EXAMPLES", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 279, + 633 + ], + "score": 1.0, + "content": "We now give various examples of mapping", + "type": "text" + }, + { + "bbox": [ + 280, + 621, + 289, + 631 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 619, + 381, + 633 + ], + "score": 1.0, + "content": "or fixed point iteration", + "type": "text" + }, + { + "bbox": [ + 381, + 621, + 389, + 630 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 619, + 506, + 633 + ], + "score": 1.0, + "content": ", recovering existing implicit", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 356, + 644 + ], + "score": 1.0, + "content": "differentiation methods and creating new ones. Each choice of", + "type": "text" + }, + { + "bbox": [ + 356, + 632, + 365, + 641 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 631, + 376, + 644 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 377, + 632, + 385, + 641 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "implies different trade-offs in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 642, + 481, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 481, + 655 + ], + "score": 1.0, + "content": "terms of computational oracles; see Table 1. Source code examples are given in Appendix A.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 503, + 677 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "Stationary point condition. The simplest example is to differentiate through the implicit function", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 679, + 356, + 701 + ], + "lines": [ + { + "bbox": [ + 254, + 679, + 356, + 701 + ], + "spans": [ + { + "bbox": [ + 254, + 679, + 356, + 701 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta ) = \\underset { x \\in \\mathbb { R } ^ { d } } { \\operatorname { a r g m i n } } f ( x , \\theta ) ,", + "type": "interline_equation", + "image_path": "07e627d53dddc2d6519bd8cf9549e9a0d802100c83341d4451b04e472450ec43.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 254, + 679, + 356, + 701 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 705, + 485, + 717 + ], + "lines": [ + { + "bbox": [ + 105, + 702, + 486, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 133, + 720 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 704, + 207, + 717 + ], + "score": 0.91, + "content": "f \\colon { \\mathbb { R } ^ { d } } \\times { \\mathbb { R } ^ { n } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 702, + 350, + 720 + ], + "score": 1.0, + "content": "is twice differentiable. 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This can be seen as a particular case of (1) by defining the residual", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 487, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 97, + 353, + 110 + ], + "lines": [ + { + "bbox": [ + 256, + 97, + 353, + 110 + ], + "spans": [ + { + "bbox": [ + 256, + 97, + 353, + 110 + ], + "score": 0.92, + "content": "F ( x , { \\boldsymbol { \\theta } } ) = T ( x , { \\boldsymbol { \\theta } } ) - x .", + "type": "interline_equation", + "image_path": "0c4605ac9c9419470cbe650e2534a0d735361af273562369bccdd0b35057c5d0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 256, + 97, + 353, + 110 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 113, + 274, + 124 + ], + "lines": [ + { + "bbox": [ + 106, + 113, + 275, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 275, + 126 + ], + "score": 1.0, + "content": "In this case, using the chain rule, we have", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 113, + 275, + 126 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 127, + 487, + 141 + ], + "lines": [ + { + "bbox": [ + 123, + 127, + 487, + 141 + ], + "spans": [ + { + "bbox": [ + 123, + 127, + 487, + 141 + ], + "score": 0.9, + "content": "A = - \\partial _ { 1 } F ( x ^ { \\star } ( \\theta ) , \\theta ) = I - \\partial _ { 1 } T ( x ^ { \\star } ( \\theta ) , \\theta ) \\quad \\mathrm { a n d } \\quad B = \\partial _ { 2 } F ( x ^ { \\star } ( \\theta ) , \\theta ) = \\partial _ { 2 } T ( x ^ { \\star } ( \\theta ) , \\theta ) .", + "type": "interline_equation", + "image_path": "ce8daec59d9a8a25ff95d9a7bb2cbf9a992612eca05cfdcaccc14e50d953adaa.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 123, + 127, + 487, + 141 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 162 + ], + "score": 1.0, + "content": "Computing JVPs and VJPs. In most practical scenarios, it is not necessary to explicitly form", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 161, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 446, + 173 + ], + "score": 1.0, + "content": "the Jacobian matrix, and instead it is sufficient to left-multiply or right-multiply by", + "type": "text" + }, + { + "bbox": [ + 447, + 161, + 466, + 172 + ], + "score": 0.91, + "content": "\\partial _ { 1 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 161, + 484, + 173 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 484, + 161, + 503, + 172 + ], + "score": 0.9, + "content": "\\partial _ { 2 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 161, + 506, + 173 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "These are called vector-Jacobian product (VJP) and Jacobian-vector product (JVP), and are useful for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 151, + 196 + ], + "score": 1.0, + "content": "integrating", + "type": "text" + }, + { + "bbox": [ + 151, + 183, + 176, + 195 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 183, + 465, + 196 + ], + "score": 1.0, + "content": "with reverse-mode and forward-mode autodiff, respectively. Oftentimes,", + "type": "text" + }, + { + "bbox": [ + 465, + 183, + 474, + 193 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "will be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 504, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 495, + 207 + ], + "score": 1.0, + "content": "explicitly defined. In this case, computing the VJP or JVP can be done via autodiff. In some cases,", + "type": "text" + }, + { + "bbox": [ + 495, + 194, + 504, + 204 + ], + "score": 0.81, + "content": "F", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 310, + 217 + ], + "score": 1.0, + "content": "may itself be implicitly defined, for instance when", + "type": "text" + }, + { + "bbox": [ + 310, + 205, + 319, + 215 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "involves the solution of a variational problem.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 216, + 432, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 432, + 228 + ], + "score": 1.0, + "content": "In this case, computing the VJP or JVP will itself involve implicit differentiation.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 150, + 506, + 228 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 265, + 245 + ], + "score": 1.0, + "content": "The right-multiplication (JVP) between", + "type": "text" + }, + { + "bbox": [ + 265, + 232, + 315, + 244 + ], + "score": 0.92, + "content": "J = \\partial x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 232, + 367, + 245 + ], + "score": 1.0, + "content": "and a vector", + "type": "text" + }, + { + "bbox": [ + 367, + 235, + 373, + 243 + ], + "score": 0.66, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 232, + 377, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 377, + 233, + 390, + 243 + ], + "score": 0.77, + "content": "J v _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 232, + 505, + 245 + ], + "score": 1.0, + "content": ", can be computed efficiently", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 150, + 256 + ], + "score": 1.0, + "content": "by solving", + "type": "text" + }, + { + "bbox": [ + 151, + 243, + 205, + 255 + ], + "score": 0.92, + "content": "A ( \\bar { J v } ) = B v", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 243, + 336, + 256 + ], + "score": 1.0, + "content": ". The left-multiplication (VJP) of", + "type": "text" + }, + { + "bbox": [ + 337, + 243, + 349, + 253 + ], + "score": 0.88, + "content": "v ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 243, + 382, + 256 + ], + "score": 1.0, + "content": "with J ,", + "type": "text" + }, + { + "bbox": [ + 383, + 243, + 402, + 254 + ], + "score": 0.56, + "content": "v ^ { \\top } J", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 243, + 505, + 256 + ], + "score": 1.0, + "content": ", can be computed by first", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 252, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 138, + 268 + ], + "score": 1.0, + "content": "solving", + "type": "text" + }, + { + "bbox": [ + 139, + 254, + 178, + 265 + ], + "score": 0.91, + "content": "A ^ { \\top } u = v", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 252, + 265, + 268 + ], + "score": 1.0, + "content": ". Then, we can obtain", + "type": "text" + }, + { + "bbox": [ + 266, + 254, + 285, + 265 + ], + "score": 0.91, + "content": "\\mathbf { \\Delta } _ { v } \\top { } _ { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 252, + 299, + 268 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 299, + 254, + 393, + 264 + ], + "score": 0.9, + "content": "\\boldsymbol { v } ^ { \\top } \\boldsymbol { J } = \\boldsymbol { u } ^ { \\top } \\boldsymbol { A } \\boldsymbol { J } = \\boldsymbol { u } ^ { \\top } \\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 252, + 460, + 268 + ], + "score": 1.0, + "content": ". Note that when", + "type": "text" + }, + { + "bbox": [ + 460, + 255, + 469, + 264 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 252, + 506, + 268 + ], + "score": 1.0, + "content": "changes", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 121, + 279 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 122, + 266, + 130, + 275 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 264, + 147, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 148, + 267, + 155, + 275 + ], + "score": 0.74, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 264, + 321, + 279 + ], + "score": 1.0, + "content": "remain the same, we do not need to solve", + "type": "text" + }, + { + "bbox": [ + 322, + 265, + 361, + 276 + ], + "score": 0.89, + "content": "A ^ { \\top } u = v", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 264, + 505, + 279 + ], + "score": 1.0, + "content": "once again. This allows to compute", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 275, + 389, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 389, + 289 + ], + "score": 1.0, + "content": "the VJP w.r.t. different variables while solving only one linear system.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 232, + 506, + 289 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "To solve these linear systems, we can use the conjugate gradient method (Hestenes et al., 1952) when", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 107, + 305, + 115, + 314 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "is symmetric positive semi-definite and GMRES (Saad & Schultz, 1986) or BiCGSTAB (Vorst &", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "van der Vorst, 1992) otherwise. These algorithms are all matrix-free: they only require matrix-vector", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 240, + 338 + ], + "score": 1.0, + "content": "products. Thus, all we need from", + "type": "text" + }, + { + "bbox": [ + 240, + 327, + 249, + 336 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "is its JVPs or VJPs. An alternative to GMRES/BiCGSTAB is to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 214, + 351 + ], + "score": 1.0, + "content": "solve the normal equation", + "type": "text" + }, + { + "bbox": [ + 214, + 337, + 268, + 348 + ], + "score": 0.91, + "content": "A A ^ { \\top } u = A v", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 336, + 506, + 351 + ], + "score": 1.0, + "content": "using conjugate gradient. We implement this using JAX’s", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 427, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 427, + 360 + ], + "score": 1.0, + "content": "automatic transpose routine jax.linear_transpose (Frostig et al., 2021).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 293, + 506, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 482, + 384 + ], + "score": 1.0, + "content": "Pre-processing and post-processing mappings. Oftentimes, the goal is not to differentiate", + "type": "text" + }, + { + "bbox": [ + 483, + 372, + 489, + 381 + ], + "score": 0.72, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "per", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 286, + 395 + ], + "score": 1.0, + "content": "se, but the parameters of a function producing", + "type": "text" + }, + { + "bbox": [ + 287, + 383, + 293, + 392 + ], + "score": 0.61, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 382, + 505, + 395 + ], + "score": 1.0, + "content": ". One example of such pre-processing is to convert the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "parameters to be differentiated from one form to another canonical form, such as a quadratic program", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 403, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 480, + 418 + ], + "score": 1.0, + "content": "(Amos & Kolter, 2017) or a conic program (Agrawal et al., 2019a). Another example is when", + "type": "text" + }, + { + "bbox": [ + 481, + 404, + 505, + 416 + ], + "score": 0.91, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 415, + 507, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 358, + 428 + ], + "score": 1.0, + "content": "is used as the output of a neural network layer, in which case", + "type": "text" + }, + { + "bbox": [ + 358, + 416, + 365, + 425 + ], + "score": 0.76, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 415, + 507, + 428 + ], + "score": 1.0, + "content": "is produced by the previous layer.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 149, + 439 + ], + "score": 1.0, + "content": "Likewise,", + "type": "text" + }, + { + "bbox": [ + 149, + 426, + 173, + 438 + ], + "score": 0.93, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "will often not be the final output we want to differentiate. One example of such", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 436, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 436, + 201, + 452 + ], + "score": 1.0, + "content": "post-processing is when", + "type": "text" + }, + { + "bbox": [ + 202, + 437, + 226, + 449 + ], + "score": 0.93, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 436, + 506, + 452 + ], + "score": 1.0, + "content": "is the solution of a dual program and we apply the dual-primal mapping", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "to recover the solution of the primal program. Another example is the application of a loss function,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 458, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 180, + 473 + ], + "score": 1.0, + "content": "in order to reduce", + "type": "text" + }, + { + "bbox": [ + 181, + 459, + 205, + 471 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 458, + 505, + 473 + ], + "score": 1.0, + "content": "to a scalar value. We leave the differentiation of such pre/post-processing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 437, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 437, + 483 + ], + "score": 1.0, + "content": "mappings to the autodiff system, allowing to compose functions in complex ways.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 371, + 507, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "Implementation details. When a solver function is decorated with @custom_root, we use", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "jax.custom_jvp and jax.custom_vjp to automatically add custom JVP and VJP rules to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "the function, overriding JAX’s default behavior. As mentioned above, we use linear system solvers", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 394, + 538 + ], + "score": 1.0, + "content": "based on matrix-vector products and therefore we only need access to", + "type": "text" + }, + { + "bbox": [ + 395, + 527, + 404, + 536 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "through the JVP or VJP", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 536, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 127, + 551 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 537, + 146, + 548 + ], + "score": 0.9, + "content": "\\partial _ { 1 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 536, + 165, + 551 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 166, + 537, + 185, + 548 + ], + "score": 0.89, + "content": "\\partial _ { 2 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 536, + 284, + 551 + ], + "score": 1.0, + "content": ". This is done by using", + "type": "text" + }, + { + "bbox": [ + 285, + 538, + 327, + 549 + ], + "score": 0.84, + "content": "\\ j \\mathtt { a x } . \\mathtt { j v p }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 536, + 347, + 551 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 347, + 538, + 389, + 549 + ], + "score": 0.84, + "content": "{ \\dot { \\mathsf { J } } } \\mathsf { a x } . \\mathsf { v } { \\dot { \\mathsf { J } } } { \\mathsf { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 536, + 506, + 551 + ], + "score": 1.0, + "content": ", respectively. Note that, as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 547, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 231, + 562 + ], + "score": 1.0, + "content": "in Figure 1, the definition of", + "type": "text" + }, + { + "bbox": [ + 231, + 549, + 241, + 558 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 547, + 405, + 562 + ], + "score": 1.0, + "content": "will often include a gradient mapping", + "type": "text" + }, + { + "bbox": [ + 406, + 548, + 448, + 560 + ], + "score": 0.93, + "content": "\\bar { \\nabla } _ { 1 } f ( x , \\bar { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 547, + 506, + 562 + ], + "score": 1.0, + "content": ". Thankfully,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "JAX supports second-order derivatives transparently. For convenience, our library also provides a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "@custom_fixed_point decorator, for adding implicit differentiation on top of a solver, given a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 341, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 187, + 594 + ], + "score": 1.0, + "content": "fixed point iteration", + "type": "text" + }, + { + "bbox": [ + 187, + 581, + 196, + 591 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 580, + 341, + 594 + ], + "score": 1.0, + "content": "; see code examples in Appendix A.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 492, + 506, + 594 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 599, + 179, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 180, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 180, + 613 + ], + "score": 1.0, + "content": "2.2 EXAMPLES", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 279, + 633 + ], + "score": 1.0, + "content": "We now give various examples of mapping", + "type": "text" + }, + { + "bbox": [ + 280, + 621, + 289, + 631 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 619, + 381, + 633 + ], + "score": 1.0, + "content": "or fixed point iteration", + "type": "text" + }, + { + "bbox": [ + 381, + 621, + 389, + 630 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 619, + 506, + 633 + ], + "score": 1.0, + "content": ", recovering existing implicit", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 356, + 644 + ], + "score": 1.0, + "content": "differentiation methods and creating new ones. Each choice of", + "type": "text" + }, + { + "bbox": [ + 356, + 632, + 365, + 641 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 631, + 376, + 644 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 377, + 632, + 385, + 641 + ], + "score": 0.83, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "implies different trade-offs in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 642, + 481, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 481, + 655 + ], + "score": 1.0, + "content": "terms of computational oracles; see Table 1. Source code examples are given in Appendix A.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 619, + 506, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 664, + 503, + 677 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "Stationary point condition. The simplest example is to differentiate through the implicit function", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45, + "bbox_fs": [ + 106, + 664, + 505, + 678 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 679, + 356, + 701 + ], + "lines": [ + { + "bbox": [ + 254, + 679, + 356, + 701 + ], + "spans": [ + { + "bbox": [ + 254, + 679, + 356, + 701 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta ) = \\underset { x \\in \\mathbb { R } ^ { d } } { \\operatorname { a r g m i n } } f ( x , \\theta ) ,", + "type": "interline_equation", + "image_path": "07e627d53dddc2d6519bd8cf9549e9a0d802100c83341d4451b04e472450ec43.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 254, + 679, + 356, + 701 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 705, + 485, + 717 + ], + "lines": [ + { + "bbox": [ + 105, + 702, + 486, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 133, + 720 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 704, + 207, + 717 + ], + "score": 0.91, + "content": "f \\colon { \\mathbb { R } ^ { d } } \\times { \\mathbb { R } ^ { n } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 702, + 350, + 720 + ], + "score": 1.0, + "content": "is twice differentiable. In this case,", + "type": "text" + }, + { + "bbox": [ + 350, + 706, + 359, + 715 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 702, + 486, + 720 + ], + "score": 1.0, + "content": "is simply the gradient mapping", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 702, + 486, + 720 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 719, + 351, + 733 + ], + "lines": [ + { + "bbox": [ + 260, + 719, + 351, + 733 + ], + "spans": [ + { + "bbox": [ + 260, + 719, + 351, + 733 + ], + "score": 0.91, + "content": "F ( x , \\theta ) = \\nabla _ { 1 } f ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "c61270e26b9d8b722985b0b7fad79558a593c097b0ccde92612e28bb3c880f28.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 260, + 719, + 351, + 733 + ], + "spans": [], + "index": 48 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 140, + 105, + 472, + 213 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 80, + 504, + 93 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 1: Summary of optimality condition mappings. Oracles are accessed through their JVP or VJP.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 140, + 105, + 472, + 213 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 105, + 472, + 213 + ], + "spans": [ + { + "bbox": [ + 140, + 105, + 472, + 213 + ], + "score": 0.983, + "html": "
NameEquationSolution neededOracles needed
Stationary(4), (5)PrimalV1f
KKT(6)Primal and dualVif,H,G,aH,0G
Proximal gradient(7)PrimalVif,proxng
Projected gradient(9)PrimalVif,projc
Mirror descent(11)PrimalVif,proje,
Newton(14)Primal[V²f(x,0)]-1,Vif(x,0)
Block proximal gradient(15)Primal[Vif]j,[proXnglj
Conic programming(18)Residual map rootprojRP × K* XR+
", + "type": "table", + "image_path": "944efd29488c585cb380b987bceed6d0f68e26858bcf0e5a0961d828d2468403.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 140, + 105, + 472, + 141.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 140, + 141.0, + 472, + 177.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 140, + 177.0, + 472, + 213.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 504, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 164, + 249 + ], + "score": 1.0, + "content": "We then have", + "type": "text" + }, + { + "bbox": [ + 164, + 235, + 262, + 248 + ], + "score": 0.91, + "content": "\\partial _ { 1 } F ( x , \\theta ) = \\nabla _ { 1 } ^ { 2 } f ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 235, + 281, + 249 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 236, + 388, + 248 + ], + "score": 0.91, + "content": "\\partial _ { 2 } F ( x , \\theta ) = \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 235, + 455, + 249 + ], + "score": 1.0, + "content": ", the Hessian of", + "type": "text" + }, + { + "bbox": [ + 455, + 236, + 462, + 248 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "in its first", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 330, + 259 + ], + "score": 1.0, + "content": "argument and the Jacobian in the second argument of", + "type": "text" + }, + { + "bbox": [ + 330, + 247, + 372, + 259 + ], + "score": 0.92, + "content": "\\nabla _ { 1 } f ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 247, + 506, + 259 + ], + "score": 1.0, + "content": ". In practice, we use autodiff to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "compute Jacobian products automatically. Equivalently, we can use the gradient descent fixed point", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 272, + 361, + 285 + ], + "lines": [ + { + "bbox": [ + 248, + 272, + 361, + 285 + ], + "spans": [ + { + "bbox": [ + 248, + 272, + 361, + 285 + ], + "score": 0.92, + "content": "T ( x , \\theta ) = x - \\eta \\nabla _ { 1 } f ( x , \\theta ) ,", + "type": "interline_equation", + "image_path": "ce04bff7c05cc5a9c1794520a6c422fb269f379b4b18d673f60c9e72e386ad49.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 248, + 272, + 361, + 285 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 288, + 504, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 133, + 301 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 133, + 289, + 158, + 300 + ], + "score": 0.89, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 287, + 450, + 301 + ], + "score": 1.0, + "content": ". Using (3), it is easy to check that we obtain the same linear system since", + "type": "text" + }, + { + "bbox": [ + 450, + 290, + 457, + 299 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 287, + 506, + 301 + ], + "score": 1.0, + "content": "cancels out.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 506, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "score": 1.0, + "content": "KKT conditions. We now show that the KKT conditions, manually differentiated in several works", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 495, + 334 + ], + "score": 1.0, + "content": "(Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae", + "type": "text" + }, + { + "bbox": [ + 495, + 323, + 505, + 333 + ], + "score": 0.32, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 332, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 506, + 346 + ], + "score": 1.0, + "content": "Martins, 2020), fit our framework. As we will see, the key will be to group the optimal primal and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 343, + 366, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 191, + 357 + ], + "score": 1.0, + "content": "dual variables as our", + "type": "text" + }, + { + "bbox": [ + 191, + 344, + 215, + 356 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 343, + 366, + 357 + ], + "score": 1.0, + "content": ". Let us consider the general problem", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 358, + 419, + 379 + ], + "lines": [ + { + "bbox": [ + 190, + 358, + 419, + 379 + ], + "spans": [ + { + "bbox": [ + 190, + 358, + 419, + 379 + ], + "score": 0.9, + "content": "\\operatorname { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } } f ( z , \\theta ) \\quad { \\mathrm { s u b j e c t ~ t o } } \\quad G ( z , \\theta ) \\leq 0 , \\ H ( z , \\theta ) = 0 ,", + "type": "interline_equation", + "image_path": "4dde206746cf2aba2d0f1a638f31205f33983dc509217bd52a1f81149053551e.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 190, + 358, + 419, + 379 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 504, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 503, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 132, + 394 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 382, + 162, + 392 + ], + "score": 0.91, + "content": "z \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 381, + 251, + 394 + ], + "score": 1.0, + "content": "is the primal variable,", + "type": "text" + }, + { + "bbox": [ + 251, + 382, + 323, + 393 + ], + "score": 0.78, + "content": "f \\colon { \\mathbb { R } } ^ { p } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 381, + 326, + 394 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 327, + 382, + 405, + 392 + ], + "score": 0.9, + "content": "G \\colon \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 381, + 423, + 394 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 382, + 503, + 392 + ], + "score": 0.88, + "content": "H \\colon \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { q }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 392, + 430, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 430, + 405 + ], + "score": 1.0, + "content": "The stationarity, primal feasibility and complementary slackness conditions give", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 405, + 408, + 451 + ], + "lines": [ + { + "bbox": [ + 203, + 405, + 408, + 451 + ], + "spans": [ + { + "bbox": [ + 203, + 405, + 408, + 451 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta ) + \\left[ \\partial _ { 1 } G ( z , \\theta ) \\right] ^ { \\top } \\boldsymbol { \\lambda } + \\left[ \\partial _ { 1 } H ( z , \\theta ) \\right] ^ { \\top } \\boldsymbol { \\nu } = 0 } \\\\ { H ( z , \\theta ) = 0 } \\\\ { \\boldsymbol { \\lambda } \\circ G ( z , \\theta ) = 0 , } \\end{array}", + "type": "interline_equation", + "image_path": "abb0c185cb5cc8c66837b68d239d550b3b91169c3424629a82110a4a9a606fcd.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 203, + 405, + 408, + 420.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 203, + 420.3333333333333, + 408, + 435.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 203, + 435.66666666666663, + 408, + 450.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 134, + 464 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 452, + 167, + 462 + ], + "score": 0.91, + "content": "\\nu \\in \\mathbb { R } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 452, + 187, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 452, + 223, + 465 + ], + "score": 0.93, + "content": "\\lambda \\in \\mathbb { R } _ { + } ^ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "are the dual variables, also known as KKT multipliers. The primal", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 463, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 504, + 475 + ], + "score": 1.0, + "content": "and dual feasibility conditions can be ignored almost everywhere. The system of (potentially", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "nonlinear) equations (6) fits our framework, as we can group the primal and dual solutions as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 484, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 228, + 497 + ], + "score": 0.89, + "content": "x ^ { \\star } ( \\theta ) = ( z ^ { \\star } ( { \\bar { \\theta } } ) , \\nu ^ { \\star } ( \\theta ) , \\lambda ^ { \\star } ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 484, + 345, + 497 + ], + "score": 1.0, + "content": "to form the root of a function", + "type": "text" + }, + { + "bbox": [ + 346, + 485, + 394, + 497 + ], + "score": 0.93, + "content": "F ( x ^ { \\star } ( \\theta ) , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 484, + 424, + 497 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 425, + 484, + 504, + 495 + ], + "score": 0.91, + "content": "F \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 124, + 508 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 497, + 184, + 507 + ], + "score": 0.91, + "content": "d = p + q + r", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 496, + 505, + 508 + ], + "score": 1.0, + "content": ". The primal and dual solutions can be obtained from a generic solver, such as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 338, + 520 + ], + "score": 1.0, + "content": "an interior point method. In practice, the above mapping", + "type": "text" + }, + { + "bbox": [ + 338, + 507, + 347, + 517 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "will be defined directly in Python (see", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 518, + 428, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 224, + 531 + ], + "score": 1.0, + "content": "Figure 7 in Appendix A) and", + "type": "text" + }, + { + "bbox": [ + 225, + 518, + 234, + 528 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 518, + 428, + 531 + ], + "score": 1.0, + "content": "will be differentiated automatically via autodiff.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 506, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "Proximal gradient fixed point. Unfortunately, not all algorithms return both primal and dual", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "solutions. Moreover, if the objective contains non-smooth terms, proximal gradient descent may be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "score": 1.0, + "content": "more efficient. We now discuss its fixed point (Niculae & Blondel, 2017; Bertrand et al., 2020; 2021).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 573, + 244, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 122, + 586 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 574, + 146, + 586 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 573, + 244, + 586 + ], + "score": 1.0, + "content": "be implicitly defined as", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 587, + 377, + 609 + ], + "lines": [ + { + "bbox": [ + 233, + 587, + 377, + 609 + ], + "spans": [ + { + "bbox": [ + 233, + 587, + 377, + 609 + ], + "score": 0.93, + "content": "x ^ { \\star } ( \\theta ) : = \\operatornamewithlimits { a r g m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x , \\theta ) + g ( x , \\theta ) ,", + "type": "interline_equation", + "image_path": "92f4e6a348b59884f2774aeba492d29f0d7c2a633b2cd24da89ef622723f2cd8.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 233, + 587, + 377, + 609 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 132, + 628 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 613, + 206, + 625 + ], + "score": 0.91, + "content": "f \\colon { \\mathbb { R } ^ { d } } \\times { \\mathbb { R } ^ { n } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 610, + 342, + 628 + ], + "score": 1.0, + "content": "is twice-differentiable convex and", + "type": "text" + }, + { + "bbox": [ + 343, + 613, + 415, + 625 + ], + "score": 0.91, + "content": "g \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 610, + 506, + 628 + ], + "score": 1.0, + "content": "is convex but possibly", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 623, + 389, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 369, + 637 + ], + "score": 1.0, + "content": "non-smooth. 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Oracles are accessed through their JVP or VJP.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 140, + 105, + 472, + 213 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 105, + 472, + 213 + ], + "spans": [ + { + "bbox": [ + 140, + 105, + 472, + 213 + ], + "score": 0.983, + "html": "
NameEquationSolution neededOracles needed
Stationary(4), (5)PrimalV1f
KKT(6)Primal and dualVif,H,G,aH,0G
Proximal gradient(7)PrimalVif,proxng
Projected gradient(9)PrimalVif,projc
Mirror descent(11)PrimalVif,proje,
Newton(14)Primal[V²f(x,0)]-1,Vif(x,0)
Block proximal gradient(15)Primal[Vif]j,[proXnglj
Conic programming(18)Residual map rootprojRP × K* XR+
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In practice, we use autodiff to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "compute Jacobian products automatically. Equivalently, we can use the gradient descent fixed point", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 235, + 506, + 271 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 272, + 361, + 285 + ], + "lines": [ + { + "bbox": [ + 248, + 272, + 361, + 285 + ], + "spans": [ + { + "bbox": [ + 248, + 272, + 361, + 285 + ], + "score": 0.92, + "content": "T ( x , \\theta ) = x - \\eta \\nabla _ { 1 } f ( x , \\theta ) ,", + "type": "interline_equation", + "image_path": "ce04bff7c05cc5a9c1794520a6c422fb269f379b4b18d673f60c9e72e386ad49.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 248, + 272, + 361, + 285 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 288, + 504, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 133, + 301 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 133, + 289, + 158, + 300 + ], + "score": 0.89, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 287, + 450, + 301 + ], + "score": 1.0, + "content": ". Using (3), it is easy to check that we obtain the same linear system since", + "type": "text" + }, + { + "bbox": [ + 450, + 290, + 457, + 299 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 287, + 506, + 301 + ], + "score": 1.0, + "content": "cancels out.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 287, + 506, + 301 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 506, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 322 + ], + "score": 1.0, + "content": "KKT conditions. We now show that the KKT conditions, manually differentiated in several works", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 495, + 334 + ], + "score": 1.0, + "content": "(Chapelle et al., 2002; Gould et al., 2016; Amos & Kolter, 2017; Niculae et al., 2018; Niculae", + "type": "text" + }, + { + "bbox": [ + 495, + 323, + 505, + 333 + ], + "score": 0.32, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 332, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 506, + 346 + ], + "score": 1.0, + "content": "Martins, 2020), fit our framework. As we will see, the key will be to group the optimal primal and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 343, + 366, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 191, + 357 + ], + "score": 1.0, + "content": "dual variables as our", + "type": "text" + }, + { + "bbox": [ + 191, + 344, + 215, + 356 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 343, + 366, + 357 + ], + "score": 1.0, + "content": ". Let us consider the general problem", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 311, + 506, + 357 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 358, + 419, + 379 + ], + "lines": [ + { + "bbox": [ + 190, + 358, + 419, + 379 + ], + "spans": [ + { + "bbox": [ + 190, + 358, + 419, + 379 + ], + "score": 0.9, + "content": "\\operatorname { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } } f ( z , \\theta ) \\quad { \\mathrm { s u b j e c t ~ t o } } \\quad G ( z , \\theta ) \\leq 0 , \\ H ( z , \\theta ) = 0 ,", + "type": "interline_equation", + "image_path": "4dde206746cf2aba2d0f1a638f31205f33983dc509217bd52a1f81149053551e.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 190, + 358, + 419, + 379 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 504, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 503, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 132, + 394 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 382, + 162, + 392 + ], + "score": 0.91, + "content": "z \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 381, + 251, + 394 + ], + "score": 1.0, + "content": "is the primal variable,", + "type": "text" + }, + { + "bbox": [ + 251, + 382, + 323, + 393 + ], + "score": 0.78, + "content": "f \\colon { \\mathbb { R } } ^ { p } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 381, + 326, + 394 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 327, + 382, + 405, + 392 + ], + "score": 0.9, + "content": "G \\colon \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 381, + 423, + 394 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 382, + 503, + 392 + ], + "score": 0.88, + "content": "H \\colon \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { q }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 392, + 430, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 430, + 405 + ], + "score": 1.0, + "content": "The stationarity, primal feasibility and complementary slackness conditions give", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 381, + 503, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 405, + 408, + 451 + ], + "lines": [ + { + "bbox": [ + 203, + 405, + 408, + 451 + ], + "spans": [ + { + "bbox": [ + 203, + 405, + 408, + 451 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta ) + \\left[ \\partial _ { 1 } G ( z , \\theta ) \\right] ^ { \\top } \\boldsymbol { \\lambda } + \\left[ \\partial _ { 1 } H ( z , \\theta ) \\right] ^ { \\top } \\boldsymbol { \\nu } = 0 } \\\\ { H ( z , \\theta ) = 0 } \\\\ { \\boldsymbol { \\lambda } \\circ G ( z , \\theta ) = 0 , } \\end{array}", + "type": "interline_equation", + "image_path": "abb0c185cb5cc8c66837b68d239d550b3b91169c3424629a82110a4a9a606fcd.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 203, + 405, + 408, + 420.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 203, + 420.3333333333333, + 408, + 435.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 203, + 435.66666666666663, + 408, + 450.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 134, + 464 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 452, + 167, + 462 + ], + "score": 0.91, + "content": "\\nu \\in \\mathbb { R } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 452, + 187, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 452, + 223, + 465 + ], + "score": 0.93, + "content": "\\lambda \\in \\mathbb { R } _ { + } ^ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "are the dual variables, also known as KKT multipliers. The primal", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 463, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 504, + 475 + ], + "score": 1.0, + "content": "and dual feasibility conditions can be ignored almost everywhere. The system of (potentially", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "nonlinear) equations (6) fits our framework, as we can group the primal and dual solutions as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 484, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 228, + 497 + ], + "score": 0.89, + "content": "x ^ { \\star } ( \\theta ) = ( z ^ { \\star } ( { \\bar { \\theta } } ) , \\nu ^ { \\star } ( \\theta ) , \\lambda ^ { \\star } ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 484, + 345, + 497 + ], + "score": 1.0, + "content": "to form the root of a function", + "type": "text" + }, + { + "bbox": [ + 346, + 485, + 394, + 497 + ], + "score": 0.93, + "content": "F ( x ^ { \\star } ( \\theta ) , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 484, + 424, + 497 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 425, + 484, + 504, + 495 + ], + "score": 0.91, + "content": "F \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } } ^ { d }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 124, + 508 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 497, + 184, + 507 + ], + "score": 0.91, + "content": "d = p + q + r", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 496, + 505, + 508 + ], + "score": 1.0, + "content": ". The primal and dual solutions can be obtained from a generic solver, such as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 338, + 520 + ], + "score": 1.0, + "content": "an interior point method. In practice, the above mapping", + "type": "text" + }, + { + "bbox": [ + 338, + 507, + 347, + 517 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "will be defined directly in Python (see", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 518, + 428, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 224, + 531 + ], + "score": 1.0, + "content": "Figure 7 in Appendix A) and", + "type": "text" + }, + { + "bbox": [ + 225, + 518, + 234, + 528 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 518, + 428, + 531 + ], + "score": 1.0, + "content": "will be differentiated automatically via autodiff.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 452, + 505, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 506, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "Proximal gradient fixed point. Unfortunately, not all algorithms return both primal and dual", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "solutions. Moreover, if the objective contains non-smooth terms, proximal gradient descent may be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 574 + ], + "score": 1.0, + "content": "more efficient. We now discuss its fixed point (Niculae & Blondel, 2017; Bertrand et al., 2020; 2021).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 573, + 244, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 122, + 586 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 574, + 146, + 586 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 573, + 244, + 586 + ], + "score": 1.0, + "content": "be implicitly defined as", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 540, + 506, + 586 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 587, + 377, + 609 + ], + "lines": [ + { + "bbox": [ + 233, + 587, + 377, + 609 + ], + "spans": [ + { + "bbox": [ + 233, + 587, + 377, + 609 + ], + "score": 0.93, + "content": "x ^ { \\star } ( \\theta ) : = \\operatornamewithlimits { a r g m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x , \\theta ) + g ( x , \\theta ) ,", + "type": "interline_equation", + "image_path": "92f4e6a348b59884f2774aeba492d29f0d7c2a633b2cd24da89ef622723f2cd8.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 233, + 587, + 377, + 609 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 132, + 628 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 613, + 206, + 625 + ], + "score": 0.91, + "content": "f \\colon { \\mathbb { R } ^ { d } } \\times { \\mathbb { R } ^ { n } } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 610, + 342, + 628 + ], + "score": 1.0, + "content": "is twice-differentiable convex and", + "type": "text" + }, + { + "bbox": [ + 343, + 613, + 415, + 625 + ], + "score": 0.91, + "content": "g \\colon { \\mathbb { R } } ^ { d } \\times { \\mathbb { R } } ^ { n } \\to { \\mathbb { R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 610, + 506, + 628 + ], + "score": 1.0, + "content": "is convex but possibly", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 623, + 389, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 369, + 637 + ], + "score": 1.0, + "content": "non-smooth. 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The proximity operator is 1-Lipschitz continuous (Moreau, 1965). By", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Rademacher’s theorem, it is differentiable almost everywhere. Many proximity operators enjoy a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 401, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 389, + 732 + ], + "score": 1.0, + "content": "closed form and can easily be differentiated, as discussed in Appendix", + "type": "text" + }, + { + "bbox": [ + 389, + 721, + 397, + 730 + ], + "score": 0.27, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 720, + 401, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 698, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 502, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 340, + 97 + ], + "score": 1.0, + "content": "Projected gradient fixed point. As a special case, when", + "type": "text" + }, + { + "bbox": [ + 341, + 83, + 370, + 95 + ], + "score": 0.93, + "content": "g ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 80, + 468, + 97 + ], + "score": 1.0, + "content": "is the indicator function", + "type": "text" + }, + { + "bbox": [ + 468, + 82, + 502, + 96 + ], + "score": 0.93, + "content": "I _ { C ( \\theta ) } ( x )", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 322, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 133, + 106 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 94, + 153, + 106 + ], + "score": 0.91, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 95, + 271, + 106 + ], + "score": 1.0, + "content": "is a convex set depending on", + "type": "text" + }, + { + "bbox": [ + 271, + 95, + 277, + 104 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 95, + 322, + 106 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 110, + 357, + 134 + ], + "lines": [ + { + "bbox": [ + 253, + 110, + 357, + 134 + ], + "spans": [ + { + "bbox": [ + 253, + 110, + 357, + 134 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x \\in { \\mathcal { C } } ( \\theta ) } f ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "21d0db90801565fc4db73402a11b563257f5f09cc557cb224a7c047030239af8.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 253, + 110, + 357, + 134 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 404, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 404, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 201, + 154 + ], + "score": 1.0, + "content": "The proximity operator", + "type": "text" + }, + { + "bbox": [ + 202, + 141, + 226, + 153 + ], + "score": 0.64, + "content": "\\operatorname { p r o x } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 138, + 384, + 154 + ], + "score": 1.0, + "content": "becomes the Euclidean projection onto", + "type": "text" + }, + { + "bbox": [ + 385, + 139, + 404, + 151 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 158, + 401, + 181 + ], + "lines": [ + { + "bbox": [ + 209, + 158, + 401, + 181 + ], + "spans": [ + { + "bbox": [ + 209, + 158, + 401, + 181 + ], + "score": 0.93, + "content": "{ \\mathrm { p r o x } } _ { g } ( y , \\theta ) = { \\mathrm { p r o j } } _ { \\mathcal { C } } ( y , \\theta ) : = { \\underset { x \\in { \\mathcal { C } } ( \\theta ) } { \\operatorname { a r g m i n } } } \\| x - y \\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "5f9e4e81f0c3d5dbce35a254d6cab821b0da78a8f8a04166872eeae9d80f5e9e.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 209, + 158, + 401, + 181 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 310, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 309, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 309, + 199 + ], + "score": 1.0, + "content": "and (7) becomes the projected gradient fixed point", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 203, + 381, + 217 + ], + "lines": [ + { + "bbox": [ + 229, + 203, + 381, + 217 + ], + "spans": [ + { + "bbox": [ + 229, + 203, + 381, + 217 + ], + "score": 0.92, + "content": "T ( x , \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) .", + "type": "interline_equation", + "image_path": "9d1e1bc231914c660ca72a0e10d85b08ee3097a06cd2d8c401321814985f4fc6.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 229, + 203, + 381, + 217 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "Compared to the KKT conditions, this fixed point is particularly suitable when the projection enjoys", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "a closed form. We discuss how to compute the JVP / VJP for a wealth of convex sets in Appendix B.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 506, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 370, + 270 + ], + "score": 1.0, + "content": "Mirror descent fixed point. We again consider the case when", + "type": "text" + }, + { + "bbox": [ + 370, + 257, + 394, + 269 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "is implicitly defined as the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "solution of (8). We now generalize the projected gradient fixed point beyond Euclidean geometry.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 278, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 221, + 293 + ], + "score": 1.0, + "content": "Let the Bregman divergence", + "type": "text" + }, + { + "bbox": [ + 221, + 279, + 237, + 291 + ], + "score": 0.8, + "content": "D _ { \\varphi }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 278, + 244, + 293 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 244, + 279, + 384, + 291 + ], + "score": 0.91, + "content": "\\operatorname { d o m } ( \\bar { \\varphi } ) \\cdot \\operatorname { r e l i n t } ( \\operatorname { d o m } ( \\varphi ) ) \\to \\bar { \\mathbb { R } } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 278, + 439, + 293 + ], + "score": 1.0, + "content": "generated by", + "type": "text" + }, + { + "bbox": [ + 439, + 280, + 447, + 290 + ], + "score": 0.8, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 278, + 504, + 293 + ], + "score": 1.0, + "content": "be defined by", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 295, + 397, + 310 + ], + "lines": [ + { + "bbox": [ + 212, + 295, + 397, + 310 + ], + "spans": [ + { + "bbox": [ + 212, + 295, + 397, + 310 + ], + "score": 0.91, + "content": "D _ { \\varphi } ( x , y ) : = \\varphi ( x ) - \\varphi ( y ) - \\langle \\nabla \\varphi ( y ) , x - y \\rangle .", + "type": "interline_equation", + "image_path": "58674b69bdba2abf82e66af3aac43fbaad9e6039780737464627a63de92b9fd4.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 212, + 295, + 397, + 310 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 363, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 363, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 255, + 329 + ], + "score": 1.0, + "content": "We define the Bregman projection of", + "type": "text" + }, + { + "bbox": [ + 256, + 317, + 262, + 326 + ], + "score": 0.82, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 312, + 284, + 329 + ], + "score": 1.0, + "content": "onto", + "type": "text" + }, + { + "bbox": [ + 284, + 314, + 349, + 327 + ], + "score": 0.92, + "content": "{ \\mathcal { C } } ( \\theta ) \\subseteq \\mathrm { d o m } ( \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 312, + 363, + 329 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 331, + 386, + 354 + ], + "lines": [ + { + "bbox": [ + 224, + 331, + 386, + 354 + ], + "spans": [ + { + "bbox": [ + 224, + 331, + 386, + 354 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y , \\theta ) : = \\underset { x \\in \\mathcal { C } ( \\theta ) } { \\mathrm { a r g m i n } } D _ { \\varphi } ( x , \\nabla \\varphi ^ { \\ast } ( y ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "5e5a83a15232511f9ede66f58e7405d946d6294562f410a45d6f9751f1ee6f1a.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 224, + 331, + 386, + 354 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 359, + 504, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 267, + 372 + ], + "score": 1.0, + "content": "Definition (10) includes the mirror map", + "type": "text" + }, + { + "bbox": [ + 267, + 360, + 300, + 372 + ], + "score": 0.93, + "content": "\\nabla \\varphi ^ { * } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "for convenience. It can be seen as a mapping from", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 369, + 497, + 385 + ], + "spans": [ + { + "bbox": [ + 107, + 371, + 119, + 381 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 369, + 131, + 385 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 131, + 371, + 165, + 383 + ], + "score": 0.88, + "content": "\\mathrm { d o m } ( \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 369, + 497, + 385 + ], + "score": 1.0, + "content": ", ensuring that (10) is well-defined. The mirror descent fixed point mapping is then", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 387, + 361, + 432 + ], + "lines": [ + { + "bbox": [ + 250, + 387, + 361, + 432 + ], + "spans": [ + { + "bbox": [ + 250, + 387, + 361, + 432 + ], + "score": 0.92, + "content": "\\begin{array} { c } { \\hat { x } = \\nabla \\varphi ( x ) } \\\\ { y = \\hat { x } - \\eta \\nabla _ { 1 } f ( x , \\theta ) } \\\\ { T ( x , \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y , \\theta ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6318c6ba9d6dcb60b25864ed6f1fd681198ea862437c2ff09c477c12ca89ce83.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 387, + 361, + 409.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 250, + 409.5, + 361, + 432.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 143, + 448 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 143, + 436, + 152, + 446 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "involves the composition of several functions, manually deriving its JVP/VJP is error", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "prone. This shows that our approach leveraging autodiff allows to handle more advanced fixed point", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 248, + 471 + ], + "score": 1.0, + "content": "mappings. A common example of", + "type": "text" + }, + { + "bbox": [ + 248, + 459, + 256, + 469 + ], + "score": 0.83, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 456, + 266, + 471 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 267, + 457, + 357, + 470 + ], + "score": 0.92, + "content": "\\varphi ( x ) = \\langle x , \\log x - 1 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 456, + 389, + 471 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 389, + 457, + 450, + 470 + ], + "score": 0.92, + "content": "\\mathrm { d o m } ( \\varphi ) = \\mathbb { R } _ { + } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 456, + 507, + 471 + ], + "score": 1.0, + "content": ". In this case,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 107, + 469, + 122, + 481 + ], + "score": 0.89, + "content": "D _ { \\varphi }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "is the Kullback-Leibler divergence. An advantage of the Kullback-Leibler projection is that it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 479, + 460, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 460, + 492 + ], + "score": 1.0, + "content": "sometimes easier to compute than the Euclidean projection, as we detail in Appendix B.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 387, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 388, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 388, + 516 + ], + "score": 1.0, + "content": "Other fixed points. More fixed points are described in Appendix C.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 309, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 309, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 309, + 537 + ], + "score": 1.0, + "content": "3 JACOBIAN PRECISION GUARANTEES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 506, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "In practice, either by the limitations of finite precision arithmetic or because we perform a finite", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 325, + 570 + ], + "score": 1.0, + "content": "number of iterations, we rarely reach the exact solution", + "type": "text" + }, + { + "bbox": [ + 325, + 558, + 349, + 570 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 559, + 505, + 570 + ], + "score": 1.0, + "content": ", but instead only reach an approximate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 142, + 582 + ], + "score": 1.0, + "content": "solution", + "type": "text" + }, + { + "bbox": [ + 142, + 570, + 150, + 579 + ], + "score": 0.78, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "and apply the implicit differentiation equation (2) at this approximate solution. This", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "score": 1.0, + "content": "motivates the need for precision guarantees of this approach. We introduce the following formalism.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 504, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 504, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 183, + 608 + ], + "score": 1.0, + "content": "Definition 1. Let", + "type": "text" + }, + { + "bbox": [ + 183, + 594, + 275, + 605 + ], + "score": 0.9, + "content": "F : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { n } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 593, + 450, + 608 + ], + "score": 1.0, + "content": "be an optimality criterion mapping. Let", + "type": "text" + }, + { + "bbox": [ + 450, + 595, + 504, + 606 + ], + "score": 0.92, + "content": "A : = - \\partial _ { 1 } F", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 125, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 606, + 169, + 617 + ], + "score": 0.9, + "content": "B : = \\partial _ { 2 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 605, + 324, + 618 + ], + "score": 1.0, + "content": ". We define the Jacobian estimate at", + "type": "text" + }, + { + "bbox": [ + 324, + 606, + 347, + 617 + ], + "score": 0.89, + "content": "( x , \\dot { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "as the solution to the linear equation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 616, + 372, + 629 + ], + "spans": [ + { + "bbox": [ + 107, + 617, + 211, + 629 + ], + "score": 0.91, + "content": "A ( x , \\theta ) J ( x , \\theta ) = B ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 616, + 276, + 629 + ], + "score": 1.0, + "content": ". It is a function", + "type": "text" + }, + { + "bbox": [ + 276, + 617, + 367, + 627 + ], + "score": 0.9, + "content": "J : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { \\bar { n } } \\mathbb { R } ^ { d \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 616, + 372, + 629 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 220, + 650 + ], + "score": 1.0, + "content": "It holds by construction that", + "type": "text" + }, + { + "bbox": [ + 221, + 638, + 310, + 650 + ], + "score": 0.93, + "content": "J ( x ^ { \\star } ( \\theta ) , \\theta ) = \\partial x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 638, + 361, + 650 + ], + "score": 1.0, + "content": ". Computing", + "type": "text" + }, + { + "bbox": [ + 362, + 638, + 392, + 650 + ], + "score": 0.91, + "content": "J ( \\hat { x } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "for an approximate solution", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 113, + 659 + ], + "score": 0.81, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 648, + 126, + 662 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 126, + 649, + 149, + 660 + ], + "score": 0.92, + "content": "x ^ { \\star } ( { \\boldsymbol { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 648, + 349, + 662 + ], + "score": 1.0, + "content": "therefore allows to approximate the true Jacobian", + "type": "text" + }, + { + "bbox": [ + 350, + 649, + 379, + 661 + ], + "score": 0.91, + "content": "{ \\partial x ^ { \\star } ( \\theta ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 648, + 505, + 662 + ], + "score": 1.0, + "content": ". In practice, an algorithm used", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 203, + 673 + ], + "score": 1.0, + "content": "to solve (1) depends on", + "type": "text" + }, + { + "bbox": [ + 203, + 660, + 209, + 670 + ], + "score": 0.71, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 659, + 454, + 673 + ], + "score": 1.0, + "content": ". Note however that, what we compute is not the Jacobian of", + "type": "text" + }, + { + "bbox": [ + 454, + 660, + 473, + 672 + ], + "score": 0.92, + "content": "{ \\hat { x } } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 659, + 505, + 673 + ], + "score": 1.0, + "content": ", unlike", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 669, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 417, + 685 + ], + "score": 1.0, + "content": "works differentiating through unrolled algorithm iterations, but an estimate of", + "type": "text" + }, + { + "bbox": [ + 417, + 671, + 446, + 682 + ], + "score": 0.92, + "content": "{ \\partial x ^ { \\star } ( \\theta ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 669, + 505, + 685 + ], + "score": 1.0, + "content": ". We therefore", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 334, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 172, + 695 + ], + "score": 1.0, + "content": "use the notation", + "type": "text" + }, + { + "bbox": [ + 172, + 682, + 179, + 691 + ], + "score": 0.8, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 681, + 291, + 695 + ], + "score": 1.0, + "content": ", leaving the dependence on", + "type": "text" + }, + { + "bbox": [ + 291, + 683, + 297, + 691 + ], + "score": 0.83, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 681, + 334, + 695 + ], + "score": 1.0, + "content": "implicit.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 233, + 712 + ], + "score": 1.0, + "content": "We develop bounds of the form", + "type": "text" + }, + { + "bbox": [ + 234, + 699, + 386, + 711 + ], + "score": 0.91, + "content": "\\| J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| < C \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 697, + 506, + 712 + ], + "score": 1.0, + "content": ", hence showing that the error", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 502, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 373, + 723 + ], + "score": 1.0, + "content": "on the estimated Jacobian is at most of the same order as that of", + "type": "text" + }, + { + "bbox": [ + 373, + 711, + 380, + 720 + ], + "score": 0.81, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 709, + 479, + 723 + ], + "score": 1.0, + "content": "as an approximation of", + "type": "text" + }, + { + "bbox": [ + 479, + 709, + 502, + 722 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 493, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 493, + 733 + ], + "score": 1.0, + "content": "These bounds are based on the following main theorem, whose proof is included in Appendix D.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 502, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 340, + 97 + ], + "score": 1.0, + "content": "Projected gradient fixed point. As a special case, when", + "type": "text" + }, + { + "bbox": [ + 341, + 83, + 370, + 95 + ], + "score": 0.93, + "content": "g ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 80, + 468, + 97 + ], + "score": 1.0, + "content": "is the indicator function", + "type": "text" + }, + { + "bbox": [ + 468, + 82, + 502, + 96 + ], + "score": 0.93, + "content": "I _ { C ( \\theta ) } ( x )", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 322, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 133, + 106 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 94, + 153, + 106 + ], + "score": 0.91, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 95, + 271, + 106 + ], + "score": 1.0, + "content": "is a convex set depending on", + "type": "text" + }, + { + "bbox": [ + 271, + 95, + 277, + 104 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 95, + 322, + 106 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 80, + 502, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 110, + 357, + 134 + ], + "lines": [ + { + "bbox": [ + 253, + 110, + 357, + 134 + ], + "spans": [ + { + "bbox": [ + 253, + 110, + 357, + 134 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x \\in { \\mathcal { C } } ( \\theta ) } f ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "21d0db90801565fc4db73402a11b563257f5f09cc557cb224a7c047030239af8.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 253, + 110, + 357, + 134 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 404, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 404, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 201, + 154 + ], + "score": 1.0, + "content": "The proximity operator", + "type": "text" + }, + { + "bbox": [ + 202, + 141, + 226, + 153 + ], + "score": 0.64, + "content": "\\operatorname { p r o x } _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 138, + 384, + 154 + ], + "score": 1.0, + "content": "becomes the Euclidean projection onto", + "type": "text" + }, + { + "bbox": [ + 385, + 139, + 404, + 151 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 138, + 404, + 154 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 158, + 401, + 181 + ], + "lines": [ + { + "bbox": [ + 209, + 158, + 401, + 181 + ], + "spans": [ + { + "bbox": [ + 209, + 158, + 401, + 181 + ], + "score": 0.93, + "content": "{ \\mathrm { p r o x } } _ { g } ( y , \\theta ) = { \\mathrm { p r o j } } _ { \\mathcal { C } } ( y , \\theta ) : = { \\underset { x \\in { \\mathcal { C } } ( \\theta ) } { \\operatorname { a r g m i n } } } \\| x - y \\| _ { 2 } ^ { 2 }", + "type": "interline_equation", + "image_path": "5f9e4e81f0c3d5dbce35a254d6cab821b0da78a8f8a04166872eeae9d80f5e9e.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 209, + 158, + 401, + 181 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 310, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 309, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 309, + 199 + ], + "score": 1.0, + "content": "and (7) becomes the projected gradient fixed point", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 186, + 309, + 199 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 203, + 381, + 217 + ], + "lines": [ + { + "bbox": [ + 229, + 203, + 381, + 217 + ], + "spans": [ + { + "bbox": [ + 229, + 203, + 381, + 217 + ], + "score": 0.92, + "content": "T ( x , \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) .", + "type": "interline_equation", + "image_path": "9d1e1bc231914c660ca72a0e10d85b08ee3097a06cd2d8c401321814985f4fc6.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 229, + 203, + 381, + 217 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "Compared to the KKT conditions, this fixed point is particularly suitable when the projection enjoys", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "a closed form. We discuss how to compute the JVP / VJP for a wealth of convex sets in Appendix B.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 222, + 506, + 246 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 506, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 370, + 270 + ], + "score": 1.0, + "content": "Mirror descent fixed point. We again consider the case when", + "type": "text" + }, + { + "bbox": [ + 370, + 257, + 394, + 269 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "is implicitly defined as the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "solution of (8). We now generalize the projected gradient fixed point beyond Euclidean geometry.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 278, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 221, + 293 + ], + "score": 1.0, + "content": "Let the Bregman divergence", + "type": "text" + }, + { + "bbox": [ + 221, + 279, + 237, + 291 + ], + "score": 0.8, + "content": "D _ { \\varphi }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 278, + 244, + 293 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 244, + 279, + 384, + 291 + ], + "score": 0.91, + "content": "\\operatorname { d o m } ( \\bar { \\varphi } ) \\cdot \\operatorname { r e l i n t } ( \\operatorname { d o m } ( \\varphi ) ) \\to \\bar { \\mathbb { R } } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 278, + 439, + 293 + ], + "score": 1.0, + "content": "generated by", + "type": "text" + }, + { + "bbox": [ + 439, + 280, + 447, + 290 + ], + "score": 0.8, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 278, + 504, + 293 + ], + "score": 1.0, + "content": "be defined by", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 256, + 506, + 293 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 295, + 397, + 310 + ], + "lines": [ + { + "bbox": [ + 212, + 295, + 397, + 310 + ], + "spans": [ + { + "bbox": [ + 212, + 295, + 397, + 310 + ], + "score": 0.91, + "content": "D _ { \\varphi } ( x , y ) : = \\varphi ( x ) - \\varphi ( y ) - \\langle \\nabla \\varphi ( y ) , x - y \\rangle .", + "type": "interline_equation", + "image_path": "58674b69bdba2abf82e66af3aac43fbaad9e6039780737464627a63de92b9fd4.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 212, + 295, + 397, + 310 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 363, + 327 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 363, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 255, + 329 + ], + "score": 1.0, + "content": "We define the Bregman projection of", + "type": "text" + }, + { + "bbox": [ + 256, + 317, + 262, + 326 + ], + "score": 0.82, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 312, + 284, + 329 + ], + "score": 1.0, + "content": "onto", + "type": "text" + }, + { + "bbox": [ + 284, + 314, + 349, + 327 + ], + "score": 0.92, + "content": "{ \\mathcal { C } } ( \\theta ) \\subseteq \\mathrm { d o m } ( \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 312, + 363, + 329 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 312, + 363, + 329 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 331, + 386, + 354 + ], + "lines": [ + { + "bbox": [ + 224, + 331, + 386, + 354 + ], + "spans": [ + { + "bbox": [ + 224, + 331, + 386, + 354 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y , \\theta ) : = \\underset { x \\in \\mathcal { C } ( \\theta ) } { \\mathrm { a r g m i n } } D _ { \\varphi } ( x , \\nabla \\varphi ^ { \\ast } ( y ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "5e5a83a15232511f9ede66f58e7405d946d6294562f410a45d6f9751f1ee6f1a.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 224, + 331, + 386, + 354 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 359, + 504, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 267, + 372 + ], + "score": 1.0, + "content": "Definition (10) includes the mirror map", + "type": "text" + }, + { + "bbox": [ + 267, + 360, + 300, + 372 + ], + "score": 0.93, + "content": "\\nabla \\varphi ^ { * } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "for convenience. It can be seen as a mapping from", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 369, + 497, + 385 + ], + "spans": [ + { + "bbox": [ + 107, + 371, + 119, + 381 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 369, + 131, + 385 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 131, + 371, + 165, + 383 + ], + "score": 0.88, + "content": "\\mathrm { d o m } ( \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 369, + 497, + 385 + ], + "score": 1.0, + "content": ", ensuring that (10) is well-defined. The mirror descent fixed point mapping is then", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 106, + 359, + 505, + 385 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 387, + 361, + 432 + ], + "lines": [ + { + "bbox": [ + 250, + 387, + 361, + 432 + ], + "spans": [ + { + "bbox": [ + 250, + 387, + 361, + 432 + ], + "score": 0.92, + "content": "\\begin{array} { c } { \\hat { x } = \\nabla \\varphi ( x ) } \\\\ { y = \\hat { x } - \\eta \\nabla _ { 1 } f ( x , \\theta ) } \\\\ { T ( x , \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y , \\theta ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6318c6ba9d6dcb60b25864ed6f1fd681198ea862437c2ff09c477c12ca89ce83.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 387, + 361, + 409.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 250, + 409.5, + 361, + 432.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 143, + 448 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 143, + 436, + 152, + 446 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "involves the composition of several functions, manually deriving its JVP/VJP is error", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "prone. This shows that our approach leveraging autodiff allows to handle more advanced fixed point", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 248, + 471 + ], + "score": 1.0, + "content": "mappings. A common example of", + "type": "text" + }, + { + "bbox": [ + 248, + 459, + 256, + 469 + ], + "score": 0.83, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 456, + 266, + 471 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 267, + 457, + 357, + 470 + ], + "score": 0.92, + "content": "\\varphi ( x ) = \\langle x , \\log x - 1 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 456, + 389, + 471 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 389, + 457, + 450, + 470 + ], + "score": 0.92, + "content": "\\mathrm { d o m } ( \\varphi ) = \\mathbb { R } _ { + } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 456, + 507, + 471 + ], + "score": 1.0, + "content": ". In this case,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 107, + 469, + 122, + 481 + ], + "score": 0.89, + "content": "D _ { \\varphi }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "is the Kullback-Leibler divergence. An advantage of the Kullback-Leibler projection is that it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 479, + 460, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 460, + 492 + ], + "score": 1.0, + "content": "sometimes easier to compute than the Euclidean projection, as we detail in Appendix B.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 436, + 507, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 502, + 387, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 388, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 388, + 516 + ], + "score": 1.0, + "content": "Other fixed points. More fixed points are described in Appendix C.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 502, + 388, + 516 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 522, + 309, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 309, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 309, + 537 + ], + "score": 1.0, + "content": "3 JACOBIAN PRECISION GUARANTEES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 506, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "In practice, either by the limitations of finite precision arithmetic or because we perform a finite", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 325, + 570 + ], + "score": 1.0, + "content": "number of iterations, we rarely reach the exact solution", + "type": "text" + }, + { + "bbox": [ + 325, + 558, + 349, + 570 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 559, + 505, + 570 + ], + "score": 1.0, + "content": ", but instead only reach an approximate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 142, + 582 + ], + "score": 1.0, + "content": "solution", + "type": "text" + }, + { + "bbox": [ + 142, + 570, + 150, + 579 + ], + "score": 0.78, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "and apply the implicit differentiation equation (2) at this approximate solution. This", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "score": 1.0, + "content": "motivates the need for precision guarantees of this approach. We introduce the following formalism.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 547, + 506, + 593 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 504, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 504, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 183, + 608 + ], + "score": 1.0, + "content": "Definition 1. Let", + "type": "text" + }, + { + "bbox": [ + 183, + 594, + 275, + 605 + ], + "score": 0.9, + "content": "F : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { n } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 593, + 450, + 608 + ], + "score": 1.0, + "content": "be an optimality criterion mapping. Let", + "type": "text" + }, + { + "bbox": [ + 450, + 595, + 504, + 606 + ], + "score": 0.92, + "content": "A : = - \\partial _ { 1 } F", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 125, + 618 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 606, + 169, + 617 + ], + "score": 0.9, + "content": "B : = \\partial _ { 2 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 605, + 324, + 618 + ], + "score": 1.0, + "content": ". We define the Jacobian estimate at", + "type": "text" + }, + { + "bbox": [ + 324, + 606, + 347, + 617 + ], + "score": 0.89, + "content": "( x , \\dot { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "as the solution to the linear equation", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 616, + 372, + 629 + ], + "spans": [ + { + "bbox": [ + 107, + 617, + 211, + 629 + ], + "score": 0.91, + "content": "A ( x , \\theta ) J ( x , \\theta ) = B ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 616, + 276, + 629 + ], + "score": 1.0, + "content": ". It is a function", + "type": "text" + }, + { + "bbox": [ + 276, + 617, + 367, + 627 + ], + "score": 0.9, + "content": "J : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { \\bar { n } } \\mathbb { R } ^ { d \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 616, + 372, + 629 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 593, + 505, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 220, + 650 + ], + "score": 1.0, + "content": "It holds by construction that", + "type": "text" + }, + { + "bbox": [ + 221, + 638, + 310, + 650 + ], + "score": 0.93, + "content": "J ( x ^ { \\star } ( \\theta ) , \\theta ) = \\partial x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 638, + 361, + 650 + ], + "score": 1.0, + "content": ". Computing", + "type": "text" + }, + { + "bbox": [ + 362, + 638, + 392, + 650 + ], + "score": 0.91, + "content": "J ( \\hat { x } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "for an approximate solution", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 113, + 659 + ], + "score": 0.81, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 648, + 126, + 662 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 126, + 649, + 149, + 660 + ], + "score": 0.92, + "content": "x ^ { \\star } ( { \\boldsymbol { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 648, + 349, + 662 + ], + "score": 1.0, + "content": "therefore allows to approximate the true Jacobian", + "type": "text" + }, + { + "bbox": [ + 350, + 649, + 379, + 661 + ], + "score": 0.91, + "content": "{ \\partial x ^ { \\star } ( \\theta ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 648, + 505, + 662 + ], + "score": 1.0, + "content": ". In practice, an algorithm used", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 203, + 673 + ], + "score": 1.0, + "content": "to solve (1) depends on", + "type": "text" + }, + { + "bbox": [ + 203, + 660, + 209, + 670 + ], + "score": 0.71, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 659, + 454, + 673 + ], + "score": 1.0, + "content": ". Note however that, what we compute is not the Jacobian of", + "type": "text" + }, + { + "bbox": [ + 454, + 660, + 473, + 672 + ], + "score": 0.92, + "content": "{ \\hat { x } } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 659, + 505, + 673 + ], + "score": 1.0, + "content": ", unlike", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 669, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 417, + 685 + ], + "score": 1.0, + "content": "works differentiating through unrolled algorithm iterations, but an estimate of", + "type": "text" + }, + { + "bbox": [ + 417, + 671, + 446, + 682 + ], + "score": 0.92, + "content": "{ \\partial x ^ { \\star } ( \\theta ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 669, + 505, + 685 + ], + "score": 1.0, + "content": ". We therefore", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 334, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 172, + 695 + ], + "score": 1.0, + "content": "use the notation", + "type": "text" + }, + { + "bbox": [ + 172, + 682, + 179, + 691 + ], + "score": 0.8, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 681, + 291, + 695 + ], + "score": 1.0, + "content": ", leaving the dependence on", + "type": "text" + }, + { + "bbox": [ + 291, + 683, + 297, + 691 + ], + "score": 0.83, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 681, + 334, + 695 + ], + "score": 1.0, + "content": "implicit.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 638, + 505, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 233, + 712 + ], + "score": 1.0, + "content": "We develop bounds of the form", + "type": "text" + }, + { + "bbox": [ + 234, + 699, + 386, + 711 + ], + "score": 0.91, + "content": "\\| J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| < C \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 697, + 506, + 712 + ], + "score": 1.0, + "content": ", hence showing that the error", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 502, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 373, + 723 + ], + "score": 1.0, + "content": "on the estimated Jacobian is at most of the same order as that of", + "type": "text" + }, + { + "bbox": [ + 373, + 711, + 380, + 720 + ], + "score": 0.81, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 709, + 479, + 723 + ], + "score": 1.0, + "content": "as an approximation of", + "type": "text" + }, + { + "bbox": [ + 479, + 709, + 502, + 722 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 493, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 493, + 733 + ], + "score": 1.0, + "content": "These bounds are based on the following main theorem, whose proof is included in Appendix D.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 697, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 506, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 255, + 95 + ], + "score": 1.0, + "content": "Theorem 1 (Jacobian estimate). Let", + "type": "text" + }, + { + "bbox": [ + 256, + 82, + 337, + 93 + ], + "score": 0.9, + "content": "F : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { n } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 80, + 436, + 95 + ], + "score": 1.0, + "content": ". Assume that there exist", + "type": "text" + }, + { + "bbox": [ + 437, + 83, + 504, + 94 + ], + "score": 0.88, + "content": "\\alpha , \\beta , \\gamma , \\varepsilon , R > 0", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 91, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 144, + 108 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 144, + 94, + 191, + 105 + ], + "score": 0.92, + "content": "A = - \\partial _ { 1 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 91, + 209, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 209, + 94, + 249, + 105 + ], + "score": 0.92, + "content": "B = \\partial _ { 2 } F", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 91, + 306, + 108 + ], + "score": 1.0, + "content": "satisfy, for all", + "type": "text" + }, + { + "bbox": [ + 306, + 93, + 336, + 104 + ], + "score": 0.85, + "content": "v \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 91, + 339, + 108 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 340, + 93, + 370, + 104 + ], + "score": 0.88, + "content": "\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 91, + 388, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 389, + 95, + 396, + 104 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 91, + 434, + 108 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 435, + 94, + 502, + 106 + ], + "score": 0.9, + "content": "\\| x - x ^ { \\star } ( \\theta ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 91, + 507, + 108 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 109, + 505, + 123 + ], + "lines": [ + { + "bbox": [ + 107, + 108, + 504, + 125 + ], + "spans": [ + { + "bbox": [ + 107, + 111, + 115, + 120 + ], + "score": 0.54, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 108, + 235, + 125 + ], + "score": 1.0, + "content": "is well-conditioned, Lipschitz:", + "type": "text" + }, + { + "bbox": [ + 236, + 110, + 504, + 123 + ], + "score": 0.76, + "content": "\\| A ( x , \\theta ) v \\| \\geq \\alpha \\| v \\| , \\| A ( x , \\theta ) - A ( x ^ { \\star } ( \\theta ) , \\theta ) \\| _ { \\mathrm { o p } } \\leq \\gamma \\| x - x ^ { \\star } ( \\theta ) \\| .", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 126, + 491, + 140 + ], + "lines": [ + { + "bbox": [ + 107, + 126, + 485, + 141 + ], + "spans": [ + { + "bbox": [ + 107, + 128, + 116, + 137 + ], + "score": 0.69, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 126, + 223, + 141 + ], + "score": 1.0, + "content": "is bounded and Lipschitz:", + "type": "text" + }, + { + "bbox": [ + 223, + 127, + 304, + 140 + ], + "score": 0.56, + "content": "\\| B ( x ^ { \\star } ( \\theta ) , \\theta ) \\| \\leq R", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 126, + 308, + 141 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 308, + 127, + 483, + 140 + ], + "score": 0.5, + "content": "\\| B ( x , \\theta ) - B ( x ^ { \\star } ( \\theta ) , \\theta ) \\| \\leq \\beta \\| x - x ^ { \\star } ( \\theta ) \\| .", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 126, + 485, + 141 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 144, + 336, + 156 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 335, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 227, + 157 + ], + "score": 1.0, + "content": "Under these conditions, when", + "type": "text" + }, + { + "bbox": [ + 228, + 144, + 297, + 157 + ], + "score": 0.92, + "content": "\\| { \\hat { x } } - x ^ { \\star } ( \\theta ) \\| \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 144, + 335, + 157 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 160, + 420, + 176 + ], + "lines": [ + { + "bbox": [ + 191, + 160, + 420, + 176 + ], + "spans": [ + { + "bbox": [ + 191, + 160, + 420, + 176 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\| J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| \\leq \\left( \\beta \\alpha ^ { - 1 } + \\gamma R \\alpha ^ { - 2 } \\right) \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\| . } \\end{array}", + "type": "interline_equation", + "image_path": "9f46752ccd33510fcf200f3960249df0d367cb5bb2933592959ba2cd6808261a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 191, + 160, + 420, + 176 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 506, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "score": 1.0, + "content": "This result is inspired by (Higham, 2002, Theorem 7.2), that is concerned with the stability of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 196, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 322, + 208 + ], + "score": 1.0, + "content": "solutions to inverse problems. Here we consider that", + "type": "text" + }, + { + "bbox": [ + 323, + 196, + 351, + 208 + ], + "score": 0.93, + "content": "A ( \\cdot , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 196, + 506, + 208 + ], + "score": 1.0, + "content": "is uniformly well-conditioned, rather", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 207, + 507, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 156, + 219 + ], + "score": 1.0, + "content": "than only at", + "type": "text" + }, + { + "bbox": [ + 156, + 207, + 180, + 219 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 207, + 330, + 219 + ], + "score": 1.0, + "content": ". This does not affect the first order in", + "type": "text" + }, + { + "bbox": [ + 331, + 209, + 337, + 217 + ], + "score": 0.79, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 207, + 496, + 219 + ], + "score": 1.0, + "content": "of this bound, and makes it valid for all", + "type": "text" + }, + { + "bbox": [ + 496, + 208, + 503, + 217 + ], + "score": 0.81, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 207, + 507, + 219 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 366, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 366, + 231 + ], + "score": 1.0, + "content": "It is also more tailored to applications to equation-specific cases.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 233, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 248 + ], + "score": 1.0, + "content": "Indeed, Theorem 1 can be applied to specific cases. In particular, for the gradient descent fixed point,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 401, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 133, + 259 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 245, + 242, + 258 + ], + "score": 0.93, + "content": "T ( x , \\theta ) = x - \\eta \\nabla _ { 1 } \\bar { f } ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 245, + 261, + 259 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 261, + 245, + 353, + 258 + ], + "score": 0.92, + "content": "F ( x , { \\boldsymbol { \\theta } } ) = T ( x , { \\overline { { \\theta } } } ) - x", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 245, + 401, + 259 + ], + "score": 1.0, + "content": ", this yields", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 262, + 417, + 276 + ], + "lines": [ + { + "bbox": [ + 193, + 262, + 417, + 276 + ], + "spans": [ + { + "bbox": [ + 193, + 262, + 417, + 276 + ], + "score": 0.86, + "content": "A ( x , \\theta ) = \\eta \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) \\mathrm { a n d } B ( x , \\theta ) = - \\eta \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "575276dad4e1574671a11f4d8d13f4404cf87388aceaa094861c3d08f947de79.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 193, + 262, + 417, + 276 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 506, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 398, + 293 + ], + "score": 1.0, + "content": "The guarantees on Jacobian precision under regularity conditions rely on", + "type": "text" + }, + { + "bbox": [ + 398, + 281, + 406, + 292 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "directly; see Corollary 1", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "in Appendix D. This reveals in particular that Jacobian estimation by implicit differentiation gains", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 352, + 315 + ], + "score": 1.0, + "content": "a factor of t compared to automatic differentiation, after", + "type": "text" + }, + { + "bbox": [ + 353, + 304, + 358, + 312 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "iterations of gradient descent in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "strongly-convex setting (Ablin et al., 2020, Proposition 3.2). While our guarantees concern the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 153, + 336 + ], + "score": 1.0, + "content": "Jacobian of", + "type": "text" + }, + { + "bbox": [ + 154, + 324, + 177, + 336 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 323, + 505, + 336 + ], + "score": 1.0, + "content": ", we note that other studies (Grazzi et al., 2020; Ji et al., 2021; Bertrand et al., 2021)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 405, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 405, + 347 + ], + "score": 1.0, + "content": "give guarantees on hypergradients (i.e., the gradient of an outer objective).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 333, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 333, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 333, + 365 + ], + "score": 1.0, + "content": "We illustrate these results on ridge regression, where", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 362, + 333, + 376 + ], + "spans": [ + { + "bbox": [ + 107, + 362, + 272, + 375 + ], + "score": 0.89, + "content": "\\begin{array} { r } { x ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x } \\| \\Phi x - y \\| _ { 2 } ^ { 2 } + \\sum _ { i } \\theta _ { i } x _ { i } ^ { \\beth } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 362, + 333, + 376 + ], + "score": 1.0, + "content": ". 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The results in Figure 2", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 462, + 333, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 333, + 473 + ], + "score": 1.0, + "content": "were obtained using the diabetes dataset from Efron et al.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 471, + 334, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 334, + 485 + ], + "score": 1.0, + "content": "(2004), with other datasets yielding a qualitatively similar", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 333, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 333, + 496 + ], + "score": 1.0, + "content": "behavior. 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Em-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 339, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 339, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "pirical error of implicit differentiation fol-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 339, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 339, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "lows closely the theoretical upper bound.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 340, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 340, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Unrolling achieves a much worse error", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 339, + 507, + 454, + 520 + ], + "spans": [ + { + "bbox": [ + 339, + 507, + 454, + 520 + ], + "score": 1.0, + "content": "for comparable iterate error.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "index": 39.0 + }, + { + "type": "title", + "bbox": [ + 108, + 517, + 200, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 201, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 201, + 531 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 541, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "To conclude this work, we demonstrate the ease of solving bi-level optimization problems with our", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 552, + 478, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 478, + 565 + ], + "score": 1.0, + "content": "framework. We also present an application to the sensitivity analysis of molecular dynamics.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + }, + { + "type": "title", + "bbox": [ + 109, + 577, + 380, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 380, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 380, + 590 + ], + "score": 1.0, + "content": "4.1 HYPERPARAMETER OPTIMIZATION OF MULTICLASS SVMS", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 106, + 597, + 506, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 611 + ], + "score": 1.0, + "content": "In this example, we consider the hyperparameter optimization of multiclass SVMs (Crammer &", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 280, + 621 + ], + "score": 1.0, + "content": "Singer, 2001) trained in the dual. 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The challenge in differentiating", + "type": "text" + }, + { + "bbox": [ + 367, + 631, + 391, + 642 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 631, + 470, + 643 + ], + "score": 1.0, + "content": "is that each row of", + "type": "text" + }, + { + "bbox": [ + 470, + 631, + 494, + 643 + ], + "score": 0.91, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 640, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 104, + 640, + 297, + 656 + ], + "score": 1.0, + "content": "constrained to belong to the probability simplex", + "type": "text" + }, + { + "bbox": [ + 297, + 641, + 311, + 652 + ], + "score": 0.88, + "content": "\\triangle ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 640, + 390, + 656 + ], + "score": 1.0, + "content": ". More formally, let", + "type": "text" + }, + { + "bbox": [ + 390, + 642, + 444, + 653 + ], + "score": 0.93, + "content": "X _ { \\mathsf { t r } } \\in \\mathbb { R } ^ { m \\times p }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 640, + 506, + 656 + ], + "score": 1.0, + "content": "be the training", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 650, + 507, + 667 + ], + "spans": [ + { + "bbox": [ + 104, + 650, + 186, + 667 + ], + "score": 1.0, + "content": "feature matrix and", + "type": "text" + }, + { + "bbox": [ + 186, + 652, + 258, + 664 + ], + "score": 0.9, + "content": "\\breve { Y } _ { \\mathrm { t r } } \\in \\bar { \\{ 0 , 1 \\} } ^ { m \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 650, + 507, + 667 + ], + "score": 1.0, + "content": "be the training labels (in row-wise one-hot encoding). Let", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 662, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 256, + 676 + ], + "score": 0.9, + "content": "W ( x , \\theta ) : = X _ { \\mathrm { t r } } ^ { \\top } ( Y _ { \\mathrm { t r } } - x ) / \\overset { } \\theta \\in \\overset { } \\mathbb { R } ^ { p \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 662, + 506, + 677 + ], + "score": 1.0, + "content": "be the dual-primal mapping. 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Let", + "type": "text" + }, + { + "bbox": [ + 256, + 82, + 337, + 93 + ], + "score": 0.9, + "content": "F : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { n } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 80, + 436, + 95 + ], + "score": 1.0, + "content": ". 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\\partial x ^ { \\star } ( \\theta ) \\| \\leq \\left( \\beta \\alpha ^ { - 1 } + \\gamma R \\alpha ^ { - 2 } \\right) \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\| . } \\end{array}", + "type": "interline_equation", + "image_path": "9f46752ccd33510fcf200f3960249df0d367cb5bb2933592959ba2cd6808261a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 191, + 160, + 420, + 176 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 506, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "score": 1.0, + "content": "This result is inspired by (Higham, 2002, Theorem 7.2), that is concerned with the stability of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 196, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 322, + 208 + ], + "score": 1.0, + "content": "solutions to inverse problems. Here we consider that", + "type": "text" + }, + { + "bbox": [ + 323, + 196, + 351, + 208 + ], + "score": 0.93, + "content": "A ( \\cdot , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 196, + 506, + 208 + ], + "score": 1.0, + "content": "is uniformly well-conditioned, rather", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 207, + 507, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 156, + 219 + ], + "score": 1.0, + "content": "than only at", + "type": "text" + }, + { + "bbox": [ + 156, + 207, + 180, + 219 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 207, + 330, + 219 + ], + "score": 1.0, + "content": ". This does not affect the first order in", + "type": "text" + }, + { + "bbox": [ + 331, + 209, + 337, + 217 + ], + "score": 0.79, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 207, + 496, + 219 + ], + "score": 1.0, + "content": "of this bound, and makes it valid for all", + "type": "text" + }, + { + "bbox": [ + 496, + 208, + 503, + 217 + ], + "score": 0.81, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 207, + 507, + 219 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 366, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 366, + 231 + ], + "score": 1.0, + "content": "It is also more tailored to applications to equation-specific cases.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 185, + 507, + 231 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 233, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 248 + ], + "score": 1.0, + "content": "Indeed, Theorem 1 can be applied to specific cases. In particular, for the gradient descent fixed point,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 401, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 133, + 259 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 245, + 242, + 258 + ], + "score": 0.93, + "content": "T ( x , \\theta ) = x - \\eta \\nabla _ { 1 } \\bar { f } ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 245, + 261, + 259 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 261, + 245, + 353, + 258 + ], + "score": 0.92, + "content": "F ( x , { \\boldsymbol { \\theta } } ) = T ( x , { \\overline { { \\theta } } } ) - x", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 245, + 401, + 259 + ], + "score": 1.0, + "content": ", this yields", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 233, + 506, + 259 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 262, + 417, + 276 + ], + "lines": [ + { + "bbox": [ + 193, + 262, + 417, + 276 + ], + "spans": [ + { + "bbox": [ + 193, + 262, + 417, + 276 + ], + "score": 0.86, + "content": "A ( x , \\theta ) = \\eta \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) \\mathrm { a n d } B ( x , \\theta ) = - \\eta \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "575276dad4e1574671a11f4d8d13f4404cf87388aceaa094861c3d08f947de79.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 193, + 262, + 417, + 276 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 506, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 398, + 293 + ], + "score": 1.0, + "content": "The guarantees on Jacobian precision under regularity conditions rely on", + "type": "text" + }, + { + "bbox": [ + 398, + 281, + 406, + 292 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "directly; see Corollary 1", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "in Appendix D. This reveals in particular that Jacobian estimation by implicit differentiation gains", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 352, + 315 + ], + "score": 1.0, + "content": "a factor of t compared to automatic differentiation, after", + "type": "text" + }, + { + "bbox": [ + 353, + 304, + 358, + 312 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "iterations of gradient descent in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "strongly-convex setting (Ablin et al., 2020, Proposition 3.2). While our guarantees concern the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 153, + 336 + ], + "score": 1.0, + "content": "Jacobian of", + "type": "text" + }, + { + "bbox": [ + 154, + 324, + 177, + 336 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 323, + 505, + 336 + ], + "score": 1.0, + "content": ", we note that other studies (Grazzi et al., 2020; Ji et al., 2021; Bertrand et al., 2021)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 405, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 405, + 347 + ], + "score": 1.0, + "content": "give guarantees on hypergradients (i.e., the gradient of an outer objective).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 280, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 333, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 333, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 333, + 365 + ], + "score": 1.0, + "content": "We illustrate these results on ridge regression, where", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 362, + 333, + 376 + ], + "spans": [ + { + "bbox": [ + 107, + 362, + 272, + 375 + ], + "score": 0.89, + "content": "\\begin{array} { r } { x ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x } \\| \\Phi x - y \\| _ { 2 } ^ { 2 } + \\sum _ { i } \\theta _ { i } x _ { i } ^ { \\beth } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 362, + 333, + 376 + ], + "score": 1.0, + "content": ". 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The results in Figure 2", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 462, + 333, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 333, + 473 + ], + "score": 1.0, + "content": "were obtained using the diabetes dataset from Efron et al.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 471, + 334, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 334, + 485 + ], + "score": 1.0, + "content": "(2004), with other datasets yielding a qualitatively similar", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 333, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 333, + 496 + ], + "score": 1.0, + "content": "behavior. 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Em-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 339, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 339, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "pirical error of implicit differentiation fol-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 339, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 339, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "lows closely the theoretical upper bound.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 340, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 340, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "Unrolling achieves a much worse error", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 339, + 507, + 454, + 520 + ], + "spans": [ + { + "bbox": [ + 339, + 507, + 454, + 520 + ], + "score": 1.0, + "content": "for comparable iterate error.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "index": 39.0 + }, + { + "type": "title", + "bbox": [ + 108, + 517, + 200, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 201, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 201, + 531 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 541, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "To conclude this work, we demonstrate the ease of solving bi-level optimization problems with our", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 552, + 478, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 478, + 565 + ], + "score": 1.0, + "content": "framework. 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This showcases that the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 264, + 316, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 316, + 277 + ], + "score": 1.0, + "content": "solver and fixed point can be independently chosen.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 132, + 309 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 296, + 215, + 307 + ], + "score": 0.91, + "content": "\\mathcal { C } = \\triangle ^ { k } \\times \\cdot \\cdot \\cdot \\times \\triangle ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 295, + 322, + 309 + ], + "score": 1.0, + "content": "is the Cartesian product of", + "type": "text" + }, + { + "bbox": [ + 322, + 299, + 333, + 307 + ], + "score": 0.71, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "probability simplices. 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y _ { \\mathrm { t r } } ) } _ { \\mathrm { o u t e r ~ p r o b l e m } } \\mathrm { ~ s u b j e c t ~ t o ~ } x ^ { \\star } ( \\theta ) \\in \\underbrace { x \\mathrm { g m i n } \\ f ( x , \\theta ; [ k ] ) + \\varepsilon \\| x \\| ^ { 2 } } _ { \\mathrm { i n n e r ~ p r o b l e m } } ,", + "type": "interline_equation", + "image_path": "5aaec9dff222bd5c452113222d7ddf7693baffc237f0684d7b184018d1131846.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 148, + 528, + 460, + 540.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 148, + 540.3333333333334, + 460, + 552.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 148, + 552.6666666666667, + 460, + 565.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 572, + 503, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 133, + 585 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 572, + 236, + 584 + ], + "score": 0.89, + "content": "f ( W , X ; 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For this experiment, we use the MNIST dataset. 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MethodL1 logregL2 logregDictL + L2 logregTask-driven DictL
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The proximal gradient fixed point (7) with step size", + "type": "text" + }, + { + "bbox": [ + 466, + 194, + 494, + 205 + ], + "score": 0.91, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 204, + 401, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 259, + 218 + ], + "score": 0.92, + "content": "T ( x , \\theta ) = \\bar { \\mathrm { p r o x } } _ { g } ( x - \\nabla _ { 1 } f \\bar { ( } x , \\theta _ { f } ) , \\theta _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 204, + 401, + 218 + ], + "score": 1.0, + "content": ". 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The stationarity, primal feasibility and complementary slackness conditions read", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 106, + 378, + 506, + 401 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 403, + 417, + 449 + ], + "lines": [ + { + "bbox": [ + 194, + 403, + 417, + 449 + ], + "spans": [ + { + "bbox": [ + 194, + 403, + 417, + 449 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta _ { f } ) + [ \\partial _ { 1 } G ( z , \\theta _ { G } ) ] ^ { \\top } \\lambda + [ \\partial _ { 1 } H ( z , \\theta _ { H } ) ] ^ { \\top } \\nu = 0 } \\\\ { H ( z , \\theta _ { H } ) = 0 } \\\\ { \\lambda \\circ G ( z , \\theta _ { G } ) = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "72abfbbf405af6a079f3008bbe7591c2f2d200e4f287cea7a05ce5068997ffe4.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 194, + 403, + 417, + 418.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 194, + 418.3333333333333, + 417, + 433.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 194, + 433.66666666666663, + 417, + 448.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 439, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 439, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 133, + 464 + ], + "score": 1.0, + "content": "Using", + "type": "text" + }, + { + "bbox": [ + 134, + 452, + 176, + 462 + ], + "score": 0.37, + "content": "{ \\dot { \\mathsf { J } } } \\mathsf { a x } . \\mathsf { v } { \\dot { \\mathsf { J } } } { \\mathsf { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 450, + 439, + 464 + ], + "score": 1.0, + "content": "to compute vector-Jacobian products, this can be implemented as", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 450, + 439, + 464 + ] + }, + { + "type": "image", + "bbox": [ + 128, + 473, + 484, + 669 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 473, + 484, + 669 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 128, + 473, + 484, + 669 + ], + "spans": [ + { + "bbox": [ + 128, + 473, + 484, + 669 + ], + "score": 0.953, + "type": "image", + "image_path": "4c5afa2d3ce2393d2ed5ba74f895571c86cc2a907b4316ffe454af22a9a5c182.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 128, + 473, + 484, + 538.3333333333334 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 128, + 538.3333333333334, + 484, + 603.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 128, + 603.6666666666667, + 484, + 669.0000000000001 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 236, + 678, + 375, + 691 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 235, + 677, + 374, + 692 + ], + "spans": [ + { + "bbox": [ + 235, + 677, + 342, + 692 + ], + "score": 1.0, + "content": "Figure 7: KKT conditions", + "type": "text" + }, + { + "bbox": [ + 343, + 678, + 374, + 690 + ], + "score": 0.89, + "content": "F ( x , \\theta )", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + } + ], + "index": 22.0 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 179, + 723 + ], + "score": 1.0, + "content": "Similar mappings", + "type": "text" + }, + { + "bbox": [ + 179, + 710, + 189, + 720 + ], + "score": 0.8, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "can be written if the optimization problem contains only equality constraints or", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 721, + 217, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 217, + 734 + ], + "score": 1.0, + "content": "only inequality constraints.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 708, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 492, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 493, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 262, + 96 + ], + "score": 1.0, + "content": "Mirror descent fixed point. 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As a general guideline, whenever a projection enjoys a closed form, we leave", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 454, + 282, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 282, + 468 + ], + "score": 1.0, + "content": "the Jacobian product to the autodiff system.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 432, + 505, + 468 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 481, + 298, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 299, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 299, + 494 + ], + "score": 1.0, + "content": "B.1 JACOBIAN PRODUCTS OF PROJECTIONS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "We describe in this section how to compute the Jacobian products of the projections (in the Euclidean", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "and KL senses) onto various convex sets. When the convex set does not depend on any variable, we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 524, + 247, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 173, + 537 + ], + "score": 1.0, + "content": "simply denote it", + "type": "text" + }, + { + "bbox": [ + 173, + 525, + 180, + 534 + ], + "score": 0.82, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 524, + 223, + 537 + ], + "score": 1.0, + "content": "instead of", + "type": "text" + }, + { + "bbox": [ + 223, + 524, + 243, + 536 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 524, + 247, + 537 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 501, + 505, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 240, + 563 + ], + "score": 1.0, + "content": "Non-negative orthant. When", + "type": "text" + }, + { + "bbox": [ + 241, + 550, + 248, + 559 + ], + "score": 0.78, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 547, + 367, + 563 + ], + "score": 1.0, + "content": "is the non-negative orthant,", + "type": "text" + }, + { + "bbox": [ + 368, + 549, + 406, + 562 + ], + "score": 0.92, + "content": "\\mathcal { C } = \\mathbb { R } _ { + } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 547, + 455, + 563 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + }, + { + "bbox": [ + 456, + 549, + 505, + 562 + ], + "score": 0.88, + "content": "\\mathrm { p r o j } _ { \\mathcal { C } } ( y ) =", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 148, + 572 + ], + "score": 0.88, + "content": "\\operatorname* { m a x } ( y , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 560, + 506, + 573 + ], + "score": 1.0, + "content": ", where the maximum is evaluated element-wise. This is also known as the ReLu function.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 570, + 458, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 376, + 585 + ], + "score": 1.0, + "content": "The projection in the KL sense reduces to the exponential function,", + "type": "text" + }, + { + "bbox": [ + 377, + 571, + 454, + 584 + ], + "score": 0.91, + "content": "\\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y ) = \\exp ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 570, + 458, + 585 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 547, + 506, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 597, + 483, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 482, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 211, + 611 + ], + "score": 1.0, + "content": "Box constraints. When", + "type": "text" + }, + { + "bbox": [ + 212, + 598, + 231, + 610 + ], + "score": 0.91, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 596, + 319, + 611 + ], + "score": 1.0, + "content": "is the box constraints", + "type": "text" + }, + { + "bbox": [ + 320, + 597, + 385, + 610 + ], + "score": 0.93, + "content": "\\mathcal { C } ( \\theta ) = [ \\theta _ { 1 } , \\theta _ { 2 } ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 596, + 406, + 611 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 407, + 597, + 437, + 608 + ], + "score": 0.92, + "content": "\\theta \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 596, + 482, + 611 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 596, + 482, + 611 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 616, + 412, + 630 + ], + "lines": [ + { + "bbox": [ + 197, + 616, + 412, + 630 + ], + "spans": [ + { + "bbox": [ + 197, + 616, + 412, + 630 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta ) = \\mathrm { c l i p } ( y , \\theta _ { 1 } , \\theta _ { 2 } ) : = \\operatorname* { m a x } ( \\operatorname* { m i n } ( y , \\theta _ { 2 } ) , \\theta _ { 1 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "5023ae9f47f81eca27c5171abe04c3c37deb6538bfbff98d071383acd0349193.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 197, + 616, + 412, + 630 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 493, + 650 + ], + "lines": [ + { + "bbox": [ + 104, + 636, + 491, + 650 + ], + "spans": [ + { + "bbox": [ + 104, + 636, + 451, + 650 + ], + "score": 1.0, + "content": "This is trivially extended to support different boxes for each coordinate, in which case", + "type": "text" + }, + { + "bbox": [ + 451, + 637, + 491, + 648 + ], + "score": 0.93, + "content": "\\theta \\in \\mathbb { R } ^ { d \\times 2 }", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 636, + 491, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 662, + 506, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 229, + 675 + ], + "score": 1.0, + "content": "Probability simplex. When", + "type": "text" + }, + { + "bbox": [ + 229, + 664, + 236, + 673 + ], + "score": 0.79, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 662, + 381, + 675 + ], + "score": 1.0, + "content": "is the standard probability simplex,", + "type": "text" + }, + { + "bbox": [ + 381, + 662, + 415, + 673 + ], + "score": 0.91, + "content": "{ \\mathcal { C } } = \\triangle ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 662, + 506, + 675 + ], + "score": 1.0, + "content": ", there is no analytical", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 673, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 155, + 687 + ], + "score": 1.0, + "content": "solution for", + "type": "text" + }, + { + "bbox": [ + 156, + 675, + 191, + 686 + ], + "score": 0.82, + "content": "\\operatorname { p r o j } _ { \\mathcal { C } } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 673, + 423, + 687 + ], + "score": 1.0, + "content": ". 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The projection in the KL sense, on the other hand, enjoys a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 714, + 432, + 737 + ], + "spans": [ + { + "bbox": [ + 103, + 714, + 284, + 737 + ], + "score": 1.0, + "content": "closed form: it reduces to the usual softmax", + "type": "text" + }, + { + "bbox": [ + 284, + 718, + 427, + 735 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathrm { p r o j } _ { \\mathcal { C } } ^ { \\varphi } ( y ) = \\exp ( y ) / \\sum _ { j = 1 } ^ { d } \\exp ( y _ { j } ) , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 714, + 432, + 737 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 103, + 662, + 506, + 737 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 336, + 96 + ], + "score": 1.0, + "content": "Box sections. Consider now the Euclidean projection", + "type": "text" + }, + { + "bbox": [ + 336, + 82, + 419, + 95 + ], + "score": 0.92, + "content": "z ^ { \\star } ( \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 81, + 472, + 96 + ], + "score": 1.0, + "content": "onto the set", + "type": "text" + }, + { + "bbox": [ + 472, + 82, + 505, + 95 + ], + "score": 0.9, + "content": "{ \\mathcal { C } } ( \\theta ) =", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 289, + 107 + ], + "score": 0.9, + "content": "\\{ z \\in \\mathbb { R } ^ { d } \\colon \\alpha _ { i } \\leq z _ { i } \\leq \\beta _ { i } , i \\in [ d ] ; w ^ { \\top } z = c \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 93, + 321, + 109 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 321, + 95, + 387, + 107 + ], + "score": 0.92, + "content": "\\theta = ( \\alpha , \\beta , w , c )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 93, + 507, + 109 + ], + "score": 1.0, + "content": ". 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It is easy to check (see, e.g., (Niculae & Martins, 2020))", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 291, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 264, + 129 + ], + "score": 1.0, + "content": "that an optimal solution satisfies for all", + "type": "text" + }, + { + "bbox": [ + 264, + 117, + 291, + 129 + ], + "score": 0.92, + "content": "i \\in [ d ]", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 132, + 412, + 146 + ], + "lines": [ + { + "bbox": [ + 199, + 132, + 412, + 146 + ], + "spans": [ + { + "bbox": [ + 199, + 132, + 412, + 146 + ], + "score": 0.88, + "content": "z _ { i } ^ { \\star } ( \\theta ) = [ L ( x ^ { \\star } ( \\theta ) , \\theta ) ] _ { i } : = \\mathrm { c l i p } ( w _ { i } x ^ { \\star } ( \\theta ) + y _ { i } , \\alpha _ { i } , \\beta _ { i } )", + "type": "interline_equation", + "image_path": "f785b0379b0466b943cf009cc33655fc6617a40801a0665a3c04308d86c93a17.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 199, + 132, + 412, + 146 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 150, + 504, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 148, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 133, + 163 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 149, + 208, + 160 + ], + "score": 0.91, + "content": "L \\colon \\mathbb { R } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 148, + 337, + 163 + ], + "score": 1.0, + "content": "is the dual-primal mapping and", + "type": "text" + }, + { + "bbox": [ + 338, + 150, + 381, + 162 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 148, + 506, + 163 + ], + "score": 1.0, + "content": "is the optimal dual variable of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 161, + 303, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 303, + 172 + ], + "score": 1.0, + "content": "the linear constraint, which should be the root of", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 175, + 377, + 190 + ], + "lines": [ + { + "bbox": [ + 233, + 175, + 377, + 190 + ], + "spans": [ + { + "bbox": [ + 233, + 175, + 377, + 190 + ], + "score": 0.92, + "content": "F ( x ^ { \\star } ( \\theta ) , \\theta ) = L ( x ^ { \\star } ( \\theta ) , \\theta ) ^ { \\top } w - c .", + "type": "interline_equation", + "image_path": "456d01b81789a3d57b2cac845912db216d847a471e5e54dd18a0eaab79ee20f5.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 233, + 175, + 377, + 190 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 194, + 504, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 321, + 208 + ], + "score": 1.0, + "content": "The root can be found, e.g., by bisection. The gradient", + "type": "text" + }, + { + "bbox": [ + 322, + 194, + 354, + 207 + ], + "score": 0.92, + "content": "\\nabla \\boldsymbol { x } ^ { \\star } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 193, + 400, + 208 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 401, + 194, + 473, + 207 + ], + "score": 0.94, + "content": "\\nabla x ^ { \\star } ( \\theta ) = B ^ { \\top } / A", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 193, + 505, + 208 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 372, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 144, + 218 + ], + "score": 1.0, + "content": "Jacobian", + "type": "text" + }, + { + "bbox": [ + 144, + 205, + 173, + 218 + ], + "score": 0.92, + "content": "\\partial z ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 205, + 360, + 218 + ], + "score": 1.0, + "content": "is obtained by application of the chain rule on", + "type": "text" + }, + { + "bbox": [ + 360, + 207, + 367, + 215 + ], + "score": 0.83, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 205, + 372, + 218 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 194, + 243 + ], + "score": 1.0, + "content": "Norm balls. When", + "type": "text" + }, + { + "bbox": [ + 195, + 228, + 312, + 241 + ], + "score": 0.91, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon \\| x \\| \\leq \\theta \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 227, + 344, + 243 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 344, + 229, + 362, + 241 + ], + "score": 0.91, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 227, + 423, + 243 + ], + "score": 1.0, + "content": "is a norm and", + "type": "text" + }, + { + "bbox": [ + 423, + 230, + 456, + 241 + ], + "score": 0.89, + "content": "\\theta \\in \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 227, + 460, + 243 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 460, + 229, + 505, + 241 + ], + "score": 0.88, + "content": "\\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta )", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 239, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 378, + 253 + ], + "score": 1.0, + "content": "becomes the projection onto a norm ball. The projection onto the", + "type": "text" + }, + { + "bbox": [ + 379, + 240, + 388, + 251 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "-ball reduces to a projection", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 412, + 265 + ], + "score": 1.0, + "content": "onto the simplex, see, e.g., (Duchi et al., 2008). The projections onto the", + "type": "text" + }, + { + "bbox": [ + 412, + 251, + 423, + 262 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 249, + 442, + 265 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 442, + 251, + 456, + 262 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 249, + 506, + 265 + ], + "score": 1.0, + "content": "balls enjoy", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 304, + 275 + ], + "score": 1.0, + "content": "a closed-form, see, e.g., (Parikh & Boyd, 2014,", + "type": "text" + }, + { + "bbox": [ + 304, + 262, + 325, + 273 + ], + "score": 0.72, + "content": "\\ S 6 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 261, + 506, + 275 + ], + "score": 1.0, + "content": ". Since they rely on simple composition of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 272, + 412, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 412, + 286 + ], + "score": 1.0, + "content": "functions, all three projections can therefore be automatically differentiated.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 105, + 295, + 501, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 501, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 189, + 310 + ], + "score": 1.0, + "content": "Affine sets. When", + "type": "text" + }, + { + "bbox": [ + 189, + 295, + 300, + 309 + ], + "score": 0.91, + "content": "{ \\mathcal { C } } ( \\theta ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon A x = b \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 293, + 330, + 310 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 331, + 295, + 373, + 307 + ], + "score": 0.89, + "content": "A \\in \\mathbb { R } ^ { p \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 293, + 377, + 310 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 378, + 296, + 406, + 307 + ], + "score": 0.88, + "content": "b \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 293, + 425, + 310 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 425, + 296, + 468, + 308 + ], + "score": 0.93, + "content": "\\theta = ( A , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 293, + 501, + 310 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 312, + 428, + 327 + ], + "lines": [ + { + "bbox": [ + 183, + 312, + 428, + 327 + ], + "spans": [ + { + "bbox": [ + 183, + 312, + 428, + 327 + ], + "score": 0.88, + "content": "\\operatorname { p r o j } _ { \\mathcal { C } } ( y , \\theta ) = y - A ^ { \\dagger } ( A y - b ) = y - A ^ { \\top } ( A A ^ { \\top } ) ^ { - 1 } ( A y - b )", + "type": "interline_equation", + "image_path": "c04a70cad22aa1a789af6aab9449d621c81b2d4668f11a17fd1147384ac7f868.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 183, + 312, + 428, + 327 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 506, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 133, + 344 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 331, + 146, + 343 + ], + "score": 0.88, + "content": "A ^ { \\dagger }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 332, + 309, + 344 + ], + "score": 1.0, + "content": "is the Moore-Penrose pseudoinverse of", + "type": "text" + }, + { + "bbox": [ + 309, + 333, + 318, + 342 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 332, + 441, + 344 + ], + "score": 1.0, + "content": ". The second equality holds if", + "type": "text" + }, + { + "bbox": [ + 442, + 333, + 467, + 344 + ], + "score": 0.91, + "content": "p < d", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 332, + 486, + 344 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 486, + 333, + 494, + 342 + ], + "score": 0.84, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 507, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 480, + 356 + ], + "score": 1.0, + "content": "full rank. A practical implementation can pre-compute a factorization of the Gram matrix", + "type": "text" + }, + { + "bbox": [ + 480, + 343, + 502, + 354 + ], + "score": 0.89, + "content": "A A ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 342, + 507, + 356 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 354, + 311, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 311, + 366 + ], + "score": 1.0, + "content": "Alternatively, we can also use the KKT conditions.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 267, + 391 + ], + "score": 1.0, + "content": "Hyperplanes and half spaces. When", + "type": "text" + }, + { + "bbox": [ + 267, + 377, + 383, + 390 + ], + "score": 0.9, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon a ^ { \\top } x = b \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 375, + 413, + 391 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 413, + 377, + 444, + 388 + ], + "score": 0.92, + "content": "a \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 375, + 462, + 391 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 378, + 487, + 388 + ], + "score": 0.91, + "content": "b \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 375, + 506, + 391 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 388, + 181, + 402 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 147, + 401 + ], + "score": 0.92, + "content": "\\theta = ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 388, + 181, + 402 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 398, + 366, + 427 + ], + "lines": [ + { + "bbox": [ + 244, + 398, + 366, + 427 + ], + "spans": [ + { + "bbox": [ + 244, + 398, + 366, + 427 + ], + "score": 0.94, + "content": "\\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta ) = y - \\frac { { a ^ { \\top } y - b } } { \\Vert a \\Vert _ { 2 } ^ { 2 } } a .", + "type": "interline_equation", + "image_path": "117a5d69337ae35d7b97e3b7471c1416afc6aeace0cd0ef458716fda906109a0.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 398, + 366, + 412.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 244, + 412.5, + 366, + 427.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 503, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 131, + 443 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 132, + 428, + 246, + 442 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon a ^ { \\top } x \\leq b \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 426, + 322, + 443 + ], + "score": 1.0, + "content": ", we simply replace", + "type": "text" + }, + { + "bbox": [ + 323, + 429, + 356, + 441 + ], + "score": 0.92, + "content": "a ^ { \\top } y - b", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 426, + 434, + 443 + ], + "score": 1.0, + "content": "in the numerator by", + "type": "text" + }, + { + "bbox": [ + 435, + 428, + 503, + 442 + ], + "score": 0.89, + "content": "\\operatorname* { m a x } ( a ^ { \\top } y - b , 0 )", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 506, + 543 + ], + "lines": [ + { + "bbox": [ + 104, + 450, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 318, + 468 + ], + "score": 1.0, + "content": "Transportation and Birkhoff polytopes. When", + "type": "text" + }, + { + "bbox": [ + 318, + 452, + 504, + 465 + ], + "score": 0.88, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ X \\in \\mathbb { R } ^ { p \\times d } \\colon X \\mathbf { 1 } _ { d } = \\boldsymbol { \\theta } _ { 1 } , X ^ { \\top } \\mathbf { 1 } _ { p } =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 107, + 465, + 154, + 478 + ], + "score": 0.92, + "content": "\\theta _ { 2 } , X \\geq 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 465, + 338, + 478 + ], + "score": 1.0, + "content": ", the so-called transportation polytope, where", + "type": "text" + }, + { + "bbox": [ + 339, + 466, + 375, + 477 + ], + "score": 0.91, + "content": "\\theta _ { 1 } \\in \\triangle ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 465, + 393, + 478 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 465, + 429, + 477 + ], + "score": 0.9, + "content": "\\theta _ { 2 } \\in \\triangle ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "are marginals, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "score": 1.0, + "content": "can compute approximately the projections, both in the Euclidean and KL senses, by switching to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "score": 1.0, + "content": "the dual or semi-dual (Blondel et al., 2018). Since both are unconstrained optimization problems,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "we can compute their Jacobian product by implicit differentiation using the gradient descent fixed", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "point. An advantage of the KL geometry here is that we can use Sinkhorn (Cuturi, 2013), which is a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "GPU-friendly algorithm. The Birkhoff polytope, the set of doubly stochastic matrices, is obtained by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 203, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 132, + 544 + ], + "score": 1.0, + "content": "fixing", + "type": "text" + }, + { + "bbox": [ + 132, + 531, + 199, + 543 + ], + "score": 0.92, + "content": "{ \\theta } _ { 1 } = \\bar { \\theta } _ { 2 } \\stackrel { - } { = } { \\mathbf { 1 } } _ { d } / d", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 531, + 203, + 544 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 553, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 211, + 568 + ], + "score": 1.0, + "content": "Order simplex. When", + "type": "text" + }, + { + "bbox": [ + 211, + 554, + 452, + 567 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ( \\theta ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon \\theta _ { 1 } \\geq x _ { 1 } \\geq x _ { 2 } \\geq \\cdots \\geq x _ { d } \\geq \\theta _ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 552, + 506, + 568 + ], + "score": 1.0, + "content": ", a so-called", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "order simplex (Grotzinger & Witzgall, 1984; Blondel, 2019), the projection operations, both in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "Euclidean and KL sense, reduce to isotonic optimization (Lim & Wright, 2016) and can be solved", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 150, + 599 + ], + "score": 1.0, + "content": "exactly in", + "type": "text" + }, + { + "bbox": [ + 150, + 587, + 194, + 600 + ], + "score": 0.92, + "content": "O ( d \\log d )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "time using the Pool Adjacent Violators algorithm (Best et al., 2000). The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "Jacobian of the projections and efficient product with it are derived in (Djolonga & Krause, 2017;", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 192, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 192, + 622 + ], + "score": 1.0, + "content": "Blondel et al., 2020).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 443, + 646 + ], + "score": 1.0, + "content": "Polyhedra. More generally, we can consider polyhedra, i.e., sets of the form", + "type": "text" + }, + { + "bbox": [ + 443, + 632, + 505, + 645 + ], + "score": 0.91, + "content": "{ \\mathcal { C } } ( \\theta ) ~ = ~ \\{ x ~ \\in ~", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 640, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 107, + 643, + 203, + 655 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d } \\colon A x = b , C x \\leq \\mathbf { \\bar { d } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 640, + 235, + 658 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 235, + 643, + 280, + 654 + ], + "score": 0.9, + "content": "A \\in \\mathbb { R } ^ { p \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 640, + 284, + 658 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 284, + 644, + 316, + 654 + ], + "score": 0.81, + "content": "b \\in \\bar { \\mathbb { R } } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 640, + 320, + 658 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 321, + 643, + 369, + 654 + ], + "score": 0.86, + "content": "C \\in \\mathbb { R } ^ { m \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 640, + 390, + 658 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 391, + 644, + 426, + 654 + ], + "score": 0.91, + "content": "d \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 640, + 507, + 658 + ], + "score": 1.0, + "content": ". There are several", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 471, + 667 + ], + "score": 1.0, + "content": "ways to differentiate this projection. The first is to use the KKT conditions as detailed in", + "type": "text" + }, + { + "bbox": [ + 471, + 655, + 491, + 666 + ], + "score": 0.86, + "content": "\\ S 2 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 654, + 506, + 667 + ], + "score": 1.0, + "content": ". A", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "second way is consider the dual of the projection instead, which is the maximization of a quadratic", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 385, + 689 + ], + "score": 1.0, + "content": "function subject to non-negative constraints (Parikh & Boyd, 2014,", + "type": "text" + }, + { + "bbox": [ + 386, + 677, + 407, + 688 + ], + "score": 0.67, + "content": "\\ S 6 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 676, + 506, + 689 + ], + "score": 1.0, + "content": ". That is, we can reduce", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "the projection on a polyhedron to a problem of the form (8) with non-negative constraints, which we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "can in turn implicitly differentiate easily using the projected gradient fixed point, combined with the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "projection on the non-negative orthant. 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Consider now the Euclidean projection", + "type": "text" + }, + { + "bbox": [ + 336, + 82, + 419, + 95 + ], + "score": 0.92, + "content": "z ^ { \\star } ( \\theta ) = \\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 81, + 472, + 96 + ], + "score": 1.0, + "content": "onto the set", + "type": "text" + }, + { + "bbox": [ + 472, + 82, + 505, + 95 + ], + "score": 0.9, + "content": "{ \\mathcal { C } } ( \\theta ) =", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 289, + 107 + ], + "score": 0.9, + "content": "\\{ z \\in \\mathbb { R } ^ { d } \\colon \\alpha _ { i } \\leq z _ { i } \\leq \\beta _ { i } , i \\in [ d ] ; w ^ { \\top } z = c \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 93, + 321, + 109 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 321, + 95, + 387, + 107 + ], + "score": 0.92, + "content": "\\theta = ( \\alpha , \\beta , w , c )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 93, + 507, + 109 + ], + "score": 1.0, + "content": ". This projection is a singly-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "constrained bounded quadratic program. It is easy to check (see, e.g., (Niculae & Martins, 2020))", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 291, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 264, + 129 + ], + "score": 1.0, + "content": "that an optimal solution satisfies for all", + "type": "text" + }, + { + "bbox": [ + 264, + 117, + 291, + 129 + ], + "score": 0.92, + "content": "i \\in [ d ]", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 81, + 507, + 129 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 132, + 412, + 146 + ], + "lines": [ + { + "bbox": [ + 199, + 132, + 412, + 146 + ], + "spans": [ + { + "bbox": [ + 199, + 132, + 412, + 146 + ], + "score": 0.88, + "content": "z _ { i } ^ { \\star } ( \\theta ) = [ L ( x ^ { \\star } ( \\theta ) , \\theta ) ] _ { i } : = \\mathrm { c l i p } ( w _ { i } x ^ { \\star } ( \\theta ) + y _ { i } , \\alpha _ { i } , \\beta _ { i } )", + "type": "interline_equation", + "image_path": "f785b0379b0466b943cf009cc33655fc6617a40801a0665a3c04308d86c93a17.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 199, + 132, + 412, + 146 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 150, + 504, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 148, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 133, + 163 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 149, + 208, + 160 + ], + "score": 0.91, + "content": "L \\colon \\mathbb { R } \\times \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 148, + 337, + 163 + ], + "score": 1.0, + "content": "is the dual-primal mapping and", + "type": "text" + }, + { + "bbox": [ + 338, + 150, + 381, + 162 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 148, + 506, + 163 + ], + "score": 1.0, + "content": "is the optimal dual variable of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 161, + 303, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 303, + 172 + ], + "score": 1.0, + "content": "the linear constraint, which should be the root of", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 148, + 506, + 172 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 175, + 377, + 190 + ], + "lines": [ + { + "bbox": [ + 233, + 175, + 377, + 190 + ], + "spans": [ + { + "bbox": [ + 233, + 175, + 377, + 190 + ], + "score": 0.92, + "content": "F ( x ^ { \\star } ( \\theta ) , \\theta ) = L ( x ^ { \\star } ( \\theta ) , \\theta ) ^ { \\top } w - c .", + "type": "interline_equation", + "image_path": "456d01b81789a3d57b2cac845912db216d847a471e5e54dd18a0eaab79ee20f5.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 233, + 175, + 377, + 190 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 194, + 504, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 321, + 208 + ], + "score": 1.0, + "content": "The root can be found, e.g., by bisection. The gradient", + "type": "text" + }, + { + "bbox": [ + 322, + 194, + 354, + 207 + ], + "score": 0.92, + "content": "\\nabla \\boldsymbol { x } ^ { \\star } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 193, + 400, + 208 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 401, + 194, + 473, + 207 + ], + "score": 0.94, + "content": "\\nabla x ^ { \\star } ( \\theta ) = B ^ { \\top } / A", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 193, + 505, + 208 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 372, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 144, + 218 + ], + "score": 1.0, + "content": "Jacobian", + "type": "text" + }, + { + "bbox": [ + 144, + 205, + 173, + 218 + ], + "score": 0.92, + "content": "\\partial z ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 205, + 360, + 218 + ], + "score": 1.0, + "content": "is obtained by application of the chain rule on", + "type": "text" + }, + { + "bbox": [ + 360, + 207, + 367, + 215 + ], + "score": 0.83, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 205, + 372, + 218 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 193, + 505, + 218 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 194, + 243 + ], + "score": 1.0, + "content": "Norm balls. When", + "type": "text" + }, + { + "bbox": [ + 195, + 228, + 312, + 241 + ], + "score": 0.91, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon \\| x \\| \\leq \\theta \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 227, + 344, + 243 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 344, + 229, + 362, + 241 + ], + "score": 0.91, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 227, + 423, + 243 + ], + "score": 1.0, + "content": "is a norm and", + "type": "text" + }, + { + "bbox": [ + 423, + 230, + 456, + 241 + ], + "score": 0.89, + "content": "\\theta \\in \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 227, + 460, + 243 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 460, + 229, + 505, + 241 + ], + "score": 0.88, + "content": "\\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta )", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 239, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 378, + 253 + ], + "score": 1.0, + "content": "becomes the projection onto a norm ball. The projection onto the", + "type": "text" + }, + { + "bbox": [ + 379, + 240, + 388, + 251 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "-ball reduces to a projection", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 412, + 265 + ], + "score": 1.0, + "content": "onto the simplex, see, e.g., (Duchi et al., 2008). The projections onto the", + "type": "text" + }, + { + "bbox": [ + 412, + 251, + 423, + 262 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 249, + 442, + 265 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 442, + 251, + 456, + 262 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 249, + 506, + 265 + ], + "score": 1.0, + "content": "balls enjoy", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 304, + 275 + ], + "score": 1.0, + "content": "a closed-form, see, e.g., (Parikh & Boyd, 2014,", + "type": "text" + }, + { + "bbox": [ + 304, + 262, + 325, + 273 + ], + "score": 0.72, + "content": "\\ S 6 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 261, + 506, + 275 + ], + "score": 1.0, + "content": ". Since they rely on simple composition of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 272, + 412, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 412, + 286 + ], + "score": 1.0, + "content": "functions, all three projections can therefore be automatically differentiated.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 227, + 506, + 286 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 295, + 501, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 501, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 189, + 310 + ], + "score": 1.0, + "content": "Affine sets. When", + "type": "text" + }, + { + "bbox": [ + 189, + 295, + 300, + 309 + ], + "score": 0.91, + "content": "{ \\mathcal { C } } ( \\theta ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon A x = b \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 293, + 330, + 310 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 331, + 295, + 373, + 307 + ], + "score": 0.89, + "content": "A \\in \\mathbb { R } ^ { p \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 293, + 377, + 310 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 378, + 296, + 406, + 307 + ], + "score": 0.88, + "content": "b \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 293, + 425, + 310 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 425, + 296, + 468, + 308 + ], + "score": 0.93, + "content": "\\theta = ( A , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 293, + 501, + 310 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 293, + 501, + 310 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 312, + 428, + 327 + ], + "lines": [ + { + "bbox": [ + 183, + 312, + 428, + 327 + ], + "spans": [ + { + "bbox": [ + 183, + 312, + 428, + 327 + ], + "score": 0.88, + "content": "\\operatorname { p r o j } _ { \\mathcal { C } } ( y , \\theta ) = y - A ^ { \\dagger } ( A y - b ) = y - A ^ { \\top } ( A A ^ { \\top } ) ^ { - 1 } ( A y - b )", + "type": "interline_equation", + "image_path": "c04a70cad22aa1a789af6aab9449d621c81b2d4668f11a17fd1147384ac7f868.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 183, + 312, + 428, + 327 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 506, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 133, + 344 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 331, + 146, + 343 + ], + "score": 0.88, + "content": "A ^ { \\dagger }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 332, + 309, + 344 + ], + "score": 1.0, + "content": "is the Moore-Penrose pseudoinverse of", + "type": "text" + }, + { + "bbox": [ + 309, + 333, + 318, + 342 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 332, + 441, + 344 + ], + "score": 1.0, + "content": ". 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A practical implementation can pre-compute a factorization of the Gram matrix", + "type": "text" + }, + { + "bbox": [ + 480, + 343, + 502, + 354 + ], + "score": 0.89, + "content": "A A ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 342, + 507, + 356 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 354, + 311, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 311, + 366 + ], + "score": 1.0, + "content": "Alternatively, we can also use the KKT conditions.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 331, + 507, + 366 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 267, + 391 + ], + "score": 1.0, + "content": "Hyperplanes and half spaces. When", + "type": "text" + }, + { + "bbox": [ + 267, + 377, + 383, + 390 + ], + "score": 0.9, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon a ^ { \\top } x = b \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 375, + 413, + 391 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 413, + 377, + 444, + 388 + ], + "score": 0.92, + "content": "a \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 375, + 462, + 391 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 378, + 487, + 388 + ], + "score": 0.91, + "content": "b \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 375, + 506, + 391 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 388, + 181, + 402 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 147, + 401 + ], + "score": 0.92, + "content": "\\theta = ( a , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 388, + 181, + 402 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 375, + 506, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 398, + 366, + 427 + ], + "lines": [ + { + "bbox": [ + 244, + 398, + 366, + 427 + ], + "spans": [ + { + "bbox": [ + 244, + 398, + 366, + 427 + ], + "score": 0.94, + "content": "\\mathrm { p r o j } _ { \\mathcal { C } } ( y , \\theta ) = y - \\frac { { a ^ { \\top } y - b } } { \\Vert a \\Vert _ { 2 } ^ { 2 } } a .", + "type": "interline_equation", + "image_path": "117a5d69337ae35d7b97e3b7471c1416afc6aeace0cd0ef458716fda906109a0.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 398, + 366, + 412.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 244, + 412.5, + 366, + 427.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 503, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 131, + 443 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 132, + 428, + 246, + 442 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon a ^ { \\top } x \\leq b \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 426, + 322, + 443 + ], + "score": 1.0, + "content": ", we simply replace", + "type": "text" + }, + { + "bbox": [ + 323, + 429, + 356, + 441 + ], + "score": 0.92, + "content": "a ^ { \\top } y - b", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 426, + 434, + 443 + ], + "score": 1.0, + "content": "in the numerator by", + "type": "text" + }, + { + "bbox": [ + 435, + 428, + 503, + 442 + ], + "score": 0.89, + "content": "\\operatorname* { m a x } ( a ^ { \\top } y - b , 0 )", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 426, + 503, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 506, + 543 + ], + "lines": [ + { + "bbox": [ + 104, + 450, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 318, + 468 + ], + "score": 1.0, + "content": "Transportation and Birkhoff polytopes. When", + "type": "text" + }, + { + "bbox": [ + 318, + 452, + 504, + 465 + ], + "score": 0.88, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } ) = \\{ X \\in \\mathbb { R } ^ { p \\times d } \\colon X \\mathbf { 1 } _ { d } = \\boldsymbol { \\theta } _ { 1 } , X ^ { \\top } \\mathbf { 1 } _ { p } =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 107, + 465, + 154, + 478 + ], + "score": 0.92, + "content": "\\theta _ { 2 } , X \\geq 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 465, + 338, + 478 + ], + "score": 1.0, + "content": ", the so-called transportation polytope, where", + "type": "text" + }, + { + "bbox": [ + 339, + 466, + 375, + 477 + ], + "score": 0.91, + "content": "\\theta _ { 1 } \\in \\triangle ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 465, + 393, + 478 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 465, + 429, + 477 + ], + "score": 0.9, + "content": "\\theta _ { 2 } \\in \\triangle ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "are marginals, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 506, + 489 + ], + "score": 1.0, + "content": "can compute approximately the projections, both in the Euclidean and KL senses, by switching to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "score": 1.0, + "content": "the dual or semi-dual (Blondel et al., 2018). Since both are unconstrained optimization problems,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "we can compute their Jacobian product by implicit differentiation using the gradient descent fixed", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "point. An advantage of the KL geometry here is that we can use Sinkhorn (Cuturi, 2013), which is a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "GPU-friendly algorithm. The Birkhoff polytope, the set of doubly stochastic matrices, is obtained by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 203, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 132, + 544 + ], + "score": 1.0, + "content": "fixing", + "type": "text" + }, + { + "bbox": [ + 132, + 531, + 199, + 543 + ], + "score": 0.92, + "content": "{ \\theta } _ { 1 } = \\bar { \\theta } _ { 2 } \\stackrel { - } { = } { \\mathbf { 1 } } _ { d } / d", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 531, + 203, + 544 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 450, + 506, + 544 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 553, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 211, + 568 + ], + "score": 1.0, + "content": "Order simplex. When", + "type": "text" + }, + { + "bbox": [ + 211, + 554, + 452, + 567 + ], + "score": 0.86, + "content": "{ \\mathcal { C } } ( \\theta ) = \\{ x \\in \\mathbb { R } ^ { d } \\colon \\theta _ { 1 } \\geq x _ { 1 } \\geq x _ { 2 } \\geq \\cdots \\geq x _ { d } \\geq \\theta _ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 552, + 506, + 568 + ], + "score": 1.0, + "content": ", a so-called", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "order simplex (Grotzinger & Witzgall, 1984; Blondel, 2019), the projection operations, both in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "Euclidean and KL sense, reduce to isotonic optimization (Lim & Wright, 2016) and can be solved", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 150, + 599 + ], + "score": 1.0, + "content": "exactly in", + "type": "text" + }, + { + "bbox": [ + 150, + 587, + 194, + 600 + ], + "score": 0.92, + "content": "O ( d \\log d )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "time using the Pool Adjacent Violators algorithm (Best et al., 2000). The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "Jacobian of the projections and efficient product with it are derived in (Djolonga & Krause, 2017;", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 192, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 192, + 622 + ], + "score": 1.0, + "content": "Blondel et al., 2020).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 552, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 443, + 646 + ], + "score": 1.0, + "content": "Polyhedra. More generally, we can consider polyhedra, i.e., sets of the form", + "type": "text" + }, + { + "bbox": [ + 443, + 632, + 505, + 645 + ], + "score": 0.91, + "content": "{ \\mathcal { C } } ( \\theta ) ~ = ~ \\{ x ~ \\in ~", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 640, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 107, + 643, + 203, + 655 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d } \\colon A x = b , C x \\leq \\mathbf { \\bar { d } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 640, + 235, + 658 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 235, + 643, + 280, + 654 + ], + "score": 0.9, + "content": "A \\in \\mathbb { R } ^ { p \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 640, + 284, + 658 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 284, + 644, + 316, + 654 + ], + "score": 0.81, + "content": "b \\in \\bar { \\mathbb { R } } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 640, + 320, + 658 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 321, + 643, + 369, + 654 + ], + "score": 0.86, + "content": "C \\in \\mathbb { R } ^ { m \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 640, + 390, + 658 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 391, + 644, + 426, + 654 + ], + "score": 0.91, + "content": "d \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 640, + 507, + 658 + ], + "score": 1.0, + "content": ". There are several", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 471, + 667 + ], + "score": 1.0, + "content": "ways to differentiate this projection. The first is to use the KKT conditions as detailed in", + "type": "text" + }, + { + "bbox": [ + 471, + 655, + 491, + 666 + ], + "score": 0.86, + "content": "\\ S 2 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 654, + 506, + 667 + ], + "score": 1.0, + "content": ". A", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "second way is consider the dual of the projection instead, which is the maximization of a quadratic", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 385, + 689 + ], + "score": 1.0, + "content": "function subject to non-negative constraints (Parikh & Boyd, 2014,", + "type": "text" + }, + { + "bbox": [ + 386, + 677, + 407, + 688 + ], + "score": 0.67, + "content": "\\ S 6 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 676, + 506, + 689 + ], + "score": 1.0, + "content": ". That is, we can reduce", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "the projection on a polyhedron to a problem of the form (8) with non-negative constraints, which we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "can in turn implicitly differentiate easily using the projected gradient fixed point, combined with the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "projection on the non-negative orthant. Finally, we apply the dual-primal mapping , which enjoys a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 434, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 434, + 732 + ], + "score": 1.0, + "content": "closed form and is therefore amenable to autodiff, to obtain the primal projection.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 632, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 342, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 342, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 342, + 95 + ], + "score": 1.0, + "content": "B.2 JACOBIAN PRODUCTS OF PROXIMITY OPERATORS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 506, + 170 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 115 + ], + "score": 1.0, + "content": "We provide several proximity operators, including for the lasso (soft thresholding), elastic net and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "score": 1.0, + "content": "group lasso (block soft thresholding). All satisfy closed form expressions and can be differentiated", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "score": 1.0, + "content": "automatically via autodiff. For more advanced proximity operators, which do not enjoy a closed form,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "recent works have derived their Jacobians. The Jacobians of fused lasso and OSCAR were derived in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "(Niculae & Blondel, 2017). For general total variation, the Jacobians were derived in (Vaiter et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 227, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 227, + 170 + ], + "score": 1.0, + "content": "2013; Cherkaoui et al., 2020).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 186, + 452, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 455, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 455, + 201 + ], + "score": 1.0, + "content": "C MORE EXAMPLES OF OPTIMALITY CRITERIA AND FIXED POINTS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 104, + 208, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 506, + 226 + ], + "score": 1.0, + "content": "To demonstrate the generality of our approach, we describe in this section more optimality mapping", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 222, + 221, + 235 + ], + "spans": [ + { + "bbox": [ + 107, + 224, + 115, + 232 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 222, + 208, + 235 + ], + "score": 1.0, + "content": "or fixed point iteration", + "type": "text" + }, + { + "bbox": [ + 209, + 223, + 217, + 232 + ], + "score": 0.84, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 222, + 221, + 235 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 504, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 218, + 259 + ], + "score": 1.0, + "content": "Newton fixed point. Let", + "type": "text" + }, + { + "bbox": [ + 219, + 249, + 226, + 257 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 245, + 279, + 259 + ], + "score": 1.0, + "content": "be a root of", + "type": "text" + }, + { + "bbox": [ + 280, + 246, + 308, + 259 + ], + "score": 0.94, + "content": "G ( \\cdot , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 245, + 331, + 259 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 331, + 246, + 384, + 259 + ], + "score": 0.93, + "content": "G ( x , \\theta ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 245, + 506, + 259 + ], + "score": 1.0, + "content": ". The fixed point iteration of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 256, + 253, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 253, + 270 + ], + "score": 1.0, + "content": "Newton’s method for root-finding is", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 274, + 385, + 289 + ], + "lines": [ + { + "bbox": [ + 226, + 274, + 385, + 289 + ], + "spans": [ + { + "bbox": [ + 226, + 274, + 385, + 289 + ], + "score": 0.9, + "content": "T ( x , \\theta ) = x - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } G ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "4438f43ac03ecd4a13b305550b2d53894c16f57a38c6d94bec19af984e5cb5b3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 226, + 274, + 385, + 289 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 268, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 269, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 269, + 307 + ], + "score": 1.0, + "content": "By the chain and product rules, we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 311, + 451, + 325 + ], + "lines": [ + { + "bbox": [ + 160, + 311, + 451, + 325 + ], + "spans": [ + { + "bbox": [ + 160, + 311, + 451, + 325 + ], + "score": 0.88, + "content": "\\partial _ { 1 } T ( x , \\theta ) = I - \\eta ( . . . ) G ( x , \\theta ) - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } \\partial _ { 1 } G ( x , \\theta ) = ( 1 - \\eta ) I .", + "type": "interline_equation", + "image_path": "357418c817bcdf725ae771c319609759511fd44e9c79cb633cd32f955df2cca6.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 160, + 311, + 451, + 325 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 317, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 318, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 178, + 346 + ], + "score": 1.0, + "content": "Using (3), we get", + "type": "text" + }, + { + "bbox": [ + 178, + 331, + 272, + 343 + ], + "score": 0.92, + "content": "A = - \\partial _ { 1 } F ( x , \\theta ) = \\eta I", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 327, + 318, + 346 + ], + "score": 1.0, + "content": ". Similarly,", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 348, + 427, + 363 + ], + "lines": [ + { + "bbox": [ + 184, + 348, + 427, + 363 + ], + "spans": [ + { + "bbox": [ + 184, + 348, + 427, + 363 + ], + "score": 0.88, + "content": "B = \\partial _ { 2 } T ( x , \\theta ) = \\partial _ { 2 } F ( x , \\theta ) = - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } \\partial _ { 2 } G ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "f704a3247fc0e7fec8441bbd283a79a54d1c785071cf979cf4c52597ed198315.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 184, + 348, + 427, + 363 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 368, + 482, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 484, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 343, + 382 + ], + "score": 1.0, + "content": "Newton’s method for optimization is obtained by choosing", + "type": "text" + }, + { + "bbox": [ + 343, + 368, + 429, + 381 + ], + "score": 0.94, + "content": "G ( x , \\theta ) = \\nabla _ { 1 } f ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 367, + 484, + 382 + ], + "score": 1.0, + "content": ", which gives", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 386, + 391, + 401 + ], + "lines": [ + { + "bbox": [ + 220, + 386, + 391, + 401 + ], + "spans": [ + { + "bbox": [ + 220, + 386, + 391, + 401 + ], + "score": 0.93, + "content": "T ( x , \\theta ) = x - \\eta [ \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) ] ^ { - 1 } \\nabla _ { 1 } f ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "e7763fd63af2daa4b5024ad287cbdeaa5a94a907a09ffe43677f3a312d16f24e.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 220, + 386, + 391, + 401 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 407, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "It is easy to check that we recover the same linear system as for the gradient descent fixed point", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 402, + 430 + ], + "score": 1.0, + "content": "(5). 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We now consider the case when", + "type": "text" + }, + { + "bbox": [ + 432, + 453, + 455, + 465 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 452, + 504, + 466 + ], + "score": 1.0, + "content": "is implicitly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 463, + 199, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 199, + 476 + ], + "score": 1.0, + "content": "defined as the solution", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 471, + 388, + 505 + ], + "lines": [ + { + "bbox": [ + 222, + 471, + 388, + 505 + ], + "spans": [ + { + "bbox": [ + 222, + 471, + 388, + 505 + ], + "score": 0.95, + "content": "x ^ { \\star } ( \\theta ) : = \\operatorname * { a r g m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x , \\theta ) + \\sum _ { i = 1 } ^ { m } g _ { i } ( x _ { i } , \\theta ) ,", + "type": "interline_equation", + "image_path": "e0304c49bb0bd86b1c474ae72fe12a17971e2de1cc9f3b422a5e2645f37b585d.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 222, + 471, + 388, + 488.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 222, + 488.0, + 388, + 505.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 132, + 521 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 510, + 177, + 519 + ], + "score": 0.9, + "content": "g _ { 1 } , \\ldots , g _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 506, + 446, + 521 + ], + "score": 1.0, + "content": "are possibly non-smooth functions operating on subvectors (blocks)", + "type": "text" + }, + { + "bbox": [ + 446, + 509, + 493, + 519 + ], + "score": 0.89, + "content": "x _ { 1 } , \\ldots , x _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 506, + 506, + 521 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 518, + 320, + 530 + ], + "spans": [ + { + "bbox": [ + 107, + 521, + 114, + 528 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 519, + 228, + 530 + ], + "score": 1.0, + "content": ". In this case, we can use for", + "type": "text" + }, + { + "bbox": [ + 228, + 518, + 259, + 530 + ], + "score": 0.93, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 519, + 320, + 530 + ], + "score": 1.0, + "content": "the fixed point", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 535, + 410, + 550 + ], + "lines": [ + { + "bbox": [ + 200, + 535, + 410, + 550 + ], + "spans": [ + { + "bbox": [ + 200, + 535, + 410, + 550 + ], + "score": 0.89, + "content": "\\begin{array} { r } { x _ { i } = [ T ( \\boldsymbol { x } , \\theta ) ] _ { i } = \\mathrm { p r o x } _ { \\eta _ { i } g _ { i } } ( x _ { i } - \\eta _ { i } [ \\nabla _ { 1 } f ( \\boldsymbol { x } , \\theta ) ] _ { i } , \\theta ) , } \\end{array}", + "type": "interline_equation", + "image_path": "864791e9ff2ab5e109fafb0c9bdd5b925e3e34f2d216a79b3b14e4b7259c1249.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 200, + 535, + 410, + 550 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 506, + 589 + ], + "lines": [ + { + "bbox": [ + 104, + 553, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 134, + 569 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 557, + 180, + 567 + ], + "score": 0.88, + "content": "\\eta _ { 1 } , \\dots , \\eta _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 553, + 480, + 569 + ], + "score": 1.0, + "content": "are block-wise step sizes. Clearly, when the step sizes are shared, i.e.,", + "type": "text" + }, + { + "bbox": [ + 481, + 556, + 505, + 566 + ], + "score": 0.84, + "content": "\\eta _ { 1 } =", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 108, + 564, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 108, + 568, + 164, + 577 + ], + "score": 0.84, + "content": "\\cdots = \\eta _ { m } = \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 564, + 464, + 595 + ], + "score": 1.0, + "content": "this fixed point is equivalent to the proximal gradient fixed point (7) with .", + "type": "text" + }, + { + "bbox": [ + 464, + 566, + 505, + 578 + ], + "score": 0.91, + "content": "g ( x , \\theta ) =", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 577, + 167, + 590 + ], + "spans": [ + { + "bbox": [ + 107, + 577, + 167, + 590 + ], + "score": 0.89, + "content": "\\textstyle \\sum _ { i = 1 } ^ { n } g _ { i } ( x _ { i } , \\theta )", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 474, + 613 + ], + "score": 1.0, + "content": "Quadratic programming. We now show how to use the KKT conditions discussed in", + "type": "text" + }, + { + "bbox": [ + 474, + 601, + 493, + 612 + ], + "score": 0.87, + "content": "\\ S 2 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "differentiate quadratic programs, recovering Optnet (Amos & Kolter, 2017) as a special case. To give", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 624, + 449, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 449, + 635 + ], + "score": 1.0, + "content": "some intuition, let us start with a simple equality-constrained quadratic program (QP)", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 639, + 451, + 666 + ], + "lines": [ + { + "bbox": [ + 159, + 639, + 451, + 666 + ], + "spans": [ + { + "bbox": [ + 159, + 639, + 451, + 666 + ], + "score": 0.94, + "content": "\\operatorname * { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } } f ( z , \\theta ) = \\frac { 1 } { 2 } z ^ { \\top } Q z + c ^ { \\top } z \\quad \\mathrm { s u b j e c t t o } \\quad H ( z , \\theta ) = E z - d = 0 ,", + "type": "interline_equation", + "image_path": "4df53a57f56b2ff7f07ea428fbf7530d7e372e67dd72edacf790c0890ffb5d09.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 159, + 639, + 451, + 648.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 159, + 648.0, + 451, + 657.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 159, + 657.0, + 451, + 666.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 672, + 506, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 133, + 686 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 672, + 177, + 684 + ], + "score": 0.8, + "content": "Q \\in \\mathbb { R } ^ { p \\times p }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 671, + 181, + 686 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 181, + 672, + 224, + 684 + ], + "score": 0.8, + "content": "\\ b { E } \\in \\mathbb { R } ^ { q \\times p }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 671, + 228, + 686 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 228, + 673, + 258, + 683 + ], + "score": 0.85, + "content": "d \\in \\mathbb { R } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 671, + 437, + 686 + ], + "score": 1.0, + "content": ". We gather the differentiable parameters as", + "type": "text" + }, + { + "bbox": [ + 437, + 673, + 502, + 685 + ], + "score": 0.92, + "content": "\\theta = ( Q , E , c , d )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 671, + 506, + 686 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 682, + 324, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 324, + 698 + ], + "score": 1.0, + "content": "The stationarity and primal feasibility conditions give", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 700, + 412, + 731 + ], + "lines": [ + { + "bbox": [ + 200, + 700, + 412, + 731 + ], + "spans": [ + { + "bbox": [ + 200, + 700, + 412, + 731 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta ) + [ \\partial _ { 1 } H ( z , \\theta ) ] ^ { \\top } \\boldsymbol { \\nu } = Q z + c + E ^ { \\top } \\boldsymbol { \\nu } = 0 } \\\\ { H ( z , \\theta ) = E z - d = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "10a7a8c2bfa8548f59ed7dbf2fb7e892b123dcf70faeda1083e6446022af3e3a.jpg" + } + ] + } + ], + "index": 40.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 700, + 412, + 715.5 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 200, + 715.5, + 412, + 731.0 + ], + "spans": [], + "index": 41 + } + ] + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 342, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 342, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 342, + 95 + ], + "score": 1.0, + "content": "B.2 JACOBIAN PRODUCTS OF PROXIMITY OPERATORS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 506, + 170 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 115 + ], + "score": 1.0, + "content": "We provide several proximity operators, including for the lasso (soft thresholding), elastic net and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "score": 1.0, + "content": "group lasso (block soft thresholding). All satisfy closed form expressions and can be differentiated", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "score": 1.0, + "content": "automatically via autodiff. For more advanced proximity operators, which do not enjoy a closed form,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "recent works have derived their Jacobians. The Jacobians of fused lasso and OSCAR were derived in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "(Niculae & Blondel, 2017). For general total variation, the Jacobians were derived in (Vaiter et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 227, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 227, + 170 + ], + "score": 1.0, + "content": "2013; Cherkaoui et al., 2020).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 104, + 506, + 170 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 186, + 452, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 455, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 455, + 201 + ], + "score": 1.0, + "content": "C MORE EXAMPLES OF OPTIMALITY CRITERIA AND FIXED POINTS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 104, + 208, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 506, + 226 + ], + "score": 1.0, + "content": "To demonstrate the generality of our approach, we describe in this section more optimality mapping", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 222, + 221, + 235 + ], + "spans": [ + { + "bbox": [ + 107, + 224, + 115, + 232 + ], + "score": 0.85, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 222, + 208, + 235 + ], + "score": 1.0, + "content": "or fixed point iteration", + "type": "text" + }, + { + "bbox": [ + 209, + 223, + 217, + 232 + ], + "score": 0.84, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 222, + 221, + 235 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 208, + 506, + 235 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 504, + 269 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 218, + 259 + ], + "score": 1.0, + "content": "Newton fixed point. Let", + "type": "text" + }, + { + "bbox": [ + 219, + 249, + 226, + 257 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 245, + 279, + 259 + ], + "score": 1.0, + "content": "be a root of", + "type": "text" + }, + { + "bbox": [ + 280, + 246, + 308, + 259 + ], + "score": 0.94, + "content": "G ( \\cdot , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 245, + 331, + 259 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 331, + 246, + 384, + 259 + ], + "score": 0.93, + "content": "G ( x , \\theta ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 245, + 506, + 259 + ], + "score": 1.0, + "content": ". The fixed point iteration of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 256, + 253, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 253, + 270 + ], + "score": 1.0, + "content": "Newton’s method for root-finding is", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 245, + 506, + 270 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 274, + 385, + 289 + ], + "lines": [ + { + "bbox": [ + 226, + 274, + 385, + 289 + ], + "spans": [ + { + "bbox": [ + 226, + 274, + 385, + 289 + ], + "score": 0.9, + "content": "T ( x , \\theta ) = x - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } G ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "4438f43ac03ecd4a13b305550b2d53894c16f57a38c6d94bec19af984e5cb5b3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 226, + 274, + 385, + 289 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 268, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 269, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 269, + 307 + ], + "score": 1.0, + "content": "By the chain and product rules, we have", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 293, + 269, + 307 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 311, + 451, + 325 + ], + "lines": [ + { + "bbox": [ + 160, + 311, + 451, + 325 + ], + "spans": [ + { + "bbox": [ + 160, + 311, + 451, + 325 + ], + "score": 0.88, + "content": "\\partial _ { 1 } T ( x , \\theta ) = I - \\eta ( . . . ) G ( x , \\theta ) - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } \\partial _ { 1 } G ( x , \\theta ) = ( 1 - \\eta ) I .", + "type": "interline_equation", + "image_path": "357418c817bcdf725ae771c319609759511fd44e9c79cb633cd32f955df2cca6.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 160, + 311, + 451, + 325 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 317, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 318, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 178, + 346 + ], + "score": 1.0, + "content": "Using (3), we get", + "type": "text" + }, + { + "bbox": [ + 178, + 331, + 272, + 343 + ], + "score": 0.92, + "content": "A = - \\partial _ { 1 } F ( x , \\theta ) = \\eta I", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 327, + 318, + 346 + ], + "score": 1.0, + "content": ". Similarly,", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 327, + 318, + 346 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 348, + 427, + 363 + ], + "lines": [ + { + "bbox": [ + 184, + 348, + 427, + 363 + ], + "spans": [ + { + "bbox": [ + 184, + 348, + 427, + 363 + ], + "score": 0.88, + "content": "B = \\partial _ { 2 } T ( x , \\theta ) = \\partial _ { 2 } F ( x , \\theta ) = - \\eta [ \\partial _ { 1 } G ( x , \\theta ) ] ^ { - 1 } \\partial _ { 2 } G ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "f704a3247fc0e7fec8441bbd283a79a54d1c785071cf979cf4c52597ed198315.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 184, + 348, + 427, + 363 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 368, + 482, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 484, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 343, + 382 + ], + "score": 1.0, + "content": "Newton’s method for optimization is obtained by choosing", + "type": "text" + }, + { + "bbox": [ + 343, + 368, + 429, + 381 + ], + "score": 0.94, + "content": "G ( x , \\theta ) = \\nabla _ { 1 } f ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 367, + 484, + 382 + ], + "score": 1.0, + "content": ", which gives", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 367, + 484, + 382 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 386, + 391, + 401 + ], + "lines": [ + { + "bbox": [ + 220, + 386, + 391, + 401 + ], + "spans": [ + { + "bbox": [ + 220, + 386, + 391, + 401 + ], + "score": 0.93, + "content": "T ( x , \\theta ) = x - \\eta [ \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) ] ^ { - 1 } \\nabla _ { 1 } f ( x , \\theta ) .", + "type": "interline_equation", + "image_path": "e7763fd63af2daa4b5024ad287cbdeaa5a94a907a09ffe43677f3a312d16f24e.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 220, + 386, + 391, + 401 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 407, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "It is easy to check that we recover the same linear system as for the gradient descent fixed point", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 402, + 430 + ], + "score": 1.0, + "content": "(5). 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In this case, we can use for", + "type": "text" + }, + { + "bbox": [ + 228, + 518, + 259, + 530 + ], + "score": 0.93, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 519, + 320, + 530 + ], + "score": 1.0, + "content": "the fixed point", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 506, + 506, + 530 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 535, + 410, + 550 + ], + "lines": [ + { + "bbox": [ + 200, + 535, + 410, + 550 + ], + "spans": [ + { + "bbox": [ + 200, + 535, + 410, + 550 + ], + "score": 0.89, + "content": "\\begin{array} { r } { x _ { i } = [ T ( \\boldsymbol { x } , \\theta ) ] _ { i } = \\mathrm { p r o x } _ { \\eta _ { i } g _ { i } } ( x _ { i } - \\eta _ { i } [ \\nabla _ { 1 } f ( \\boldsymbol { x } , \\theta ) ] _ { i } , \\theta ) , } \\end{array}", + "type": "interline_equation", + "image_path": "864791e9ff2ab5e109fafb0c9bdd5b925e3e34f2d216a79b3b14e4b7259c1249.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 200, + 535, + 410, + 550 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 506, + 589 + ], + "lines": [ + { + "bbox": [ + 104, + 553, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 134, + 569 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 557, + 180, + 567 + ], + "score": 0.88, + "content": "\\eta _ { 1 } , \\dots , \\eta _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 553, + 480, + 569 + ], + "score": 1.0, + "content": "are block-wise step sizes. 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We now show how to use the KKT conditions discussed in", + "type": "text" + }, + { + "bbox": [ + 474, + 601, + 493, + 612 + ], + "score": 0.87, + "content": "\\ S 2 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "differentiate quadratic programs, recovering Optnet (Amos & Kolter, 2017) as a special case. 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More", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 140, + 333, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 333, + 152 + ], + "score": 1.0, + "content": "generally, the QP can also include inequality constraints", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 155, + 498, + 179 + ], + "lines": [ + { + "bbox": [ + 111, + 155, + 498, + 179 + ], + "spans": [ + { + "bbox": [ + 111, + 155, + 498, + 179 + ], + "score": 0.91, + "content": "\\operatorname * { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } } f ( z , \\theta ) = \\frac { 1 } { 2 } z ^ { \\top } Q z + c ^ { \\top } z \\quad \\mathrm { s u b j e c t ~ t o } \\quad H ( z , \\theta ) = E z - d = 0 , G ( z , \\theta ) = M z - h \\leq 0 .", + "type": "interline_equation", + "image_path": "c35a0e0d7ffcbd3d0a0fb863709e3ede718b8f9c734aa13959026b5398a7264f.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 155, + 498, + 179 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 184, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 133, + 199 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 186, + 181, + 196 + ], + "score": 0.93, + "content": "M \\in \\mathbb { R } ^ { r \\times p }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 184, + 199, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 200, + 186, + 231, + 196 + ], + "score": 0.91, + "content": "h \\in \\mathbb { R } ^ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 184, + 411, + 199 + ], + "score": 1.0, + "content": ". 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With our framework, no derivation is needed. We simply define", + "type": "text" + }, + { + "bbox": [ + 496, + 330, + 502, + 340 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 327, + 507, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 339, + 222, + 352 + ], + "spans": [ + { + "bbox": [ + 107, + 340, + 117, + 349 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 339, + 135, + 352 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 135, + 340, + 144, + 349 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 339, + 222, + 352 + ], + "score": 1.0, + "content": "directly in Python.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 506, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "Conic programming. We now show that the differentiation of conic linear programs (Agrawal", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "et al., 2019b; Amos, 2019), at the heart of differentiating through cvxpy layers (Agrawal et al., 2019a),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 385, + 300, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 300, + 397 + ], + "score": 1.0, + "content": "easily fits our framework. Consider the problem", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 400, + 437, + 422 + ], + "lines": [ + { + "bbox": [ + 172, + 400, + 437, + 422 + ], + "spans": [ + { + "bbox": [ + 172, + 400, + 437, + 422 + ], + "score": 0.9, + "content": "z ^ { \\star } ( \\lambda ) , s ^ { \\star } ( \\lambda ) = \\operatorname * { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } , s \\in \\mathbb { R } ^ { m } } c ^ { \\top } z \\quad \\mathrm { s u b j e c t } \\mathrm { t o } \\quad E z + s = d , s \\in \\mathcal { K } ,", + "type": "interline_equation", + "image_path": "82bea159c2902c9b51e155aef2c56c618e8802e5c479299db514216d84430c7d.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 172, + 400, + 437, + 422 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 133, + 440 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 428, + 189, + 440 + ], + "score": 0.81, + "content": "\\lambda = ( c , E , d )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 427, + 193, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 193, + 428, + 240, + 439 + ], + "score": 0.86, + "content": "\\ b { E } \\in \\mathbb { R } ^ { m \\times p }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 427, + 244, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 244, + 428, + 279, + 439 + ], + "score": 0.83, + "content": "d \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 427, + 282, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 428, + 313, + 439 + ], + "score": 0.88, + "content": "c \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 427, + 332, + 440 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 428, + 370, + 439 + ], + "score": 0.91, + "content": "{ \\mathcal { K } } \\subseteq \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 427, + 412, + 440 + ], + "score": 1.0, + "content": "is a cone;", + "type": "text" + }, + { + "bbox": [ + 413, + 430, + 420, + 438 + ], + "score": 0.78, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 427, + 438, + 440 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 430, + 444, + 438 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "are the primal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "and slack variables, respectively. Every convex optimization problem can be reduced to the form (17).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 450, + 267, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 267, + 461 + ], + "score": 1.0, + "content": "Let us form the skew-symmetric matrix", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 464, + 382, + 505 + ], + "lines": [ + { + "bbox": [ + 228, + 464, + 382, + 505 + ], + "spans": [ + { + "bbox": [ + 228, + 464, + 382, + 505 + ], + "score": 0.94, + "content": "\\theta ( \\lambda ) = \\left[ \\begin{array} { c c c } { 0 } & { E ^ { \\top } } & { c } \\\\ { - E } & { 0 } & { d } \\\\ { - c ^ { \\top } } & { - d ^ { \\top } } & { 0 } \\end{array} \\right] \\in \\mathbb { R } ^ { N \\times N } ,", + "type": "interline_equation", + "image_path": "e6f11544cc1ef48ceee7670cade7f5e32f50afd42af4e8f955f08d94e946be27.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 228, + 464, + 382, + 484.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 228, + 484.5, + 382, + 505.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 508, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 132, + 520 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 509, + 191, + 519 + ], + "score": 0.92, + "content": "N = p { + } m { + } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 507, + 505, + 520 + ], + "score": 1.0, + "content": ". Following (Agrawal et al., 2019b;a; Amos, 2019), we can use the homogeneous", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 518, + 484, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 484, + 532 + ], + "score": 1.0, + "content": "self-dual embedding to reduce the process of solving (17) to finding a root of the residual map", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 534, + 394, + 549 + ], + "lines": [ + { + "bbox": [ + 216, + 534, + 394, + 549 + ], + "spans": [ + { + "bbox": [ + 216, + 534, + 394, + 549 + ], + "score": 0.92, + "content": "F ( x , \\theta ) = \\theta \\Pi x + \\Pi ^ { * } x = ( ( \\theta - I ) \\Pi + I ) x ,", + "type": "interline_equation", + "image_path": "e7370eb1ef21e658f878423743d609b8492228a2bdefd5ea26267aa20b282eac.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 216, + 534, + 394, + 549 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 506, + 588 + ], + "lines": [ + { + "bbox": [ + 104, + 551, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 133, + 567 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 553, + 214, + 566 + ], + "score": 0.91, + "content": "\\Pi = \\mathrm { p r o j } _ { \\mathbb { R } ^ { p } \\times K ^ { * } \\times \\mathbb { R } _ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 551, + 232, + 567 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 552, + 274, + 564 + ], + "score": 0.92, + "content": "{ K } ^ { * } \\subseteq \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 551, + 506, + 567 + ], + "score": 1.0, + "content": "is the dual cone. The splitting conic solver (O’Donoghue", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 339, + 577 + ], + "score": 1.0, + "content": "et al., 2016), which is based on ADMM, outputs a solution", + "type": "text" + }, + { + "bbox": [ + 339, + 565, + 407, + 577 + ], + "score": 0.93, + "content": "F ( x ^ { \\star } ( \\theta ) , \\theta ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "which is decomposed as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 575, + 455, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 230, + 588 + ], + "score": 0.9, + "content": "x ^ { \\star } ( \\theta ) = ( u ^ { \\star } ( \\theta ) , v ^ { \\star } ( \\theta ) , w ^ { \\star } ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 575, + 455, + 590 + ], + "score": 1.0, + "content": ". We can then recover the optimal solution of (17) using", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 591, + 439, + 605 + ], + "lines": [ + { + "bbox": [ + 171, + 591, + 439, + 605 + ], + "spans": [ + { + "bbox": [ + 171, + 591, + 439, + 605 + ], + "score": 0.88, + "content": "z ^ { \\star } ( \\lambda ) = u ^ { \\star } ( \\theta ( \\lambda ) ) \\quad \\mathrm { a n d } \\quad s ^ { \\star } ( \\lambda ) = \\mathrm { p r o j } _ { \\mathcal { K } ^ { * } } ( v ^ { \\star } ( \\theta ( \\lambda ) ) ) - v ^ { \\star } ( \\theta ( \\lambda ) ) .", + "type": "interline_equation", + "image_path": "a39afbb580d55a04c8d6fed81e605b9a50a902c08cb7b3eeb30b80e966811c90.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 171, + 591, + 439, + 605 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 504, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 321, + 621 + ], + "score": 1.0, + "content": "The key oracle whose JVP/VJP we need is therefore", + "type": "text" + }, + { + "bbox": [ + 321, + 610, + 330, + 619 + ], + "score": 0.67, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 609, + 504, + 621 + ], + "score": 1.0, + "content": ", which is studied in (Ali et al., 2017). The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 442, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 429, + 632 + ], + "score": 1.0, + "content": "projection onto a few cones is available in our library and can be used to express", + "type": "text" + }, + { + "bbox": [ + 430, + 621, + 438, + 630 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 620, + 442, + 632 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 244, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 245, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 245, + 657 + ], + "score": 1.0, + "content": "Frank-Wolfe. We now consider", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 658, + 361, + 682 + ], + "lines": [ + { + "bbox": [ + 249, + 658, + 361, + 682 + ], + "spans": [ + { + "bbox": [ + 249, + 658, + 361, + 682 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x \\in \\mathcal { C } ( \\theta ) \\subset \\mathbb { R } ^ { d } } f ( x , \\theta ) ,", + "type": "interline_equation", + "image_path": "de83a2d74258321559a6fe2ada43cbc330f287c42f57a80b8100146a0d523fb5.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 249, + 658, + 361, + 682 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 686, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 133, + 699 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 686, + 153, + 698 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 685, + 384, + 699 + ], + "score": 1.0, + "content": "is a convex polytope, i.e., it is the convex hull of vertices", + "type": "text" + }, + { + "bbox": [ + 385, + 686, + 455, + 698 + ], + "score": 0.95, + "content": "v _ { 1 } ( \\theta ) , \\dots , v _ { m } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 685, + 506, + 699 + ], + "score": 1.0, + "content": ". The Frank-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 695, + 354, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 354, + 711 + ], + "score": 1.0, + "content": "Wolfe algorithm requires a linear minimization oracle (LMO)", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 265, + 713, + 346, + 735 + ], + "lines": [ + { + "bbox": [ + 265, + 713, + 346, + 735 + ], + "spans": [ + { + "bbox": [ + 265, + 713, + 346, + 735 + ], + "score": 0.94, + "content": "s \\mapsto \\operatorname * { a r g m i n } _ { x \\in { \\mathcal { C } } ( \\theta ) } ~ \\langle s , x \\rangle", + "type": "interline_equation", + "image_path": "b0ecff09defea20f90f6d8bffffd7944f812d68a890aa7a03f671cc0e5efdcb9.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 265, + 713, + 346, + 735 + ], + "spans": [], + "index": 38 + } + ] + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 277, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 277, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 277, + 95 + ], + "score": 1.0, + "content": "In matrix notation, this can be rewritten as", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 277, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 97, + 357, + 125 + ], + "lines": [ + { + "bbox": [ + 253, + 97, + 357, + 125 + ], + "spans": [ + { + "bbox": [ + 253, + 97, + 357, + 125 + ], + "score": 0.94, + "content": "\\left[ { Q \\quad E ^ { \\top } } \\right] \\left[ { z } \\right] = \\left[ { - c } \\right] .", + "type": "interline_equation", + "image_path": "ca8b212b581e40473dd4dc5169335875be5456fbcda798bdd8c189bfbfb8985c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 253, + 97, + 357, + 125 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 128, + 504, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 349, + 141 + ], + "score": 1.0, + "content": "We can write the solution of the linear system (16) as the root", + "type": "text" + }, + { + "bbox": [ + 349, + 128, + 392, + 141 + ], + "score": 0.93, + "content": "\\boldsymbol { x } = \\left( z , \\nu \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 128, + 445, + 141 + ], + "score": 1.0, + "content": "of a function", + "type": "text" + }, + { + "bbox": [ + 446, + 128, + 477, + 141 + ], + "score": 0.94, + "content": "F ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 128, + 505, + 141 + ], + "score": 1.0, + "content": ". More", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 140, + 333, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 333, + 152 + ], + "score": 1.0, + "content": "generally, the QP can also include inequality constraints", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 128, + 505, + 152 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 155, + 498, + 179 + ], + "lines": [ + { + "bbox": [ + 111, + 155, + 498, + 179 + ], + "spans": [ + { + "bbox": [ + 111, + 155, + 498, + 179 + ], + "score": 0.91, + "content": "\\operatorname * { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } } f ( z , \\theta ) = \\frac { 1 } { 2 } z ^ { \\top } Q z + c ^ { \\top } z \\quad \\mathrm { s u b j e c t ~ t o } \\quad H ( z , \\theta ) = E z - d = 0 , G ( z , \\theta ) = M z - h \\leq 0 .", + "type": "interline_equation", + "image_path": "c35a0e0d7ffcbd3d0a0fb863709e3ede718b8f9c734aa13959026b5398a7264f.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 155, + 498, + 179 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 184, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 133, + 199 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 186, + 181, + 196 + ], + "score": 0.93, + "content": "M \\in \\mathbb { R } ^ { r \\times p }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 184, + 199, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 200, + 186, + 231, + 196 + ], + "score": 0.91, + "content": "h \\in \\mathbb { R } ^ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 184, + 411, + 199 + ], + "score": 1.0, + "content": ". We gather the differentiable parameters as", + "type": "text" + }, + { + "bbox": [ + 412, + 186, + 502, + 198 + ], + "score": 0.91, + "content": "\\theta = ( Q , E , M , c , d , h )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 184, + 506, + 199 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 196, + 429, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 429, + 210 + ], + "score": 1.0, + "content": "The stationarity, primal feasibility and complementary slackness conditions give", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 184, + 506, + 210 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 211, + 464, + 257 + ], + "lines": [ + { + "bbox": [ + 148, + 211, + 464, + 257 + ], + "spans": [ + { + "bbox": [ + 148, + 211, + 464, + 257 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\nabla _ { 1 } f ( z , \\theta ) + [ \\partial _ { 1 } H ( z , \\theta ) ] ^ { \\top } \\boldsymbol { \\nu } + [ \\partial _ { 1 } G ( z , \\theta ) ] ^ { \\top } \\boldsymbol { \\lambda } = Q z + c + E ^ { \\top } \\boldsymbol { \\nu } + M ^ { \\top } \\boldsymbol { \\lambda } = 0 } \\\\ { H ( z , \\theta ) = E z - d = 0 } \\\\ { \\boldsymbol { \\lambda } \\circ G ( z , \\theta ) = \\mathrm { d i a g } ( \\boldsymbol { \\lambda } ) ( M z - h ) = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "fbb65e91f54b8309ea2c307445fc7907e0a5a3e89babace5166113674103c56f.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 148, + 211, + 464, + 226.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 148, + 226.33333333333334, + 464, + 241.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 148, + 241.66666666666669, + 464, + 257.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 269, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 270, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 270, + 272 + ], + "score": 1.0, + "content": "In matrix notation, this can be written as", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 258, + 270, + 272 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 274, + 394, + 314 + ], + "lines": [ + { + "bbox": [ + 218, + 274, + 394, + 314 + ], + "spans": [ + { + "bbox": [ + 218, + 274, + 394, + 314 + ], + "score": 0.93, + "content": "{ \\left[ \\begin{array} { l l l } { \\quad Q } & { E ^ { \\top } } & { M ^ { \\top } } \\\\ { E } & { 0 } & { 0 } \\\\ { \\operatorname { d i a g } ( \\lambda ) M } & { 0 } & { \\quad 0 } \\end{array} \\right] } \\left[ { \\begin{array} { l } { z } \\\\ { \\nu } \\\\ { \\lambda } \\end{array} } \\right] = { \\left[ \\begin{array} { l } { - c } \\\\ { \\ d } \\\\ { \\lambda \\circ h } \\end{array} \\right] }", + "type": "interline_equation", + "image_path": "d1d2a4995d1154f6f0c22d70597483f1fee03f4565de7025a05cd5d71b00b9ab.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 274, + 394, + 294.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 218, + 294.0, + 394, + 314.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 506, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 133, + 330 + ], + "score": 1.0, + "content": "While", + "type": "text" + }, + { + "bbox": [ + 133, + 317, + 186, + 330 + ], + "score": 0.93, + "content": "\\boldsymbol { x } = ( z , \\nu , \\lambda )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 316, + 473, + 330 + ], + "score": 1.0, + "content": "is no longer the solution of a linear system, it is the root of a function", + "type": "text" + }, + { + "bbox": [ + 473, + 318, + 505, + 329 + ], + "score": 0.92, + "content": "F ( x , \\theta )", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 327, + 507, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 495, + 342 + ], + "score": 1.0, + "content": "and therefore fits our framework. With our framework, no derivation is needed. We simply define", + "type": "text" + }, + { + "bbox": [ + 496, + 330, + 502, + 340 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 327, + 507, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 339, + 222, + 352 + ], + "spans": [ + { + "bbox": [ + 107, + 340, + 117, + 349 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 339, + 135, + 352 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 135, + 340, + 144, + 349 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 339, + 222, + 352 + ], + "score": 1.0, + "content": "directly in Python.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 316, + 507, + 352 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 506, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "Conic programming. We now show that the differentiation of conic linear programs (Agrawal", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "et al., 2019b; Amos, 2019), at the heart of differentiating through cvxpy layers (Agrawal et al., 2019a),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 385, + 300, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 300, + 397 + ], + "score": 1.0, + "content": "easily fits our framework. Consider the problem", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 362, + 506, + 397 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 400, + 437, + 422 + ], + "lines": [ + { + "bbox": [ + 172, + 400, + 437, + 422 + ], + "spans": [ + { + "bbox": [ + 172, + 400, + 437, + 422 + ], + "score": 0.9, + "content": "z ^ { \\star } ( \\lambda ) , s ^ { \\star } ( \\lambda ) = \\operatorname * { a r g m i n } _ { z \\in \\mathbb { R } ^ { p } , s \\in \\mathbb { R } ^ { m } } c ^ { \\top } z \\quad \\mathrm { s u b j e c t } \\mathrm { t o } \\quad E z + s = d , s \\in \\mathcal { K } ,", + "type": "interline_equation", + "image_path": "82bea159c2902c9b51e155aef2c56c618e8802e5c479299db514216d84430c7d.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 172, + 400, + 437, + 422 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 133, + 440 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 428, + 189, + 440 + ], + "score": 0.81, + "content": "\\lambda = ( c , E , d )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 427, + 193, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 193, + 428, + 240, + 439 + ], + "score": 0.86, + "content": "\\ b { E } \\in \\mathbb { R } ^ { m \\times p }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 427, + 244, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 244, + 428, + 279, + 439 + ], + "score": 0.83, + "content": "d \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 427, + 282, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 283, + 428, + 313, + 439 + ], + "score": 0.88, + "content": "c \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 427, + 332, + 440 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 428, + 370, + 439 + ], + "score": 0.91, + "content": "{ \\mathcal { K } } \\subseteq \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 427, + 412, + 440 + ], + "score": 1.0, + "content": "is a cone;", + "type": "text" + }, + { + "bbox": [ + 413, + 430, + 420, + 438 + ], + "score": 0.78, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 427, + 438, + 440 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 430, + 444, + 438 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "are the primal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "and slack variables, respectively. Every convex optimization problem can be reduced to the form (17).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 450, + 267, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 267, + 461 + ], + "score": 1.0, + "content": "Let us form the skew-symmetric matrix", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 427, + 506, + 461 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 464, + 382, + 505 + ], + "lines": [ + { + "bbox": [ + 228, + 464, + 382, + 505 + ], + "spans": [ + { + "bbox": [ + 228, + 464, + 382, + 505 + ], + "score": 0.94, + "content": "\\theta ( \\lambda ) = \\left[ \\begin{array} { c c c } { 0 } & { E ^ { \\top } } & { c } \\\\ { - E } & { 0 } & { d } \\\\ { - c ^ { \\top } } & { - d ^ { \\top } } & { 0 } \\end{array} \\right] \\in \\mathbb { R } ^ { N \\times N } ,", + "type": "interline_equation", + "image_path": "e6f11544cc1ef48ceee7670cade7f5e32f50afd42af4e8f955f08d94e946be27.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 228, + 464, + 382, + 484.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 228, + 484.5, + 382, + 505.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 508, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 132, + 520 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 509, + 191, + 519 + ], + "score": 0.92, + "content": "N = p { + } m { + } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 507, + 505, + 520 + ], + "score": 1.0, + "content": ". Following (Agrawal et al., 2019b;a; Amos, 2019), we can use the homogeneous", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 518, + 484, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 484, + 532 + ], + "score": 1.0, + "content": "self-dual embedding to reduce the process of solving (17) to finding a root of the residual map", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 507, + 505, + 532 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 534, + 394, + 549 + ], + "lines": [ + { + "bbox": [ + 216, + 534, + 394, + 549 + ], + "spans": [ + { + "bbox": [ + 216, + 534, + 394, + 549 + ], + "score": 0.92, + "content": "F ( x , \\theta ) = \\theta \\Pi x + \\Pi ^ { * } x = ( ( \\theta - I ) \\Pi + I ) x ,", + "type": "interline_equation", + "image_path": "e7370eb1ef21e658f878423743d609b8492228a2bdefd5ea26267aa20b282eac.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 216, + 534, + 394, + 549 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 506, + 588 + ], + "lines": [ + { + "bbox": [ + 104, + 551, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 133, + 567 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 553, + 214, + 566 + ], + "score": 0.91, + "content": "\\Pi = \\mathrm { p r o j } _ { \\mathbb { R } ^ { p } \\times K ^ { * } \\times \\mathbb { R } _ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 551, + 232, + 567 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 552, + 274, + 564 + ], + "score": 0.92, + "content": "{ K } ^ { * } \\subseteq \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 551, + 506, + 567 + ], + "score": 1.0, + "content": "is the dual cone. The splitting conic solver (O’Donoghue", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 339, + 577 + ], + "score": 1.0, + "content": "et al., 2016), which is based on ADMM, outputs a solution", + "type": "text" + }, + { + "bbox": [ + 339, + 565, + 407, + 577 + ], + "score": 0.93, + "content": "F ( x ^ { \\star } ( \\theta ) , \\theta ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "which is decomposed as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 575, + 455, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 230, + 588 + ], + "score": 0.9, + "content": "x ^ { \\star } ( \\theta ) = ( u ^ { \\star } ( \\theta ) , v ^ { \\star } ( \\theta ) , w ^ { \\star } ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 575, + 455, + 590 + ], + "score": 1.0, + "content": ". We can then recover the optimal solution of (17) using", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 551, + 506, + 590 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 591, + 439, + 605 + ], + "lines": [ + { + "bbox": [ + 171, + 591, + 439, + 605 + ], + "spans": [ + { + "bbox": [ + 171, + 591, + 439, + 605 + ], + "score": 0.88, + "content": "z ^ { \\star } ( \\lambda ) = u ^ { \\star } ( \\theta ( \\lambda ) ) \\quad \\mathrm { a n d } \\quad s ^ { \\star } ( \\lambda ) = \\mathrm { p r o j } _ { \\mathcal { K } ^ { * } } ( v ^ { \\star } ( \\theta ( \\lambda ) ) ) - v ^ { \\star } ( \\theta ( \\lambda ) ) .", + "type": "interline_equation", + "image_path": "a39afbb580d55a04c8d6fed81e605b9a50a902c08cb7b3eeb30b80e966811c90.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 171, + 591, + 439, + 605 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 504, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 321, + 621 + ], + "score": 1.0, + "content": "The key oracle whose JVP/VJP we need is therefore", + "type": "text" + }, + { + "bbox": [ + 321, + 610, + 330, + 619 + ], + "score": 0.67, + "content": "\\Pi", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 609, + 504, + 621 + ], + "score": 1.0, + "content": ", which is studied in (Ali et al., 2017). The", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 442, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 429, + 632 + ], + "score": 1.0, + "content": "projection onto a few cones is available in our library and can be used to express", + "type": "text" + }, + { + "bbox": [ + 430, + 621, + 438, + 630 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 620, + 442, + 632 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 609, + 504, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 244, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 245, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 245, + 657 + ], + "score": 1.0, + "content": "Frank-Wolfe. We now consider", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 642, + 245, + 657 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 658, + 361, + 682 + ], + "lines": [ + { + "bbox": [ + 249, + 658, + 361, + 682 + ], + "spans": [ + { + "bbox": [ + 249, + 658, + 361, + 682 + ], + "score": 0.94, + "content": "x ^ { \\star } ( \\theta ) = \\operatorname * { a r g m i n } _ { x \\in \\mathcal { C } ( \\theta ) \\subset \\mathbb { R } ^ { d } } f ( x , \\theta ) ,", + "type": "interline_equation", + "image_path": "de83a2d74258321559a6fe2ada43cbc330f287c42f57a80b8100146a0d523fb5.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 249, + 658, + 361, + 682 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 686, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 133, + 699 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 686, + 153, + 698 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 685, + 384, + 699 + ], + "score": 1.0, + "content": "is a convex polytope, i.e., it is the convex hull of vertices", + "type": "text" + }, + { + "bbox": [ + 385, + 686, + 455, + 698 + ], + "score": 0.95, + "content": "v _ { 1 } ( \\theta ) , \\dots , v _ { m } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 685, + 506, + 699 + ], + "score": 1.0, + "content": ". The Frank-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 695, + 354, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 354, + 711 + ], + "score": 1.0, + "content": "Wolfe algorithm requires a linear minimization oracle (LMO)", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 685, + 506, + 711 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 265, + 713, + 346, + 735 + ], + "lines": [ + { + "bbox": [ + 265, + 713, + 346, + 735 + ], + "spans": [ + { + "bbox": [ + 265, + 713, + 346, + 735 + ], + "score": 0.94, + "content": "s \\mapsto \\operatorname * { a r g m i n } _ { x \\in { \\mathcal { C } } ( \\theta ) } ~ \\langle s , x \\rangle", + "type": "interline_equation", + "image_path": "b0ecff09defea20f90f6d8bffffd7944f812d68a890aa7a03f671cc0e5efdcb9.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 265, + 713, + 346, + 735 + ], + "spans": [], + "index": 38 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 483, + 95 + ], + "score": 1.0, + "content": "and is a popular algorithm when this LMO is easier to compute than the projection onto", + "type": "text" + }, + { + "bbox": [ + 483, + 82, + 503, + 95 + ], + "score": 0.9, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 82, + 506, + 95 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "However, since this LMO is piecewise constant, its Jacobian is null almost everywhere. Inspired by", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 393, + 117 + ], + "score": 1.0, + "content": "SparseMAP (Niculae et al., 2018), which corresponds to the case when", + "type": "text" + }, + { + "bbox": [ + 394, + 105, + 401, + 116 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "is a quadratic, we rewrite", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 138, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 138, + 128 + ], + "score": 1.0, + "content": "(19) as", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 125, + 387, + 146 + ], + "lines": [ + { + "bbox": [ + 223, + 125, + 387, + 146 + ], + "spans": [ + { + "bbox": [ + 223, + 125, + 387, + 146 + ], + "score": 0.93, + "content": "p ^ { \\star } ( \\theta ) = \\underset { p \\in \\triangle ^ { m } } { \\operatorname { a r g m i n } } g ( p , \\theta ) : = f ( V ( \\theta ) p , \\theta ) ,", + "type": "interline_equation", + "image_path": "2806c56d422ef608e306f21ee869a7464829a762b80d5487090e07b932111bb9.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 223, + 125, + 387, + 146 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 148, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 134, + 162 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 149, + 156, + 160 + ], + "score": 0.9, + "content": "V ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 147, + 177, + 162 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 177, + 149, + 205, + 159 + ], + "score": 0.91, + "content": "d \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 147, + 330, + 162 + ], + "score": 1.0, + "content": "matrix gathering the vertices", + "type": "text" + }, + { + "bbox": [ + 330, + 149, + 400, + 161 + ], + "score": 0.92, + "content": "v _ { 1 } ( \\theta ) , \\dots , v _ { m } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 147, + 467, + 162 + ], + "score": 1.0, + "content": ". We then have", + "type": "text" + }, + { + "bbox": [ + 468, + 149, + 505, + 161 + ], + "score": 0.91, + "content": "x ^ { \\star } ( \\theta ) =", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 150, + 172 + ], + "score": 0.92, + "content": "V ( \\theta ) p ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 159, + 506, + 172 + ], + "score": 1.0, + "content": ". Since we have reduced (19) to minimization over the simplex, we can use the projected", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 225, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 225, + 182 + ], + "score": 1.0, + "content": "gradient fixed point to obtain", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 186, + 402, + 200 + ], + "lines": [ + { + "bbox": [ + 208, + 186, + 402, + 200 + ], + "spans": [ + { + "bbox": [ + 208, + 186, + 402, + 200 + ], + "score": 0.89, + "content": "T ( p ^ { \\star } ( \\boldsymbol { \\theta } ) , \\boldsymbol { \\theta } ) \\stackrel { } { = } \\mathrm { p r o j } _ { \\triangle ^ { m } } \\big ( p ^ { \\star } ( \\boldsymbol { \\theta } ) - \\nabla _ { 1 } g ( p ^ { \\ * } ( \\boldsymbol { \\theta } ) , \\boldsymbol { \\theta } ) \\big ) .", + "type": "interline_equation", + "image_path": "26335ef9f5b29258f3409890b9591693786eddfffde92b77eeb7b1ce764136f2.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 208, + 186, + 402, + 200 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 506, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 285, + 217 + ], + "score": 1.0, + "content": "We can therefore compute the derivatives of", + "type": "text" + }, + { + "bbox": [ + 286, + 205, + 309, + 217 + ], + "score": 0.93, + "content": "p ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "by implicit differentiation and the derivatives of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 107, + 216, + 131, + 228 + ], + "score": 0.91, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "by product rule. Frank-Wolfe implementations typically maintain the convex weights of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 504, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 315, + 240 + ], + "score": 1.0, + "content": "vertices, which we use to get an approximation of", + "type": "text" + }, + { + "bbox": [ + 315, + 227, + 338, + 239 + ], + "score": 0.92, + "content": "p ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 226, + 498, + 240 + ], + "score": 1.0, + "content": ". Moreover, it is well-known that after", + "type": "text" + }, + { + "bbox": [ + 499, + 228, + 504, + 237 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 178, + 250 + ], + "score": 1.0, + "content": "iterations, at most", + "type": "text" + }, + { + "bbox": [ + 179, + 239, + 183, + 248 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 238, + 506, + 250 + ], + "score": 1.0, + "content": "vertices are visited. We can leverage this sparsity to solve a smaller linear system.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 361, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 333, + 262 + ], + "score": 1.0, + "content": "Moreover, in practice, we only need to compute VJPs of", + "type": "text" + }, + { + "bbox": [ + 334, + 249, + 357, + 261 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 248, + 361, + 262 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 276, + 281, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 282, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 282, + 291 + ], + "score": 1.0, + "content": "D JACOBIAN PRECISION PROOFS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 506, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 179, + 313 + ], + "score": 1.0, + "content": "Proof of Theorem", + "type": "text" + }, + { + "bbox": [ + 180, + 302, + 185, + 311 + ], + "score": 0.46, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 299, + 312, + 313 + ], + "score": 1.0, + "content": ". To simplify notations, we note", + "type": "text" + }, + { + "bbox": [ + 313, + 300, + 376, + 313 + ], + "score": 0.93, + "content": "A _ { \\star } : = A ( x ^ { \\star } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 299, + 393, + 313 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 394, + 299, + 448, + 313 + ], + "score": 0.93, + "content": "\\hat { A } : = A ( \\hat { x } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 299, + 506, + 313 + ], + "score": 1.0, + "content": ", and similarly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 312, + 507, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 121, + 325 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 122, + 314, + 131, + 324 + ], + "score": 0.82, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 312, + 150, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 150, + 314, + 158, + 324 + ], + "score": 0.78, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 312, + 394, + 325 + ], + "score": 1.0, + "content": ". We have by definition of the Jacobian estimate function", + "type": "text" + }, + { + "bbox": [ + 395, + 314, + 446, + 325 + ], + "score": 0.91, + "content": "A _ { \\star } J _ { \\star } = B _ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 312, + 465, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 465, + 312, + 503, + 324 + ], + "score": 0.92, + "content": "\\hat { A } \\hat { J } = \\hat { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 312, + 507, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 324, + 183, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 183, + 336 + ], + "score": 1.0, + "content": "Therefore we have", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 337, + 426, + 388 + ], + "lines": [ + { + "bbox": [ + 185, + 337, + 426, + 388 + ], + "spans": [ + { + "bbox": [ + 185, + 337, + 426, + 388 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) = \\hat { A } ^ { - 1 } \\hat { B } - A _ { \\star } ^ { - 1 } B _ { \\star } } \\\\ & { \\qquad = \\hat { A } ^ { - 1 } \\hat { B } - \\hat { A } ^ { - 1 } B _ { \\star } + \\hat { A } ^ { - 1 } B _ { \\star } - A _ { \\star } ^ { - 1 } B _ { \\star } } \\\\ & { \\qquad = \\hat { A } ^ { - 1 } ( \\hat { B } - B _ { \\star } ) + ( \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } ) B _ { \\star } . } \\end{array}", + "type": "interline_equation", + "image_path": "ea19f87c8539544562bd4b7de5e16b2fd89fdc51869d3b5430f2a2930582992e.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 185, + 337, + 426, + 354.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 185, + 354.0, + 426, + 371.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 185, + 371.0, + 426, + 388.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 478, + 405 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 480, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 216, + 408 + ], + "score": 1.0, + "content": "For any invertible matrices", + "type": "text" + }, + { + "bbox": [ + 216, + 393, + 249, + 404 + ], + "score": 0.92, + "content": "M _ { 1 } , M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 389, + 303, + 408 + ], + "score": 1.0, + "content": ", it holds that", + "type": "text" + }, + { + "bbox": [ + 303, + 391, + 464, + 405 + ], + "score": 0.92, + "content": "M _ { 1 } ^ { - 1 } - M _ { 2 } ^ { - 1 } = M _ { 1 } ^ { - 1 } ( M _ { 2 } - M _ { 1 } ) M _ { 2 } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 389, + 480, + 408 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 408, + 419, + 424 + ], + "lines": [ + { + "bbox": [ + 191, + 408, + 419, + 424 + ], + "spans": [ + { + "bbox": [ + 191, + 408, + 419, + 424 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\| M _ { 2 } ^ { - 1 } - M _ { 2 } ^ { - 1 } \\| _ { \\mathrm { o p } } \\leq \\| M _ { 1 } ^ { - 1 } \\| _ { \\mathrm { o p } } \\| M _ { 2 } - M _ { 1 } \\| _ { \\mathrm { o p } } \\| M _ { 2 } ^ { - 1 } \\| _ { \\mathrm { o p } } . } \\end{array}", + "type": "interline_equation", + "image_path": "f6fdde9603b4506641a20ca93b4ced9e9979aafe4a2e4ecfb32bdf5f13d112f5.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 191, + 408, + 419, + 424 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 149, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 151, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 151, + 440 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 436, + 418, + 460 + ], + "lines": [ + { + "bbox": [ + 192, + 436, + 418, + 460 + ], + "spans": [ + { + "bbox": [ + 192, + 436, + 418, + 460 + ], + "score": 0.89, + "content": "\\| \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } \\| _ { \\mathrm { o p } } \\leq \\frac { 1 } { \\alpha ^ { 2 } } \\| \\hat { A } - A _ { \\star } \\| _ { \\mathrm { o p } } \\leq \\frac { \\gamma } { \\alpha ^ { 2 } } \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\| .", + "type": "interline_equation", + "image_path": "8d2dfeee4d4a36ec4a8042a257662c7a48d0017fa610f8c68a701ac7abbbe53c.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 192, + 436, + 418, + 460 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 479, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 480, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 258, + 474 + ], + "score": 1.0, + "content": "As a consequence, the second term in", + "type": "text" + }, + { + "bbox": [ + 259, + 461, + 330, + 473 + ], + "score": 0.93, + "content": "J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 460, + 480, + 474 + ], + "score": 1.0, + "content": "can be upper bounded and we obtain", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 476, + 445, + 516 + ], + "lines": [ + { + "bbox": [ + 165, + 476, + 445, + 516 + ], + "spans": [ + { + "bbox": [ + 165, + 476, + 445, + 516 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\| J ( \\hat { \\boldsymbol { x } } , \\theta ) - \\partial \\boldsymbol { x } ^ { \\star } ( \\theta ) \\| \\leq \\| \\hat { A } ^ { - 1 } ( \\hat { B } - B _ { \\star } ) \\| + \\| ( \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } ) B _ { \\star } \\| } \\\\ & { \\qquad \\leq \\| \\hat { A } ^ { - 1 } \\| _ { \\mathsf { o p } } \\| \\hat { B } - B _ { \\star } \\| + \\frac { \\gamma } { \\alpha ^ { 2 } } \\| \\hat { \\boldsymbol { x } } - \\boldsymbol { x } ^ { \\star } ( \\theta ) \\| \\ \\| B _ { \\star } \\| , } \\end{array}", + "type": "interline_equation", + "image_path": "04a120c2c8d35d56824e45203ba451c01c1e663a25dc0d466c67e313514736dc.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 165, + 476, + 445, + 489.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 165, + 489.3333333333333, + 445, + 502.66666666666663 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 165, + 502.66666666666663, + 445, + 516.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 232, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 231, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 231, + 529 + ], + "score": 1.0, + "content": "which yields the desired result.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 506, + 579 + ], + "lines": [ + { + "bbox": [ + 107, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 402, + 548 + ], + "score": 1.0, + "content": "Corollary 1 (Jacobian precision for gradient descent fixed point). 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The estimated Jacobian evaluated at", + "type": "text" + }, + { + "bbox": [ + 488, + 559, + 494, + 568 + ], + "score": 0.79, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 162, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 162, + 582 + ], + "score": 1.0, + "content": "then given by", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 577, + 389, + 592 + ], + "lines": [ + { + "bbox": [ + 221, + 577, + 389, + 592 + ], + "spans": [ + { + "bbox": [ + 221, + 577, + 389, + 592 + ], + "score": 0.91, + "content": "\\boldsymbol { J } ( \\boldsymbol { \\hat { x } } , \\theta ) = - ( \\nabla _ { 1 } ^ { 2 } f ( \\boldsymbol { \\hat { x } } , \\theta ) ) ^ { - 1 } \\partial _ { 2 } \\nabla _ { 1 } f ( \\boldsymbol { \\hat { x } } , \\theta ) .", + "type": "interline_equation", + "image_path": "13e69f1b0fef0bf383fb6263539535ab5dabbdb6037c601d6674f7dce9189dd2.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 221, + 577, + 389, + 592 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 504, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 137, + 607 + ], + "score": 1.0, + "content": "For all", + "type": "text" + }, + { + "bbox": [ + 137, + 594, + 169, + 604 + ], + "score": 0.9, + "content": "\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 592, + 209, + 607 + ], + "score": 1.0, + "content": ", and any", + "type": "text" + }, + { + "bbox": [ + 209, + 594, + 216, + 603 + ], + "score": 0.79, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 592, + 263, + 607 + ], + "score": 1.0, + "content": "estimating", + "type": "text" + }, + { + "bbox": [ + 263, + 593, + 287, + 605 + ], + "score": 0.91, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 592, + 506, + 607 + ], + "score": 1.0, + "content": ", we have the following bound for the approximation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 604, + 234, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 234, + 617 + ], + "score": 1.0, + "content": "error of the estimated Jacobian", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 619, + 408, + 647 + ], + "lines": [ + { + "bbox": [ + 203, + 619, + 408, + 647 + ], + "spans": [ + { + "bbox": [ + 203, + 619, + 408, + 647 + ], + "score": 0.92, + "content": "\\| J ( { \\hat { x } } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) \\| \\leq \\left( { \\frac { \\beta } { \\alpha } } + { \\frac { \\gamma R } { \\alpha ^ { 2 } } } \\right) \\| { \\hat { x } } - x ^ { \\star } ( \\theta ) \\| .", + "type": "interline_equation", + "image_path": "dcad6b4bd5caf54e48896a9959995a9337c2e18bd05e29aacdfd1f541fbb2309.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 203, + 619, + 408, + 647 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 657, + 487, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 488, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 183, + 672 + ], + "score": 1.0, + "content": "Proof of Corollary", + "type": "text" + }, + { + "bbox": [ + 183, + 659, + 189, + 668 + ], + "score": 0.43, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 656, + 403, + 672 + ], + "score": 1.0, + "content": ". This follows from Theorem 1, applied to this specific", + "type": "text" + }, + { + "bbox": [ + 403, + 658, + 434, + 670 + ], + "score": 0.93, + "content": "A ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 656, + 451, + 672 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 452, + 658, + 483, + 670 + ], + "score": 0.93, + "content": "B ( x , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 656, + 488, + 672 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 105, + 681, + 459, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 459, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 258, + 696 + ], + "score": 1.0, + "content": "For proximal gradient descent, where", + "type": "text" + }, + { + "bbox": [ + 258, + 682, + 412, + 695 + ], + "score": 0.93, + "content": "T ( x , \\theta ) = \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 680, + 459, + 696 + ], + "score": 1.0, + "content": ", this yields", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 699, + 485, + 731 + ], + "lines": [ + { + "bbox": [ + 126, + 699, + 485, + 731 + ], + "spans": [ + { + "bbox": [ + 126, + 699, + 485, + 731 + ], + "score": 0.89, + "content": "\\begin{array} { r l } & { A ( x , \\theta ) = I - \\partial _ { 1 } T ( x , \\theta ) = I - \\partial _ { 1 } \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) ( I - \\eta \\nabla _ { 1 } ^ { 2 } f ( x , \\theta ) ) } \\\\ & { B ( x , \\theta ) = \\partial _ { 2 } \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) - \\eta \\partial _ { 1 } \\mathrm { p r o x } _ { \\eta g } ( x - \\eta \\nabla _ { 1 } f ( x , \\theta ) , \\theta ) \\partial _ { 2 } \\nabla _ { 1 } f ( x , \\theta ) . } \\end{array}", + "type": "interline_equation", + "image_path": "4bf082d2069c3545cb03be603d1c989ae677864cd1a72cf06dd97beee3d24bc0.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 126, + 699, + 485, + 709.6666666666666 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 126, + 709.6666666666666, + 485, + 720.3333333333333 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 126, + 720.3333333333333, + 485, + 730.9999999999999 + ], + "spans": [], + "index": 42 + } + ] + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 518, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 495, + 520, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 495, + 520, + 505, + 529 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 658, + 504, + 668 + ], + "lines": [ + { + "bbox": [ + 496, + 660, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 496, + 660, + 505, + 669 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 483, + 95 + ], + "score": 1.0, + "content": "and is a popular algorithm when this LMO is easier to compute than the projection onto", + "type": "text" + }, + { + "bbox": [ + 483, + 82, + 503, + 95 + ], + "score": 0.9, + "content": "\\mathcal { C } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 82, + 506, + 95 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "However, since this LMO is piecewise constant, its Jacobian is null almost everywhere. Inspired by", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 393, + 117 + ], + "score": 1.0, + "content": "SparseMAP (Niculae et al., 2018), which corresponds to the case when", + "type": "text" + }, + { + "bbox": [ + 394, + 105, + 401, + 116 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "is a quadratic, we rewrite", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 138, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 138, + 128 + ], + "score": 1.0, + "content": "(19) as", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 104, + 82, + 506, + 128 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 125, + 387, + 146 + ], + "lines": [ + { + "bbox": [ + 223, + 125, + 387, + 146 + ], + "spans": [ + { + "bbox": [ + 223, + 125, + 387, + 146 + ], + "score": 0.93, + "content": "p ^ { \\star } ( \\theta ) = \\underset { p \\in \\triangle ^ { m } } { \\operatorname { a r g m i n } } g ( p , \\theta ) : = f ( V ( \\theta ) p , \\theta ) ,", + "type": "interline_equation", + "image_path": "2806c56d422ef608e306f21ee869a7464829a762b80d5487090e07b932111bb9.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 223, + 125, + 387, + 146 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 148, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 134, + 162 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 149, + 156, + 160 + ], + "score": 0.9, + "content": "V ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 147, + 177, + 162 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 177, + 149, + 205, + 159 + ], + "score": 0.91, + "content": "d \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 147, + 330, + 162 + ], + "score": 1.0, + "content": "matrix gathering the vertices", + "type": "text" + }, + { + "bbox": [ + 330, + 149, + 400, + 161 + ], + "score": 0.92, + "content": "v _ { 1 } ( \\theta ) , \\dots , v _ { m } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 147, + 467, + 162 + ], + "score": 1.0, + "content": ". We then have", + "type": "text" + }, + { + "bbox": [ + 468, + 149, + 505, + 161 + ], + "score": 0.91, + "content": "x ^ { \\star } ( \\theta ) =", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 150, + 172 + ], + "score": 0.92, + "content": "V ( \\theta ) p ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 159, + 506, + 172 + ], + "score": 1.0, + "content": ". 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Frank-Wolfe implementations typically maintain the convex weights of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 504, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 315, + 240 + ], + "score": 1.0, + "content": "vertices, which we use to get an approximation of", + "type": "text" + }, + { + "bbox": [ + 315, + 227, + 338, + 239 + ], + "score": 0.92, + "content": "p ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 226, + 498, + 240 + ], + "score": 1.0, + "content": ". Moreover, it is well-known that after", + "type": "text" + }, + { + "bbox": [ + 499, + 228, + 504, + 237 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 178, + 250 + ], + "score": 1.0, + "content": "iterations, at most", + "type": "text" + }, + { + "bbox": [ + 179, + 239, + 183, + 248 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 238, + 506, + 250 + ], + "score": 1.0, + "content": "vertices are visited. We can leverage this sparsity to solve a smaller linear system.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 361, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 333, + 262 + ], + "score": 1.0, + "content": "Moreover, in practice, we only need to compute VJPs of", + "type": "text" + }, + { + "bbox": [ + 334, + 249, + 357, + 261 + ], + "score": 0.92, + "content": "x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 248, + 361, + 262 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 203, + 506, + 262 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 276, + 281, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 282, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 282, + 291 + ], + "score": 1.0, + "content": "D JACOBIAN PRECISION PROOFS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 506, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 179, + 313 + ], + "score": 1.0, + "content": "Proof of Theorem", + "type": "text" + }, + { + "bbox": [ + 180, + 302, + 185, + 311 + ], + "score": 0.46, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 299, + 312, + 313 + ], + "score": 1.0, + "content": ". To simplify notations, we note", + "type": "text" + }, + { + "bbox": [ + 313, + 300, + 376, + 313 + ], + "score": 0.93, + "content": "A _ { \\star } : = A ( x ^ { \\star } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 299, + 393, + 313 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 394, + 299, + 448, + 313 + ], + "score": 0.93, + "content": "\\hat { A } : = A ( \\hat { x } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 299, + 506, + 313 + ], + "score": 1.0, + "content": ", and similarly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 312, + 507, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 121, + 325 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 122, + 314, + 131, + 324 + ], + "score": 0.82, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 312, + 150, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 150, + 314, + 158, + 324 + ], + "score": 0.78, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 312, + 394, + 325 + ], + "score": 1.0, + "content": ". We have by definition of the Jacobian estimate function", + "type": "text" + }, + { + "bbox": [ + 395, + 314, + 446, + 325 + ], + "score": 0.91, + "content": "A _ { \\star } J _ { \\star } = B _ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 312, + 465, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 465, + 312, + 503, + 324 + ], + "score": 0.92, + "content": "\\hat { A } \\hat { J } = \\hat { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 312, + 507, + 325 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 324, + 183, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 183, + 336 + ], + "score": 1.0, + "content": "Therefore we have", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 299, + 507, + 336 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 337, + 426, + 388 + ], + "lines": [ + { + "bbox": [ + 185, + 337, + 426, + 388 + ], + "spans": [ + { + "bbox": [ + 185, + 337, + 426, + 388 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta ) = \\hat { A } ^ { - 1 } \\hat { B } - A _ { \\star } ^ { - 1 } B _ { \\star } } \\\\ & { \\qquad = \\hat { A } ^ { - 1 } \\hat { B } - \\hat { A } ^ { - 1 } B _ { \\star } + \\hat { A } ^ { - 1 } B _ { \\star } - A _ { \\star } ^ { - 1 } B _ { \\star } } \\\\ & { \\qquad = \\hat { A } ^ { - 1 } ( \\hat { B } - B _ { \\star } ) + ( \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } ) B _ { \\star } . } \\end{array}", + "type": "interline_equation", + "image_path": "ea19f87c8539544562bd4b7de5e16b2fd89fdc51869d3b5430f2a2930582992e.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 185, + 337, + 426, + 354.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 185, + 354.0, + 426, + 371.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 185, + 371.0, + 426, + 388.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 478, + 405 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 480, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 216, + 408 + ], + "score": 1.0, + "content": "For any invertible matrices", + "type": "text" + }, + { + "bbox": [ + 216, + 393, + 249, + 404 + ], + "score": 0.92, + "content": "M _ { 1 } , M _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 389, + 303, + 408 + ], + "score": 1.0, + "content": ", it holds that", + "type": "text" + }, + { + "bbox": [ + 303, + 391, + 464, + 405 + ], + "score": 0.92, + "content": "M _ { 1 } ^ { - 1 } - M _ { 2 } ^ { - 1 } = M _ { 1 } ^ { - 1 } ( M _ { 2 } - M _ { 1 } ) M _ { 2 } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 389, + 480, + 408 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 389, + 480, + 408 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 408, + 419, + 424 + ], + "lines": [ + { + "bbox": [ + 191, + 408, + 419, + 424 + ], + "spans": [ + { + "bbox": [ + 191, + 408, + 419, + 424 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\| M _ { 2 } ^ { - 1 } - M _ { 2 } ^ { - 1 } \\| _ { \\mathrm { o p } } \\leq \\| M _ { 1 } ^ { - 1 } \\| _ { \\mathrm { o p } } \\| M _ { 2 } - M _ { 1 } \\| _ { \\mathrm { o p } } \\| M _ { 2 } ^ { - 1 } \\| _ { \\mathrm { o p } } . } \\end{array}", + "type": "interline_equation", + "image_path": "f6fdde9603b4506641a20ca93b4ced9e9979aafe4a2e4ecfb32bdf5f13d112f5.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 191, + 408, + 419, + 424 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 149, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 151, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 151, + 440 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 425, + 151, + 440 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 436, + 418, + 460 + ], + "lines": [ + { + "bbox": [ + 192, + 436, + 418, + 460 + ], + "spans": [ + { + "bbox": [ + 192, + 436, + 418, + 460 + ], + "score": 0.89, + "content": "\\| \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } \\| _ { \\mathrm { o p } } \\leq \\frac { 1 } { \\alpha ^ { 2 } } \\| \\hat { A } - A _ { \\star } \\| _ { \\mathrm { o p } } \\leq \\frac { \\gamma } { \\alpha ^ { 2 } } \\| \\hat { x } - x ^ { \\star } ( \\theta ) \\| .", + "type": "interline_equation", + "image_path": "8d2dfeee4d4a36ec4a8042a257662c7a48d0017fa610f8c68a701ac7abbbe53c.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 192, + 436, + 418, + 460 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 479, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 480, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 258, + 474 + ], + "score": 1.0, + "content": "As a consequence, the second term in", + "type": "text" + }, + { + "bbox": [ + 259, + 461, + 330, + 473 + ], + "score": 0.93, + "content": "J ( \\hat { x } , \\theta ) - \\partial x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 460, + 480, + 474 + ], + "score": 1.0, + "content": "can be upper bounded and we obtain", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 460, + 480, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 476, + 445, + 516 + ], + "lines": [ + { + "bbox": [ + 165, + 476, + 445, + 516 + ], + "spans": [ + { + "bbox": [ + 165, + 476, + 445, + 516 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\| J ( \\hat { \\boldsymbol { x } } , \\theta ) - \\partial \\boldsymbol { x } ^ { \\star } ( \\theta ) \\| \\leq \\| \\hat { A } ^ { - 1 } ( \\hat { B } - B _ { \\star } ) \\| + \\| ( \\hat { A } ^ { - 1 } - A _ { \\star } ^ { - 1 } ) B _ { \\star } \\| } \\\\ & { \\qquad \\leq \\| \\hat { A } ^ { - 1 } \\| _ { \\mathsf { o p } } \\| \\hat { B } - B _ { \\star } \\| + \\frac { \\gamma } { \\alpha ^ { 2 } } \\| \\hat { \\boldsymbol { x } } - \\boldsymbol { x } ^ { \\star } ( \\theta ) \\| \\ \\| B _ { \\star } \\| , } \\end{array}", + "type": "interline_equation", + "image_path": "04a120c2c8d35d56824e45203ba451c01c1e663a25dc0d466c67e313514736dc.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 165, + 476, + 445, + 489.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 165, + 489.3333333333333, + 445, + 502.66666666666663 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 165, + 502.66666666666663, + 445, + 516.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 232, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 231, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 231, + 529 + ], + "score": 1.0, + "content": "which yields the desired result.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 517, + 231, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 506, + 579 + ], + "lines": [ + { + "bbox": [ + 107, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 402, + 548 + ], + "score": 1.0, + "content": "Corollary 1 (Jacobian precision for gradient descent fixed point). 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This is the case in our experiments in", + "type": "text" + }, + { + "bbox": [ + 407, + 95, + 426, + 105 + ], + "score": 0.84, + "content": "\\ S 4 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ". Recent work also", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 504, + 117 + ], + "score": 1.0, + "content": "exploits local smoothness of solutions to derive similar bounds (Bertrand et al., 2021, Theorem 13)", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 118, + 506, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 118, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 442, + 131 + ], + "score": 1.0, + "content": "Corollary 2 (Jacobian precision for proximal gradient descent fixed point). 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We therefore have that", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 366, + 506, + 400 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 403, + 394, + 438 + ], + "lines": [ + { + "bbox": [ + 217, + 403, + 394, + 438 + ], + "spans": [ + { + "bbox": [ + 217, + 403, + 394, + 438 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\quad \\mathrm { p r o x } _ { \\eta g } ( y ) = ( I + \\eta \\nabla g ) ^ { - 1 } ( y ) } \\\\ & { \\quad \\partial \\mathrm { p r o x } _ { \\eta g } ( y ) = ( I _ { d } + \\eta \\nabla ^ { 2 } g ( \\mathrm { p r o x } _ { \\eta g } ( y ) ) ) ^ { - 1 } , } \\end{array}", + "type": "interline_equation", + "image_path": "66ff28fd2cd139035577be6245ccb3f4bae0f07ccf77ded77c4a4ce4c4b8c8d6.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 217, + 403, + 394, + 420.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 217, + 420.5, + 394, + 438.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 140, + 452 + ], + "score": 1.0, + "content": "the first", + "type": "text" + }, + { + "bbox": [ + 140, + 441, + 146, + 450 + ], + "score": 0.81, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 439, + 410, + 452 + ], + "score": 1.0, + "content": "and inverse being functional identity and inverse, and the second", + "type": "text" + }, + { + "bbox": [ + 410, + 441, + 420, + 451 + ], + "score": 0.89, + "content": "I _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "and inverse being in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 484, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 293, + 464 + ], + "score": 1.0, + "content": "the matrix sense, by inverse rule for Jacobians", + "type": "text" + }, + { + "bbox": [ + 293, + 450, + 396, + 463 + ], + "score": 0.92, + "content": "\\partial h ( z ) = [ \\partial h ^ { - 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For each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "dataset size, we report the average runtime of an individual iteration in the outer problem, alongside a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 419, + 424, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 126, + 430 + ], + "score": 0.84, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 419, + 424, + 432 + ], + "score": 1.0, + "content": "confidence interval estimated from the corresponding 150 runtime values.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "Additional results Figure 13 compares the runtime of implicit differentiation and unrolling on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "GPU. 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In all cases, an additional", + "type": "text" + }, + { + "bbox": [ + 375, + 219, + 425, + 230 + ], + "score": 0.9, + "content": "m _ { \\mathrm { v a l } } = 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "validation samples", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 229, + 369, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 369, + 241 + ], + "score": 1.0, + "content": "were generated from the same model to define the outer problem.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 163, + 506, + 241 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 258 + ], + "score": 1.0, + "content": "For the inner problem, we employed three different solvers: (i) mirror descent, (ii) (accelerated)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "proximal gradient descent and (iii) block coordinate descent. Hyperparameters for all solvers were", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "individually tuned manually to ensure convergence across the range of problem sizes. For mirror", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "descent, a stepsize of 1.0 was used for the first 100 steps, following a inverse square root decay", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 432, + 302 + ], + "score": 1.0, + "content": "afterwards up to a total of 2500 steps. For proximal gradient descent, a stepsize of", + "type": "text" + }, + { + "bbox": [ + 432, + 290, + 466, + 300 + ], + "score": 0.91, + "content": "5 \\cdot 1 0 ^ { - 4 } ", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "was used", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "for 2500 steps. The block coordinate descent solver was run for 500 iterations. All solvers used the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 311, + 429, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 218, + 325 + ], + "score": 1.0, + "content": "same initialization, namely,", + "type": "text" + }, + { + "bbox": [ + 219, + 311, + 281, + 325 + ], + "score": 0.94, + "content": "\\begin{array} { r } { x _ { \\mathrm { i n i t } } = \\frac { 1 } { k } 1 _ { m \\times k } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 312, + 429, + 325 + ], + "score": 1.0, + "content": ", which satisfies the dual constraints.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 246, + 506, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 504, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 507, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 381, + 345 + ], + "score": 1.0, + "content": "For the outer problem, gradient descent was used with a stepsize of", + "type": "text" + }, + { + "bbox": [ + 381, + 330, + 415, + 341 + ], + "score": 0.9, + "content": "5 \\cdot 1 0 ^ { - 3 } ~ ", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 327, + 507, + 345 + ], + "score": 1.0, + "content": "for the first 100 steps,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 405, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 405, + 354 + ], + "score": 1.0, + "content": "following a inverse square root decay afterwards up to a total of 150 steps.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 327, + 507, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 504, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "Conjugate gradient was used to solve the linear systems in implicit differentiation for at most 2500", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 368, + 149, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 149, + 381 + ], + "score": 1.0, + "content": "iterations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 358, + 506, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "All results reported pertaining CPU runtimes were obtained using an internal compute cluster. GPU", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 398, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 409 + ], + "score": 1.0, + "content": "results were obtained using a single NVIDIA P100 GPU with 16GB of memory per dataset. For each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "dataset size, we report the average runtime of an individual iteration in the outer problem, alongside a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 419, + 424, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 126, + 430 + ], + "score": 0.84, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 419, + 424, + 432 + ], + "score": 1.0, + "content": "confidence interval estimated from the corresponding 150 runtime values.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 386, + 505, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "Additional results Figure 13 compares the runtime of implicit differentiation and unrolling on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "GPU. These results highlight a fundamental limitation of the unrolling approach in memory-limited", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "systems such as accelerators, as the inner solver suffered from out-of-memory errors for most problem", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 477, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 132, + 489 + ], + "score": 1.0, + "content": "sizes", + "type": "text" + }, + { + "bbox": [ + 132, + 477, + 173, + 488 + ], + "score": 0.87, + "content": "{ \\it p } \\geq 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 477, + 253, + 489 + ], + "score": 1.0, + "content": "for mirror descent,", + "type": "text" + }, + { + "bbox": [ + 253, + 477, + 289, + 488 + ], + "score": 0.91, + "content": "p \\geq 7 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 477, + 506, + 489 + ], + "score": 1.0, + "content": "for proximal gradient and block coordinate descent).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "While it might be possible to ameliorate this limitation by reducing the maximum number of iterations", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "in the inner solver, doing so might lead to additional challenges (Wu et al., 2018) and require careful", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 508, + 138, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 138, + 524 + ], + "score": 1.0, + "content": "tuning.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 443, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 526, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 539 + ], + "score": 1.0, + "content": "Figure 14 depicts the validation loss (value of the outer problem objective function) at convergence. It", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "shows that all approaches were able to solve the outer problem, with solutions produced by different", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "approaches being qualitatively indistinguishable from each other across the range of problem sizes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 154, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 154, + 570 + ], + "score": 1.0, + "content": "considered.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 526, + 506, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 575, + 504, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "Figure 15 shows the Jacobian error achieved as a function of the solution error, when varying the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 587, + 187, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 187, + 599 + ], + "score": 1.0, + "content": "number of features.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 574, + 506, + 599 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 612, + 296, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 298, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 298, + 624 + ], + "score": 1.0, + "content": "E.2 TASK-DRIVEN DICTIONARY LEARNING", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 123, + 643 + ], + "score": 1.0, + "content": "We", + "type": "text" + }, + { + "bbox": [ + 130, + 632, + 219, + 644 + ], + "score": 1.0, + "content": "downloaded from", + "type": "text" + }, + { + "bbox": [ + 227, + 633, + 504, + 645 + ], + "score": 1.0, + "content": "http://acgt.cs.tau.ac.il/multi_omic_benchmark/", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "download.html a set of breast cancer gene expression data together with survival information", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 506, + 668 + ], + "score": 1.0, + "content": "generated by the TCGA Research Network (https://www.cancer.gov/tcga) and processed", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "as explained by (Rappoport & Shamir, 2018). 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Note that non-selected patients", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 381, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 381, + 733 + ], + "score": 1.0, + "content": "are those who are marked as alive but were not followed for 5 years.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44, + "bbox_fs": [ + 104, + 632, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "To evaluate different binary classification methods on this cohort, we repeated 10 times a random split", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 234, + 106 + ], + "score": 1.0, + "content": "of the full cohort into a training", + "type": "text" + }, + { + "bbox": [ + 234, + 94, + 259, + 105 + ], + "score": 0.87, + "content": "( 6 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 93, + 304, + 106 + ], + "score": 1.0, + "content": ", validation", + "type": "text" + }, + { + "bbox": [ + 305, + 94, + 330, + 105 + ], + "score": 0.87, + "content": "( 2 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 93, + 365, + 106 + ], + "score": 1.0, + "content": "and test", + "type": "text" + }, + { + "bbox": [ + 365, + 94, + 390, + 105 + ], + "score": 0.88, + "content": "( 2 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "sets. 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We then report, for each method, the mean test AUC over the 10 repeats,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 481, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 168, + 161 + ], + "score": 1.0, + "content": "together with a", + "type": "text" + }, + { + "bbox": [ + 169, + 149, + 189, + 159 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 148, + 330, + 161 + ], + "score": 1.0, + "content": "confidence interval defined a mean", + "type": "text" + }, + { + "bbox": [ + 330, + 149, + 370, + 159 + ], + "score": 0.78, + "content": "\\pm \\ : 1 . 9 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 148, + 481, + 161 + ], + "score": 1.0, + "content": "standard error of the mean.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 506, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 460, + 177 + ], + "score": 1.0, + "content": "We used Scikit Learn’s implementation of logistic regression regularized by", + "type": "text" + }, + { + "bbox": [ + 460, + 165, + 470, + 177 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "(lasso)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 128, + 188 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 128, + 176, + 138, + 187 + ], + "score": 0.83, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "(ridge) penalty from sklearn.linear_model.LogisticRegression, and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 184, + 507, + 201 + ], + "spans": [ + { + "bbox": [ + 104, + 184, + 409, + 201 + ], + "score": 1.0, + "content": "varied the C regularization parameter over a grid of 10 values:", + "type": "text" + }, + { + "bbox": [ + 410, + 187, + 503, + 199 + ], + "score": 0.93, + "content": "\\{ 1 0 ^ { - 5 } , 1 0 ^ { - 3 } , \\dots , 1 0 ^ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 184, + 507, + 201 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 507, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 507, + 211 + ], + "score": 1.0, + "content": "For the unsupervised dictionary learning experiment method, we estimated a dic-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 508, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 508, + 222 + ], + "score": 1.0, + "content": "tionary from the gene expression data in the training and validation sets, us-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 122, + 233 + ], + "score": 1.0, + "content": "ing", + "type": "text" + }, + { + "bbox": [ + 164, + 221, + 476, + 233 + ], + "score": 1.0, + "content": "sklearn.decomposition.DictionaryLearning(n_component", + "type": "text" + }, + { + "bbox": [ + 476, + 221, + 501, + 230 + ], + "score": 0.85, + "content": "s { = } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 221, + 506, + 233 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 163, + 243 + ], + "score": 0.27, + "content": "\\mathtt { a l p h a } { = } 2 . 0 ", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 231, + 322, + 242 + ], + "score": 1.0, + "content": "), which produces sparse codes in", + "type": "text" + }, + { + "bbox": [ + 323, + 231, + 367, + 241 + ], + "score": 0.89, + "content": "\\begin{array} { r l r } { k } & { { } = } & { 1 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 231, + 484, + 242 + ], + "score": 1.0, + "content": "dimensions with roughly", + "type": "text" + }, + { + "bbox": [ + 484, + 231, + 505, + 242 + ], + "score": 0.87, + "content": "5 0 \\%", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "score": 1.0, + "content": "nonzero coefficients by minimizing the squared Frobenius reconstruction distance with lasso", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "regularization on the code. We then use sklearn.linear_model.LogisticRegression", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "to train a logistic regression on the codes, varying the ridge regularization parameter C over a grid of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 272, + 237, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 146, + 288 + ], + "score": 1.0, + "content": "10 values", + "type": "text" + }, + { + "bbox": [ + 146, + 274, + 233, + 287 + ], + "score": 0.93, + "content": "\\{ \\breve { 1 } 0 ^ { - 1 } , 1 \\breve { 0 ^ { 0 } } , \\dots , 1 0 ^ { 8 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 272, + 237, + 288 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "score": 1.0, + "content": "Finally, we implemented the task-driven dictionary learning model (13) with our toolbox, following", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "the pseudo-code in Figure 10. Like for the unsupervised dictionary learning experiment, we set the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 214, + 326 + ], + "score": 1.0, + "content": "dimension of the codes to", + "type": "text" + }, + { + "bbox": [ + 214, + 314, + 245, + 324 + ], + "score": 0.89, + "content": "k = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 313, + 505, + 326 + ], + "score": 1.0, + "content": ", and a fixed elastic net regularization on the inner optimization", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 290, + 338 + ], + "score": 1.0, + "content": "problem to ensure that the codes have roughly", + "type": "text" + }, + { + "bbox": [ + 291, + 325, + 311, + 336 + ], + "score": 0.88, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "sparsity. For the outer optimization problem, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 142, + 349 + ], + "score": 1.0, + "content": "solve an", + "type": "text" + }, + { + "bbox": [ + 142, + 336, + 152, + 347 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "regularized ridge regression problem, varying again the ridge regularization parameter", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 344, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 114, + 357 + ], + "score": 0.27, + "content": "\\mathrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 344, + 219, + 360 + ], + "score": 1.0, + "content": "over a grid of 10 values", + "type": "text" + }, + { + "bbox": [ + 219, + 346, + 306, + 358 + ], + "score": 0.92, + "content": "\\{ 1 0 ^ { - 1 } , 1 0 ^ { 0 } , \\dot { ~ } . ~ . ~ . ~ , 1 0 ^ { 8 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 344, + 506, + 360 + ], + "score": 1.0, + "content": ". 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Unrolled differentiation took instead 8h:05 (4 times", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "more) to run the same number of iterations. As can be seen in Figure 16, the output is the same in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 479, + 176, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 176, + 491 + ], + "score": 1.0, + "content": "both approaches.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "image", + "bbox": [ + 104, + 504, + 504, + 549 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 504, + 504, + 549 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 504, + 504, + 549 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 504, + 549 + ], + "score": 0.934, + "type": "image", + "image_path": "d14a67dc9aa1e9ca11d55deb01b59e8b2000ea8205e7371b8fdd576c7bf9e062.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 104, + 504, + 504, + 519.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 104, + 519.0, + 504, + 534.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 104, + 534.0, + 504, + 549.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 556, + 504, + 591 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 254, + 570 + ], + "score": 1.0, + "content": "Figure 16: Distilled MNIST dataset", + "type": "text" + }, + { + "bbox": [ + 254, + 556, + 295, + 568 + ], + "score": 0.92, + "content": "\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { k \\times p }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "obtained by solving (12) through unrolled differen-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "tiation. 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We then report, for each method, the mean test AUC over the 10 repeats,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 481, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 168, + 161 + ], + "score": 1.0, + "content": "together with a", + "type": "text" + }, + { + "bbox": [ + 169, + 149, + 189, + 159 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 148, + 330, + 161 + ], + "score": 1.0, + "content": "confidence interval defined a mean", + "type": "text" + }, + { + "bbox": [ + 330, + 149, + 370, + 159 + ], + "score": 0.78, + "content": "\\pm \\ : 1 . 9 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 148, + 481, + 161 + ], + "score": 1.0, + "content": "standard error of the mean.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 82, + 506, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 506, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 460, + 177 + ], + "score": 1.0, + "content": "We used Scikit Learn’s implementation of logistic regression regularized by", + "type": "text" + }, + { + "bbox": [ + 460, + 165, + 470, + 177 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "(lasso)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 128, + 188 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 128, + 176, + 138, + 187 + ], + "score": 0.83, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "(ridge) penalty from sklearn.linear_model.LogisticRegression, and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 184, + 507, + 201 + ], + "spans": [ + { + "bbox": [ + 104, + 184, + 409, + 201 + ], + "score": 1.0, + "content": "varied the C regularization parameter over a grid of 10 values:", + "type": "text" + }, + { + "bbox": [ + 410, + 187, + 503, + 199 + ], + "score": 0.93, + "content": "\\{ 1 0 ^ { - 5 } , 1 0 ^ { - 3 } , \\dots , 1 0 ^ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 184, + 507, + 201 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 507, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 507, + 211 + ], + "score": 1.0, + "content": "For the unsupervised dictionary learning experiment method, we estimated a dic-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 508, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 508, + 222 + ], + "score": 1.0, + "content": "tionary from the gene expression data in the training and validation sets, us-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 122, + 233 + ], + "score": 1.0, + "content": "ing", + "type": "text" + }, + { + "bbox": [ + 164, + 221, + 476, + 233 + ], + "score": 1.0, + "content": "sklearn.decomposition.DictionaryLearning(n_component", + "type": "text" + }, + { + "bbox": [ + 476, + 221, + 501, + 230 + ], + "score": 0.85, + "content": "s { = } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 221, + 506, + 233 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 163, + 243 + ], + "score": 0.27, + "content": "\\mathtt { a l p h a } { = } 2 . 0 ", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 231, + 322, + 242 + ], + "score": 1.0, + "content": "), which produces sparse codes in", + "type": "text" + }, + { + "bbox": [ + 323, + 231, + 367, + 241 + ], + "score": 0.89, + "content": "\\begin{array} { r l r } { k } & { { } = } & { 1 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 231, + 484, + 242 + ], + "score": 1.0, + "content": "dimensions with roughly", + "type": "text" + }, + { + "bbox": [ + 484, + 231, + 505, + 242 + ], + "score": 0.87, + "content": "5 0 \\%", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "score": 1.0, + "content": "nonzero coefficients by minimizing the squared Frobenius reconstruction distance with lasso", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "regularization on the code. We then use sklearn.linear_model.LogisticRegression", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "to train a logistic regression on the codes, varying the ridge regularization parameter C over a grid of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 272, + 237, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 146, + 288 + ], + "score": 1.0, + "content": "10 values", + "type": "text" + }, + { + "bbox": [ + 146, + 274, + 233, + 287 + ], + "score": 0.93, + "content": "\\{ \\breve { 1 } 0 ^ { - 1 } , 1 \\breve { 0 ^ { 0 } } , \\dots , 1 0 ^ { 8 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 272, + 237, + 288 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 164, + 508, + 288 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "score": 1.0, + "content": "Finally, we implemented the task-driven dictionary learning model (13) with our toolbox, following", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "the pseudo-code in Figure 10. Like for the unsupervised dictionary learning experiment, we set the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 214, + 326 + ], + "score": 1.0, + "content": "dimension of the codes to", + "type": "text" + }, + { + "bbox": [ + 214, + 314, + 245, + 324 + ], + "score": 0.89, + "content": "k = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 313, + 505, + 326 + ], + "score": 1.0, + "content": ", and a fixed elastic net regularization on the inner optimization", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 290, + 338 + ], + "score": 1.0, + "content": "problem to ensure that the codes have roughly", + "type": "text" + }, + { + "bbox": [ + 291, + 325, + 311, + 336 + ], + "score": 0.88, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "sparsity. For the outer optimization problem, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 142, + 349 + ], + "score": 1.0, + "content": "solve an", + "type": "text" + }, + { + "bbox": [ + 142, + 336, + 152, + 347 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "regularized ridge regression problem, varying again the ridge regularization parameter", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 344, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 114, + 357 + ], + "score": 0.27, + "content": "\\mathrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 344, + 219, + 360 + ], + "score": 1.0, + "content": "over a grid of 10 values", + "type": "text" + }, + { + "bbox": [ + 219, + 346, + 306, + 358 + ], + "score": 0.92, + "content": "\\{ 1 0 ^ { - 1 } , 1 0 ^ { 0 } , \\dot { ~ } . ~ . ~ . ~ , 1 0 ^ { 8 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 344, + 506, + 360 + ], + "score": 1.0, + "content": ". Because the outer problem is non-convex, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 450, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 450, + 371 + ], + "score": 1.0, + "content": "minimize it using the Adam optimizer (Kingma & Ba, 2014) with default parameters.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 290, + 506, + 371 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 385, + 236, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 237, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 237, + 398 + ], + "score": 1.0, + "content": "E.3 DATASET DISTILLATION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "Experimental setup. For the inner problem, we used gradient descent with backtracking line-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 419, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 430 + ], + "score": 1.0, + "content": "search, while for the outer problem we used gradient descent with momentum and a fixed step-size.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 400, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 400, + 442 + ], + "score": 1.0, + "content": "The momentum parameter was set to 0.9 while the step-size was set to 1.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 407, + 506, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "Figure 4 was produced after 4000 iterations of the outer loop on CPU (Intel(R) Xeon(R) Platinum", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 161, + 469 + ], + "score": 1.0, + "content": "P-8136 CPU", + "type": "text" + }, + { + "bbox": [ + 161, + 457, + 212, + 468 + ], + "score": 0.46, + "content": " @ ~ 2 . 0 0 \\mathrm { G H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "), which took 1h55. 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This is because", + "type": "text" + }, + { + "bbox": [ + 246, + 699, + 306, + 711 + ], + "score": 0.93, + "content": "\\nabla _ { 1 } f ( x , \\theta ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 698, + 317, + 711 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 317, + 699, + 360, + 711 + ], + "score": 0.91, + "content": "x = x ^ { \\star } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 698, + 365, + 711 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 672, + 506, + 711 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 718, + 463, + 733 + ], + "lines": [ + { + "bbox": [ + 147, + 718, + 463, + 733 + ], + "spans": [ + { + "bbox": [ + 147, + 718, + 463, + 733 + ], + "score": 0.9, + "content": "\\nabla _ { \\boldsymbol { \\theta } } f ( { \\boldsymbol x } ^ { \\star } ( { \\boldsymbol { \\theta } } ) , { \\boldsymbol { \\theta } } ) = \\partial { \\boldsymbol x } ^ { \\star } ( { \\boldsymbol { \\theta } } ) ^ { \\top } \\nabla _ { 1 } f ( { \\boldsymbol x } ^ { \\star } ( { \\boldsymbol { \\theta } } ) , { \\boldsymbol { \\theta } } ) + \\nabla _ { 2 } f ( { \\boldsymbol x } ^ { \\star } ( { \\boldsymbol { \\theta } } ) , { \\boldsymbol { \\theta } } ) = \\nabla _ { 2 } f ( { \\boldsymbol { x } } ^ { \\star } ( { \\boldsymbol { \\theta } } ) , { \\boldsymbol { \\theta } } ) .", + "type": "interline_equation", + "image_path": "f072a260627c1312a19fa6e1bd114fcea22e76534d25592c79aa7c3632d8d965.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 147, + 718, + 463, + 733 + ], + "spans": [], + "index": 46 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "This is known as Danskin’s theorem or envelope theorem. 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NameEquationSolution neededOracles needed
Stationary(4), (5)PrimalV1f
KKT(6)Primal and dualVif,H,G,aH,0G
Proximal gradient(7)PrimalVif,proxng
Projected gradient(9)PrimalVif,projc
Mirror descent(11)PrimalVif,proje,
Newton(14)Primal[V²f(x,0)]-1,Vif(x,0)
Block proximal gradient(15)Primal[Vif]j,[proXnglj
Conic programming(18)Residual map rootprojRP × K* XR+
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MethodL1 logregL2 logregDictL + L2 logregTask-driven DictL
AUC (%)71.6± 2.072.4 ± 2.868.3 ± 2.373.2 ± 2.1
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diff --git a/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z.md b/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z.md new file mode 100644 index 0000000000000000000000000000000000000000..85204daaead9a85d0e203f4a15f2c88bba2a1b32 --- /dev/null +++ b/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z.md @@ -0,0 +1,443 @@ +# SYMBOLIC PHYSICS LEARNER: DISCOVERING GOVERNING EQUATIONS VIA MONTE CARLO TREE SEARCH + +Fangzheng $\mathbf { S u n ^ { 1 } }$ , Yang $\mathbf { L i u ^ { 2 } }$ , Jian-Xun Wang3, Hao Sun4,∗ + +1Northeastern University, Boston, MA, USA; 2University of Chinese Academy of Sciences, Beijing, China; 3University of Notre Dame, Notre Dame, IN, USA; 4Renmin University of China, Beijing, China. Emails: sun.fa@northeastern.edu; liuyang22@ucas.ac.cn; jwang33@nd.edu; haosun@ruc.edu.cn + +# ABSTRACT + +Nonlinear dynamics is ubiquitous in nature and commonly seen in various science and engineering disciplines. Distilling analytical expressions that govern nonlinear dynamics from limited data remains vital but challenging. To tackle this fundamental issue, we propose a novel Symbolic Physics Learner (SPL) machine to discover the mathematical structure of nonlinear dynamics. The key concept is to interpret mathematical operations and system state variables by computational rules and symbols, establish symbolic reasoning of mathematical formulas via expression trees, and employ a Monte Carlo tree search (MCTS) agent to explore optimal expression trees based on measurement data. The MCTS agent obtains an optimistic selection policy through the traversal of expression trees, featuring the one that maps to the arithmetic expression of underlying physics. Salient features of the proposed framework include search flexibility and enforcement of parsimony for discovered equations. The efficacy and superiority of the SPL machine are demonstrated by numerical examples, compared with state-of-the-art baselines. + +# 1 INTRODUCTION + +We usually learn the behavior of a nonlinear dynamical system through its nonlinear governing differential equations. These equations can be formulated as $\dot { \mathbf { y } } ( t ) = d \mathbf { y } / d t = \mathcal { F } ( \mathbf { y } ( t ) )$ , where $\mathbf { y } ( t ) ~ = ~ \{ y _ { 1 } ( \hat { t } ) , y _ { 2 } ( t ) , . . . , y _ { n } ( t ) \} ^ { \hat { } } \in ~ \mathbb { R } ^ { 1 \times n _ { s } }$ denotes the system state at time $t$ , $\mathcal F ( \cdot )$ a nonlinear function set defining the state motions and $n _ { s }$ the system dimension. The explicit form of $\mathcal F ( \cdot )$ for some nonlinear dynamics remains underexplored. For example, in a mounted double pendulum system, the mathematical description of the underlying physics might be unclear due to unknown viscous and frictional damping forms. These uncertainties yield critical demands for the discovery of nonlinear dynamics given observational data. Nevertheless, distilling the analytical form of governing equations from limited noisy data, commonly seen in practice, is an intractable challenge. + +Ever since the early work on the data-driven discovery of nonlinear dynamics (Džeroski & Todorovski, 1993; Dzeroski & Todorovski, 1995), many scientists have stepped into this field of study. During the recent decade, the escalating advances in machine learning, data science, and computing power have enabled several milestone efforts of unearthing the governing equations for nonlinear dynamical systems. Notably, a breakthrough model named SINDy (Sparse Identification of Nonlinear Dynamics) (Brunton et al., 2016) has shed light on tackling this achallenge. SINDy was invented to determine the sparse solution among a pre-defined basis function library recursively through a sequential threshold ridge regression (STRidge) algorithm. SINDy quickly became one of the state-of-art methods and kindled significant enthusiasm in this field of study (Rudy et al., 2017; Long et al., 2018; Champion et al., 2019; Chen et al., 2021; Sun et al., 2021; Rao et al., 2022). However, the success of this sparsity-promoting approach relies on a properly defined candidate function library that requires good prior knowledge of the system. It is also restricted by the fact that a linear combination of candidate functions might be insufficient to recover complicated mathematical expressions. Moreover, when the library size is massive, it empirically fails to hold the sparsity constraint. + +At the same time, attempts have been made to tackle the nonlinear dynamics discovery problems by introducing neural networks with activation functions replaced by commonly seen mathematical operators (Martius & Lampert, 2017; Sahoo et al., 2018; Kim et al., 2019; Long et al., 2019). The intricate formulas are obtained via symbolic expansion of the well-trained network. This interpretation of physical laws results in larger candidate pools compared with the library-based representation of physics employed by SINDy. Nevertheless, since the sparsity of discovered expressions is primarily achieved by empirical pruning of the network weights, this framework exhibits sensitivity to userdefined thresholds and may fall short to produce parsimonious equations for noisy and scarce data. + +Alternatively, another inspiring work (Bongard & Lipson, 2007; Schmidt & Lipson, 2009) reenvisioned the data-driven nonlinear dynamics discovery tasks by casting them into symbolic regression problems which have been profoundly resolved by the genetic programming (GP) approach (Koza & Koza, 1992; Billard & Diday, 2003). Under this framework, a symbolic regressor is established to identify the governing equations that best describe the underlying physics through free combination of mathematical operators and symbols, leading to great flexibility in model selection. One essential weakness of this early methodology is that, driven exclusively by the goal of empirically seeking the best-fitting expression (e.g. minimizing the mean-square error) in a genetic expansion process, the GP-based model usually over-fits the target system with numerous false-positive terms under data noise, even sometimes at a subtle level, causing huge instability and uncertainty. However, this ingenious idea has inspired a series of subsequent endeavors (Cornforth & Lipson, 2012; Gaucel et al., 2014; Ly & Lipson, 2012; Quade et al., 2016; Vaddireddy et al., 2020). In a more recent work, Deep Symbolic Regression (DSR) (Petersen et al., 2021; Mundhenk et al., 2021), a reinforcement learning-based model was established and generally outperformed the GP based models including the commercial Eureqa software (Langdon & Gustafson, 2010). Additionally, the AI-Feynman methods (Udrescu & Tegmark, 2020; Udrescu et al., 2020; Udrescu & Tegmark, 2021) ameliorated symbolic regression for distilling physics laws from data by combining neural network fitting with a suite of physics-inspired techniques. This approach is also highlighted by a recursive decomposition of a complicated mathematical expression into different parts on a tree-based graph, which disentangles the original problem and speeds up the discovery. It outperformed Eureqa in the uncovering Feynman physics equations (Feynman et al., 1965). However, this approach is built upon ad-hoc steps and, to some extent, lacks flexible automation in equation discovery. + +The popularity of adopting the tree-based symbolic reasoning of mathematical formulas (Lample & Charton, 2019) has been rising recently to discover unknown mathematical expressions with a reinforcement learning agent (Kubalík et al., 2019; Petersen et al., 2021; Mundhenk et al., 2021). However, some former work attempting to apply the Monte Carlo tree search (MCTS) algorithm as an alternative to GP for symbolic regression (Cazenave, 2013; White et al., 2015; Islam et al., 2018; Lu et al., 2021) failed to leverage the full flexibility of this algorithm, resulting in the similar shortage that GP-based symbolic regressors possess as discussed earlier. Despite these outcomes, we are conscious of the strengths of the MCTS algorithm in equation discovery: it enables the flexible representation of search space with customized computational grammars to guide the search tree expansion. A sound mathematical underpinning for the trade-off between exploration and exploitation is remarkably advantageous as well. These features make it possible to inform the MCTS agent by our prior physics knowledge in nonlinear dynamics discovery rather than randomly searching in large spaces. + +Contribution. We propose a promising model named Symbolic Physics Learner (SPL) machine, empowered by MCTS, for discovery of nonlinear dynamics. This architecture relies on a grammar composed of (i) computational rules and symbols to guide the search tree spanning and (ii) a composite objective rewarding function to simultaneously evaluate the generated equations with observational data and enforce the sparsity of the expression. Moreover, we design multiple adjustments to the conventional MCTS by: (1) replacing the expected reward in UCT score with maximum reward to better fit the equation discovery objective, (2) employing an adaptive scaling in policy evaluation which would eliminate the uncertainty of the reward value range owing to the unknown error of the system state derivatives, and (3) transplanting modules with high returns to the subsequent search as a single leaf node. With these adjustments, the SPL machine is capable of efficiently uncovering the best path to formulate the complex governing equations of the target dynamical system. + +# 2 BACKGROUND + +In this section, we expand and explain the background concepts brought up in the introduction to the SPL architecture, including the expression tree (parse tree) and the MCTS algorithm. + +Expression tree. Any mathematical expression can be represented by a combinatorial set of symbols and mathematical operations, and further expressed by a parse tree structure (Hopcroft et al., 2006; Kusner et al., 2017) empowered by a context-free grammar (CFG). A CFG is a formal grammar characterized by a tuple comprised of 4 elements, namely, $\mathcal { G } = ( V , \Sigma , R , S )$ , where $V$ denotes a finite set of non-terminal nodes, $\Sigma$ a finite set of terminal nodes, $R$ a finite set of production rules, each interpreted as a mapping from a single non-terminal symbol in $V$ to one or multiple terminal/non-terminal node(s) in $( V \cup \Sigma ) ^ { * }$ where $^ *$ represents the Kleene star operation, and $S$ a single non-terminal node standing for a start symbol. In our work, equations are symbolized into parse trees: we define the start node as equation symbol $f$ , terminal symbols (leaf nodes) corresponding to the independent variables formulating the equation (e.g, $x , y )$ , and a placeholder symbol $C$ for identifiable constant coefficients that stick to specific production rules. The non-terminal nodes between root and leaf nodes are represented by some symbols distinct from the start and terminal nodes (i.e., $M _ { ☉ }$ ). The production rules denote the commonly seen mathematical operators: unary rules (one non-terminal node mapping to one node) for operators like $\cos ( \cdot ) , \exp ( \cdot ) , \bar { \log ( | \cdot | ) }$ , and binary rules (one non-terminal node mapping to two nodes) for operators such $\mathrm { ~ \imath s ~ } + , - , \times , \dot { \mathrm { ~ \cdot ~ } }$ . A parse tree is then generated via a pre-order traversal of production rules rooted at $f$ and terminates when all leaf nodes are entirely filled with terminal symbols. Each mathematical expression can be represented by such a traversal set of production rules. + +Monte Carlo tree search. Monte Carlo tree search (MCTS) (Coulom, 2006) is an algorithm for searching optimal decisions in large combinatorial spaces represented by search trees. This technique complies with the best-first search principle based on the evaluations of stochastic simulations. It has already been widely employed in and proved the spectacular success by various gaming artificial intelligence systems, including the famous AlphaGo and AlphaZero (Silver et al., 2017) for computer Go game. A basic MCTS algorithm is composed of an iterative process with four steps: + +1. Selection. The MCTS agent, starting from the root node, moves through the visited nodes of the search tree and selects the next node according to a given selection policy until it reaches an expandable node or a leaf node. +2. Expansion. At an expandable node, the MCTS agent expands the search tree by selecting one of its unvisited children. +3. Simulation. After expansion, if the current node is non-terminal, the agent performs one or multiple independent simulations starting from the current node until reaching the terminal state. In this process, actions are randomly selected. +4. Backpropagation. Statistics of nodes along the path from the current node to the root are updated with respect to search results (scores evaluated from the terminate states reached). + +To maintain a proper balance between the less-tested paths and the best policy identified so far, the MCTS agent sticks to a trade-off between exploration and exploitation by taking action that maximizes the Upper Confidence Bounds applied for Trees (UCT), formulated as (Kocsis & Szepesvári, 2006): + +$$ +U C T ( s , a ) = Q ( s , a ) + c \sqrt { \ln [ N ( s ) ] / N ( s , a ) } +$$ + +where $Q ( s , a )$ is the average result/reward of playing action $a$ in state $s$ in the simulations performed in the history, encouraging the exploitation of current best child node; $N ( s )$ is number of times state $s$ visited, $N ( s , a )$ the number of times action $a$ has been selected at state $s$ , and $\sqrt { \ln [ N ( s ) ] / N ( s , a ) }$ consequently encourages exploration of less-visited child nodes. Constant $c$ controls the balance between exploration and exploitation, empirically defined upon the specific problem. Theoretical analysis of UCT-based MCTS (e.g., convergence, guarantees) is referred to Shah et al. (2019). + +# 3 METHODS + +Existing studies show that the MCTS agent continuously gains knowledge of specified tasks via the expansion of the search tree and, based on the backpropagation of evaluation results (i.e., rewards and number of visits), render a proper selection policy on visited states to guide the upcoming searching (Silver et al., 2017). In the proposed SPL machine, such a process is integrated with the symbolic reasoning of mathematical expressions to reproduce and evaluate valid mathematical expressions of the physical laws in nonlinear dynamics step-by-step, and then obtain a favorable selection policy pointing to the best solution. This algorithm is depicted in Figure 1 with an illustrative example and its overall training scheme is shown in Algorithm 1. Discussion of the hyperparameter setting for this algorithm is given in Appendix Section A. + +![](images/a42dd81e12d9c668a474f4a45fa9cef2e188d64eac5f65dd6b946b67828ae5ab.jpg) +Figure 1: Schematic architecture of the SPL machine for nonlinear dynamics discovery. The graph explains the 4 MCTS phases of one learning episode with an illustrative example. + +Rewarding. To evaluate the mathematical expression $\tilde { f }$ projected from a parse tree, we define a numerical reward $r \in \mathcal { R } \subset \mathbb { R }$ based on this expression and input data ${ \mathcal { D } } = \{ { \bf Y } ; \dot { Y } _ { i } \}$ , serving as the search result of the current expansion or simulation. It is formulated as + +$$ +r = \frac { \eta ^ { n } } { 1 + \sqrt { \frac { 1 } { N } \left. \dot { Y } _ { i } - \tilde { f } ( \mathbf { Y } ) \right. _ { 2 } ^ { 2 } } } +$$ + +where $\mathbf { Y } = \{ \mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , . . . , \mathbf { y } _ { m } \} \in \mathbb { R } ^ { m \times N }$ is the $m$ dimensional state variables of a dynamical system, $\dot { Y } _ { i } \in \mathbb { R } ^ { 1 \times N }$ the numerically estimated state derivative for ith dimension, and $N$ the number of measurement data points. $\eta$ denotes a discount factor, assigned slightly smaller than 1; $n$ is empirically defined as the total number of production rules in the parse tree. This numerator arrangement is designated to penalize non-parsimonious solutions. This rewarding formulation outputs a reasonable assessment to the distilled equations and encourages parsimonious solution by discounting the reward of a non-parsimonious one. The rooted mean square error (RMSE) in denominator evaluates the goodness-of-fit of the discovered equation w.r.t. the measurement data. + +Training scheme. A grammar $\mathcal { G } = ( V , \Sigma , R , S )$ is defined with appropriate nodes and production rules to cover all possible forms of equations. To keep track of non-terminal nodes of the parsing tree, we apply a last-in-first-out (LIFO) strategy and denote the non-terminal node placed last on the stack $N T$ as the current node. We define the action space $A = R$ and the state space $s$ as all possible traversals of complete/incomplete parse trees (i.e., production rules selected) in ordered sequences. At the current state $s _ { t } = [ a _ { 1 } , a _ { 2 } , . . . a _ { t } ]$ where $t \in \mathbb N$ is the discrete traversal step-index of the upcoming production rule, the MCTS agent masks out the invalid production rules for current non-terminal node and on that basis selects a valid rule as action $a _ { t + 1 }$ (i.e, the left-hand side of a valid production rule is the current non-terminal symbol). Consequently, the parse tree gains a new terminal/non-terminal branch in accordance with $a _ { t + 1 }$ , meanwhile the agent finds itself in a new state $s _ { t + 1 } = [ a _ { 1 } , a _ { 2 } , . . . a _ { t } , a _ { t + 1 } ]$ . The agent subsequently pops off the current non-terminal symbol from $N T$ and pushes the non-terminal nodes, if there are any, on the right-hand side of the selected rule onto the stack. Once the agent attains an unvisited node, a certain amount of simulations are performed, where the agent starts to randomly select the next node until the parse tree is completed. + +1 Input: Grammar $G = ( V , \Sigma , R , S )$ , measurement data ${ \mathcal { D } } = \{ \mathbf { Y } ; { \dot { Y } } _ { i } \}$ ; +2 Parameters: discount/regularization factor $\eta$ , exploration rate $c$ , $t _ { m a x }$ ; # $\eta$ controls equation parsimony; +3 Output: Optimal governing equation $\tilde { f } ^ { \star }$ ; +4 for each episode do +5 Selection: Initialize $s _ { 0 } = \emptyset , t = 0 , N T = [ S ]$ ; +6 while $s _ { t }$ expandable and $t < t _ { m a x }$ do +7 Choose $a _ { t + 1 } = \arg \operatorname* { m a x } _ { \mathcal { A } } U C T ( s _ { t } , a )$ ; +8 Take action $a _ { t + 1 }$ , observe $s ^ { \prime } , N T$ ; +9 $s _ { t + 1 } \gets s ^ { \prime }$ note as visited, $t \gets t + 1$ ; +10 end +11 Expansion: Randomly take an unvisited path with action $a$ , observe $s ^ { \prime } , N T$ ; +12 $s _ { t + 1 } \gets s ^ { \prime }$ note as visited, $t \gets t + 1$ ; +13 if $N T = \emptyset$ then +14 Project $\tilde { f }$ , Backpropagate $r _ { t + 1 }$ and visited count and finish the episode; +15 end +16 Simulation: Fix the starting point $s _ { t } , N T$ ; +17 for each simulation do +18 while $s _ { t }$ non-terminal and $t < t _ { m a x }$ do +19 Randomly take an action $a$ , observe $s ^ { \prime } , N T$ ; +20 $s _ { t + 1 } \gets s ^ { \prime } , t \gets t + 1$ ; +21 end +22 if $N T = \emptyset$ then +23 Project $\tilde { f }$ and calculate $r _ { t + 1 }$ ; +24 end +25 end +26 Backpropagate simulation results; +27 end + +The reward is calculated or the maximal size is exceeded, resulting in a zero reward. The best result from the attempts counts as the reward of the current simulation phase and backpropagates from the current unvisited node all the way to the root node. + +Greedy search. Different from the MCTS-based gaming AIs where the agents are inclined to pick the action with a high expected reward (average returns), the SPL machine seeks the unique optimal solution. In the proposed training framework, we apply a greedy search heuristic to encourage the agent to explore the branch which yields the best solution in the past: $Q ( s , a )$ is defined as the maximum reward of the state-action pair, and its value is backpropagated from the highest reward in the simulations upon the selection of the pair. Meanwhile, to overcome the local minima problems due to this greedy approach in policy search, we enforce a certain level of randomness by empirically adopting the $\epsilon$ -greedy algorithm, a commonly seen approach in reinforcement learning models. + +Adaptive-scaled rewarding. Owing to the unknown level of error from the numerically estimated state derivatives, the range of the RMSE in the SPL reward function is unpredictable. This uncertainty affects the scale of rewarding values thus the balance between exploration and exploitation is presented in Eq. (1). Besides adding “1” to the denominator of Eq. (2) to avoid dramatically large numerical rewards, we also apply an adaptive scale of the reward, given by + +$$ +Q ( s , a ) = { \frac { r ^ { * } ( s , a ) } { \operatorname* { m a x } _ { s ^ { \prime } \in S , a ^ { \prime } \in A } Q ( s ^ { \prime } , a ^ { \prime } ) } } +$$ + +where $r ^ { * }$ denotes the maximum reward of the state-action pair. It is scaled by the current maximum reward among all states $s$ and actions $\mathcal { A }$ to reach an equilibrium that the $Q$ -values are stretched to the scale $[ 0 , 1 ]$ at any time. This self-adaptive fashion yields a well-scaled calculation of UCT under different value ranges of rewards throughout the training. + +Module transplantation. A function can be decomposed into smaller modules where each is simpler than the original one (Udrescu et al., 2020). This modularity feature, as shown in Figure 2, helps us develop a divide-and-conquer heuristic for distilling some complicated functions: after every certain amount of MCTS iterations, the discovered parse trees with high rewards are picked out and thereupon + +reckoned as individual production rules and appended to the set of production rules $R$ ; accordingly, these trees are “transplanted” to the future ones as their modules (i.e, the leaves). To avoid early overfitting problem, we incrementally enlarge the sizes of such modules from a baseline length + +![](images/5a25f743cfc4fd3f1a08cb871a3cdb3f18582c0ca7d6fe587e43f81249f2f211.jpg) +Figure 2: A module transplantation process: A complete parse serves as a single production rule and is appended to the grammar pool. + +to the maximum allowed size of the parse tree throughout the iterations. The augmentation part of $R$ is refreshed whenever new production rules are created, keeping only the ones engendering high rewards. This approach accelerates the policy search by capturing and locking some modules that likely contribute to, or appear as part of the optimal solution, especially in the cases of the mathematical expression containing “deep” operations (e.g., high-order polynomials) whose structures are difficult for the MCTS agent to repeatedly obtain during the selection and expansion. + +# 4 SYMBOLIC REGRESSION: FINDING MATHEMATICAL FORMULAS + +# 4.1 DATA NOISE & SCARCITY + +Data scarcity and noise are commonly seen in measurement data and become one of the bottleneck issues for discovering the governing equations of nonlinear dynamics. Tackling the challenges in high-level data scarcity and noise situations is traditionally regarded as an essential robustness indicator for a nonlinear dynamics discovery model. To this end, we present an examination of the proposed SPL machine by an equation discovery task in the presence of multiple levels of data noise and + +![](images/47b9fbce3bf68ac13532690055be70536c76e905d4ef341af2b871f31de9beea.jpg) +Figure 3: The effect of data noise/scarcity on recovery rate. The heatmaps demonstrate the recovery rate of GP and the SPL machine under different data conditions, summarized over 100 independent trials. + +volume, comparing with a GP-based symbolic regressor (implemented with gplearn python package)1. The target equation is $f ( x ) = 0 . 3 x ^ { 3 } + 0 . 5 x ^ { 2 } + 2 x$ , and the independent variable $X$ is uniformly sampled in the given range $[ - 1 0 , 1 0 ]$ . Gaussian white noise is added to the dependent variable $Y$ with the noise level defined as the root-mean-square ratio between the noise and the exact values. For discovery, the two models are fed with equivalent search space: $\{ + , - , \times , \div , c o s t , x \}$ as candidate mathematical operations and symbols. The hyperparameters of the SPL machine are set as $\eta = 0 . 9 9$ , $t _ { m a x } = 5 0$ , and 10,000 episodes of training is regarded as one trail. For the GP-based symbolic regressor, the population of programs is set as 2,000, the number of generations as 20. The range of constant coefficient values is $[ - 1 0 , 1 0 ]$ . For 16 different data noise and scarcity levels, each model was performed 100 independent trails. The recovery rates are displayed as a $4 \times 4$ mesh grid w.r.t. different noise/scarcity levels in Figure 3. It is observed that the SPL machine outperforms the GP-based symbolic regressor in all the cases. A T-test also proves that the recovery rate of the SPL machine is significantly higher than that of GP (e.g., $p$ -value $= 1 . 0 6 \times 1 0 ^ { - 7 }$ ). + +# 4.2 NGUYEN’S SYMBOLIC REGRESSION BENCHMARK + +Nguyen’s symbolic regression benchmark task (Uy et al., 2011) is widely used to test the model’s robustness in symbolic regression problems. Given a set of allowed operators, a target equation, and data generated by the specified equation (see Table 1 for example), the tested model is supposed to distill the mathematical expression that is identical to the target equation, or equivalent to it (e.g., Nguyen-7 equation can be recovered as $\log ( x ^ { 3 } + x ^ { 2 } + x + 1 )$ , Nguyen-10 equation can be recovered as $\sin ( x + y )$ , and Nguyen-11 equation can be recovered as $\exp ( y \log ( x ) ) )$ . Some variants of Nguyen’s benchmark equations are also considered in this experiment. Their discoveries require numerical estimation of the constant coefficient values. Each equation generates two datasets: one for training and another for testing. The discovered equation that perfectly fits the testing data is regarded as a successful discovery (i.e., the discovered equation should be identical or equivalent to the target one). The recovery rate is calculated based on 100 independent tests for each task. In these benchmark tasks, three algorithms are tested: GP-based symbolic regressor, the neural-guided GP (NGGP) (Mundhenk et al., 2021), and the SPL machine. Note that NGGP is an improved approach over DSR (Petersen et al., 2021). They are given the same set of candidate operations: √ $\{ + , - , \times , \div , \exp ( \cdot ) , \cos ( \cdot ) , \sin ( \cdot ) \}$ for all benchmarks and $\{ { \sqrt { \cdot } } , \ln ( \cdot ) \}$ are added to the 7, 8, 11 benchmarks. The hyperparameters of the GP-based symbolic regressor are the same as those mentioned in Section 4.1; configurations of the NGGP models are obtained from its source code2; detailed setting of the benchmark tasks and the SPL model is described in Appendix Section B. The success rates are shown in Table 1. It is observed that the SPL machine and the NGGP model both produce reliable results in Nguyen’s benchmark problems and the SPL machine slightly outperforms NGGP. This experiment betokens the capacity of the SPL machine in discovery of equations with divergent forms. + +Table 1: Recovery rate of three algorithms in Nguyen’s benchmark symbolic regression problems. The SPL machine outperforms the other two models in average recovery rate. + +
BenchmarkExpressionSPLNGGPGP
Nguyen-1x²+x²+x100%100%99%
Nguyen-2x²+x²+x²+x100%100%90%
Nguyen-3+x4+x²+x²+x 25100%100%34%
Nguyen-4x+x+x4+x²+x²+x99%100%54%
Nguyen-5sin(x²) cos(𝑥) -195%80%12%
Nguyen-6sin(x²)+sin(x+x²)100%100%11%
Nguyen-7ln(𝑥 +1) + ln(x²+ 1)100%100%17%
Nguyen-8√x100%100%100%
Nguyen-9sin(x) + sin(y²)100%100%76%
Nguyen-102 sin(x) cos(y)100%100%86%
Nguyen-11xy100%100%13%
Nguyen-12x4-x²+¹y²-y28%4%0%
Nguyen-1c3.39x +2.12x² +1.78x100%100%0%
Nguyen-2c0.48x4 +3.39x+2.12x² +1.78x94%100%0%
Nguyen-5csin(x²) cos(x) -0.7595%98%1%
Nguyen-8c√1.23x100%100%56%
Nguyen-9csin(1.5x) + sin(0.5y²)96%90%0%
Average94.5%92.4%38.2%
+ +Ablation Study: We consider four ablation studies by removing: (a) the adaptive scaling in reward calculation, (b) the discount factor $\eta ^ { n }$ that drives equation parsimony in Eq. (2), (c) module transplantation in tree generation, and (d) all of the above. The four models were tested on the first 12 Nguyen equations (see Appendix Section C). Results show the average recovery rates for these models are all smaller than that produced by SPL (see Appendix Table C.1), where the module transplantation brings the largest effect. Hence, these modules are critical to guarantee the proposed model efficacy. + +5 PHYSICAL LAW DISCOVERY: FREE FALLING BALLS WITH AIR RESISTANCE + +It is well known that, in 1589–1592, Galileo dropped two objects of unequal mass from the Leaning Tower of Pisa and drew a conclusion that their velocities were not affected by the mass. This has been well recognized globally as the “textbook” physical law for the vertical motion of a free-falling object: + +Table 2: Baseline models ( $\dot { c } _ { i }$ : unknown constants). + +
Physics ModelDerived model expression
Model-1H(t)= co+cit + c2t²+c3t3
Model-2H(t)= Co +Cit + C2eCt
Model-3H(t)= co + c1 log(cosh(c2t))
+ +the height of the object is formulated as $\begin{array} { r } { H ( t ) = h _ { 0 } + v _ { 0 } t - \frac { 1 } { 2 } g t ^ { 2 } } \end{array}$ , where $h _ { 0 }$ denotes initial height, $v _ { 0 }$ the initial velocity, and $g$ the gravitational acceleration. However, this ideal situation is rarely reached in our daily life because air resistance serves as a significant damping factor that prevents the above physical law from occurring in real-life cases. + +Many efforts have been made to uncover the effect of the air resistance and derive mathematical models to describe the free-falling objects with air resistance (Clancy, 1975; Lindemuth, 1971; Greenwood et al., 1986). This section provides data-driven discovery of the physical laws of relationships between height and time in the cases of free-falling objects with air resistance based on multiple experimental ball-drop datasets (de Silva et al., 2020), which contain the records of 11 different types of balls dropped from a + +Table 3: Mean square error (MSE) between ball motion prediction with the measurements in the test set. The SPL machine reaches the best prediction results in most (9 out of 11) cases. + +
TypeSPLModel-1Model-2Model-3
baseball0.32.79894.5893.507
blue basketball0.4570.51369.2092.227
green basketball0.0880.185.4351.604
volleyball0.1110.57480.9650.76
bowling ball0.0030.3387.023.167
golf ball0.0090.21486.0931.684
tennis ball0.0910.24672.2780.161
whiffle ball 11.581.61965.4260.21
whiffle ball 20.0990.62858.5330.966
yellow whiffle ball0.42817.34144.9842.57
orange whiffle ball0.7450.37936.7653.257
+ +bridge (see Appendix Figure D.1). For discovery, each dataset is split into a training set (records from the first 2 seconds) and a testing set (records after 2 seconds). Three mathematically derived physics models are selected from the literature as baseline models3,4,5 for this experiment (see Table 2), and the unknown constant coefficient values are estimated by Powell’s conjugate direction method (Powell, 1964). Based on our prior knowledge of the physical law that may appear in this case, we use $\{ + , - , \times , \div , \exp ( \cdot ) , \cosh ( \cdot ) , \log ( \cdot ) \}$ as the candidate grammars for the SPL discovery, with terminal nodes $\{ t , c o n s t \}$ . The hyperparameters are set as $\eta = 0 . 9 9 9 9$ , $t _ { m a x } = 2 0$ , and one single discovery is built upon 6,000 episodes of training. The physical laws distilled by SPL from training data are applied to the test data and compared with the ground truth. Their prediction errors, in terms of MSE, are presented in Table 3 (the SPL-discovered equations are shown in Appendix Table D.1). The full results can be found in Appendix Section D. It can be concluded that the data-driven discovery of physical laws leads to a better approximation of the free-falling objects with air resistance. + +# 6 CHAOTIC DYNAMICS DISCOVERY: THE LORENZ SYSTEM + +The first nonlinear dynamics discovery example is a 3-dimensional Lorenz system (Lorenz, 1963) whose dynamical behavior $( x , y , z )$ is governed by $\dot { x } = \sigma ( y -$ $x ) , \dot { y } = x ( \rho - z ) - y , \dot { z } = x y - \beta z$ with parameters $\sigma = 1 0$ , $\beta = 8 / 3$ , and $\rho = 2 8$ . The Lorenz attractor has two lobes and the system, starting from anywhere, makes cycles around one lobe before switching to the other and iterates repeatedly, exhibiting strong chaos. The synthetic system states $( x , y , z )$ are generated by solving the nonlinear differential equations using the Matlab ode113 (Shampine, 1975; Shampine & Reichelt, 1997) function. $5 \%$ Gaussian white noise is added to the clean data to generate noisy measurement. + +Table 4: Summary of the discovered governing equations for Lonrez system. Each cell concludes if target physics terms are distilled (if yes, number of false positive terms in uncovered expression). + +
Model宝y之
EureqaYes (1)Yes (3)Yes (1)
pySINDyYes (1)No (N/A)Yes (2)
NGGPYes (10)Yes (8)Yes (16)
SPLYes (0)Yes (0)Yes (0)
+ +The derivatives of the system states $( \dot { x } , \dot { y } , \dot { z } )$ are unmeasured but estimated by central difference and smoothed by the Savitzky–Golay filter (Savitzky & Golay, 1964) in order to reduce the noise effect. + +In this experiment, the proposed SPL machine is compared with three benchmark methods: Eureqa, pySINDy and NGGP. For Eureqa, NGGP, and the SPL machine, $\{ + , - , \times , \div \}$ are used as candidate operations; the upper bound of complexity is set to be 50; for pySINDy, the candidate function library includes all polynomial basis of $( x , y , z )$ from degree 1 to degree 4. Appendix Table E.1 presents the distilled governing equations by each approach and Table 4 summarizes these results: the SPL machine uncovers the explicit form of equations accurately in the context of active terms, whereas Eureqa, pySINDy and NGGP yield several false-positive terms in the governing equations. In particular, although Eureqa and NGGP are capable of uncovering the correct terms, their performance is very sensitive to the measurement noise as indicated by the redundant terms (despite with small coefficients) shown in Appendix Table E.1. Overall, the baseline methods fail to handle the large noise effect, essentially limiting their applicability in nonlinear dynamics discovery. It is evident that the SPL machine is capable of distilling the concise symbolic combination of operators and variables to correctly formulate parsimonious mathematical expressions that govern the Lorenz system, outperforming the baseline methods of Eureqa, pySINDy and NGGP. + +# 7 EXPERIMENTAL DYNAMICS DISCOVERY: DOUBLE PENDULUM + +This section shows SPL-based discovery of a chaotic double pendulum system with experimental data (Asseman et al., 2018) as shown in Appendix Figure E.3. The governing equations are given by: + +$$ +\begin{array} { r l } & { \dot { \omega } _ { 1 } = c _ { 1 } \dot { \omega } _ { 2 } \cos ( \Delta \theta ) + c _ { 2 } \omega _ { 2 } ^ { 2 } \sin ( \Delta \theta ) + c _ { 3 } \sin ( \theta _ { 1 } ) + \mathcal { R } _ { 1 } ( \theta _ { 1 } , \theta _ { 2 } , \dot { \theta } _ { 1 } , \dot { \theta } _ { 2 } ) , } \\ & { \dot { \omega } _ { 2 } = c _ { 1 } \dot { \omega } _ { 1 } \cos ( \Delta \theta ) + c _ { 2 } \omega _ { 1 } ^ { 2 } \sin ( \Delta \theta ) + c _ { 3 } \sin ( \theta _ { 2 } ) + \mathcal { R } _ { 2 } ( \theta _ { 1 } , \theta _ { 2 } , \dot { \theta } _ { 1 } , \dot { \theta } _ { 2 } ) } \end{array} +$$ + +where $\theta _ { 1 } , \theta _ { 2 }$ denote the angular displacements; $\omega _ { 1 } = \dot { \theta } _ { 1 }$ , $\omega _ { 2 } ~ = ~ { \dot { \theta } } _ { 2 }$ the velocities; $\dot { \omega } _ { 1 } , \dot { \omega } _ { 2 }$ the accelerations; $\mathcal { R } _ { 1 }$ and $\mathcal { R } _ { 2 }$ denote the unknown damping forces. Note that $\Delta \theta = \theta _ { 1 } - \theta _ { 2 }$ . + +The data source contains multiple camera-sensed datasets. Here, 5,000 denoised random sub-samples from 5 datasets are used for training, 2,000 random sub-samples from another 2 datasets for validation, and 1 dataset for testing. The derivatives of the system states are numerically estimated by the same approach discussed in the Lorenz case. Some prior physics knowledge is employed to guide the discovery: (1) the terms $\dot { \omega } _ { 2 } \cos ( \Delta \theta )$ for $\dot { \omega } _ { 1 }$ and $\dot { \omega } _ { 1 } \cos ( \Delta \theta )$ for $\dot { \omega } _ { 2 }$ are assumed to be part of the governing equations based on the Lagrange derivation; (2) the angles $( \theta _ { 1 } , \theta _ { 2 } , \Delta \theta )$ are under the trigonometric functions $\cos ( \cdot )$ and $\sin ( \cdot )$ ; (3) directions of velocities/relative velocities may appear in damping. Production rules fulfilling the above prior knowledge are exhibited in Appendix Section E.2. The hyperparameters are set as $\eta = 1$ , $t _ { m a x } = 2 0$ , and 40,000 episodes of training are regarded as one trail. 5 independent trials are performed and the equations with the highest validation scores are selected as the final results. The uncovered equations are given as follows: + +$$ +\dot { \omega } _ { 1 } = - 0 . 0 9 9 1 \dot { \omega } _ { 2 } \cos ( \Delta \theta ) - 0 . 1 0 3 \omega _ { 2 } ^ { 2 } \sin ( \Delta \theta ) - 6 9 . 2 7 4 \sin ( \theta _ { 1 } ) + 0 . 5 1 5 \cos ( \theta _ { 1 } ) , +$$ + +$$ +\begin{array} { r } { \dot { \omega } _ { 2 } = - 1 . 3 6 8 \dot { \omega } _ { 1 } \cos ( \Delta \theta ) + 1 . 3 6 3 \omega _ { 1 } ^ { 2 } \sin ( \Delta \theta ) - 9 2 . 9 1 3 \sin ( \theta _ { 2 } ) + 0 . 0 3 2 \omega _ { 1 } , } \end{array} +$$ + +where the explicit expression of physics in an ideal double pendulum system, as displayed in Eq. (5), are successfully distilled and damping terms are estimated. This set of equation is validated through interpolation on the testing set and compared with the smoothed derivatives, as shown in Appendix Figure E.4. The solution appears felicitous as the governing equations of the testing responses. + +# 8 CONCLUSION AND DISCUSSION + +This paper introduces a Symbolic Physics Learner (SPL) machine to tackle the challenge of distilling the mathematical structure of equations for physical systems (e.g., nonlinear dynamics) with scarce/noisy data. This framework is built upon the expression tree interpretation of mathematical operations and variables and an MCTS agent that searches for the optimistic policy to reconstruct the target mathematical formula. With some remarkable adjustments to the MCTS algorithms, the SPL model straightforwardly accepts our prior or domain knowledge, or any sort of constraints of the tasks in the grammar design while leveraging great flexibility in expression formulation. The robustness of the proposed SPL machine for complex target expression discovery within a large search space is indicated in the Nguyen’s symbolic regression benchmark problems, where the SPL machine outperforms state-of-the-art symbolic regression methods. Moreover, encouraging results are obtained in the tasks of discovering physical laws and nonlinear dynamics, based on synthetic or experimental datasets. While the proposed SPL machine shows huge potential in both symbolic regression and physical law discovery tasks, there are still some imperfections that can be improved: (i) the computational cost is high for constant coefficient value estimation due to repeated calls for an optimization process, (ii) graph modularity is underexamined, and (iii) robustness against extreme data noise and scarcity is not optimal. These limitations are further explained in Appendix Section F. + +# ACKNOWLEDGMENTS + +The work is supported by the National Natural Science Foundation of China (No. 92270118) and the Beijing Outstanding Young Scientist Program (No. BJJWZYJH012019100020098). + +# REFERENCES + +Alexis Asseman, Tomasz Kornuta, and Ahmet Ozcan. Learning beyond simulated physics. In Modeling and Decision-making in the Spatiotemporal Domain Workshop–NIPS, 2018. + +L Billard and E Diday. From the statistics of data to the statistics of knowledge: symbolic data analysis. 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The programming game: evaluating mcts as an alternative to gp for symbolic regression. In Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation, pp. 1521–1522, 2015. + +# APPENDIX + +# A HYPERPARAMETER SETTING + +We perform a parametric study on the value of discount factor $\eta$ based on Nguyen’s benchmark problems without measurement noise. Empirically, setting $\eta = 0 . 9 9 9 9$ ensures the scores of ground truth equations stand out and successfully enforces the sparsity in all experiments. For the discovery of very chaotic dynamical systems based on measurement data, we expect some physics terms from the governing equations to have a weak impact on the state variables (e.g., in the chaotic double pendulum system experiments, the effects from physics terms $\sin ( \theta _ { 1 } )$ and $\mathrm { s i n } ( \theta _ { 2 } )$ are hard to be captured), and the effect of data noise is unknown. Hence, we set $\eta = 1$ to leverage the full strength of data fitting to enable the detection of physics terms that are offset or overwhelmed by data noise but are pivotal to the systems. Nevertheless, we must acknowledge that this selection process is empirical, which depends on our desire for the degree of parsimony of the target equation(s). + +As for the hyperparameters in the training schema (i.e., maximum module transplantation, episodes, maximum tree size, maximum augmented grammars), we have conducted parametric convergence tests for each experiment to ensure the learning curves (i.e., maximum scores in the history) converge. For example, as discussed in Section B, Table B.2 shows the setting of these hyperparameters. + +# B NGUYEN’S BENCHMARK PROBLEMS + +This section provides more detailed experiment settings for the Nguyen’s benchmark tasks that are described in Section 4.2 of the main text, where the training hyperparameters for the SPL machine in these equation discovery experiments are also listed. Table B.1 presents the candidate mathematical operations allowed for three tested models and Table B.2 displays training hyperparameters for the SPL machine in the Nguyen’s benchmark tasks. + +Moreover, the utilization of CFG in the SPL machine facilitates the flexibility of applying some prior knowledge including the universal mathematical rules and constraints. This feature empirically turns out to be an accessible and scalable approach for avoiding meaningless mathematical expressions. In the Nguyen’s benchmark tasks, one or multiple mathematical constraints are given to the SPL machine. These constraints include + +1. Only variables and constant values are allowed in trigonometric functions. + +Table B.1: Candidate operators for each Nguyen’s benchmark task. const denotes constant values. + +
Benchmark Candidate Operations
Nguyen-1 +,-,×,÷,cos(-),sin(-),exp(·),sin(+,-,×,÷,cos(-),sin(-),exp(-),log(),√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√+,-,×,÷,cos(-),sin(-),exp(-),log()，√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√,const
Nguyen-2 +,-,×,÷,cos(·),sin(.),exp()
Nguyen-3×,÷,cos(·,sin(·
Nguyen-4X，÷,cos(,sin(·
Nguyen-5×，÷,cos(·,sin(·
Nguyen-6X,÷,cos(·,sin(·
Nguyen-7×÷,cos(·,sin(·,exp(·
Nguyen-8×÷,cos(·,sin(·,exp(·
Nguyen-9×÷cos(·,sin(·,exp(·
Nguyen-10×÷Cos,sin(·,exp(.
Nguyen-11×÷Cos,sin(·,exp(.
Nguyen-12×÷Cos,sin(·,exp(.
Nguyen-1c×÷Cos,sin(·,exp(.
Nguyen-2c×÷cos(.,sin(·,exp(.
Nguyen-5c十，一×÷Cos,sin(·,exp(·
Nguyen-8c十，一×÷cos,sin(·,exp(.
Nguyen-9c×÷cos(·,sin(·,exp(·
+ +Table B.2: Training Hyperparameter settings for the SPL model in Nguyen’s benchmark tasks. + +
BenchmarkMaximum Module TransplantationEpisodes Between Module TransplantationMaximum Tree SizeMaximum Augmented Grammars
Nguyen-12010,000505
Nguyen-22010,000505
Nguyen-320100,000505
Nguyen-420100,000505
Nguyen-520100,000505
Nguyen-62010,000505
Nguyen-7205,000505
Nguyen-8205,000505
Nguyen-92010,000505
Nguyen-102010,000505
Nguyen-112010,000505
Nguyen-1220100,000505
Nguyen-1c202,000505
Nguyen-2c2010,000505
Nguyen-5c2010.000505
Nguyen-8c202,000505
Nguyen-9c201,000505
+ +Table B.3: Mathematical constraints for the SPL machine in each Nguyen’s benchmark problem. + +
BenchmarkConstraints Utilization
Nguyen-1{1,3}
Nguyen-2{1,3}
Nguyen-3{1,3}
Nguyen-4{1,3}
Nguyen-5{2,3}
Nguyen-6{2,3}
Nguyen-7{2,3}
Nguyen-80
Nguyen-9{2,3,4}
Nguyen-10{2,3,4}
Nguyen-11{2,3}
Nguyen-12{1,3,4}
Nguyen-1c{1,3}
Nguyen-2c{1,3}
Nguyen-5c{2,3}
Nguyen-8c0
Nguyen-9c{2,3,4}
+ +2. Variables in trigonometric functions, logarithms and roots are up to the polynomial of 3. +3. Trigonometric functions, logarithms and roots are not allowed to form unreasonable composite functions with each other, such as $\sin ( \cos ( . . . ) )$ . +4. Some small integers (e.g. 1, 2) are used directly as leaves. + +They can be easily implemented into the SPL machine by defining or adjusting non-terminal nodes or production rules in the customized CFG. Constraints adopted by each task are shown in Table B.3. + +# C RESULTS OF ABLATION STUDY + +We consider four ablation studies by removing the following: + +(a) the adaptive scaling in reward calculation, (b) the discount factor $\eta ^ { n }$ that drives equation parsimony in Eq. (2), (c) module transplantation in tree generation, (d) all of the above three. + +The resulting models are denoted with Model A, Model B, Model C, and Model D (note that Model $\mathbf { D }$ is equivalent to the vanilla MCTS). The ablation study is performed on the 12 classic Nguyen’s benchmark problems, where the recovery rate of each model is calculated. The results of the ablation study are summarized in Table C.1. It is clear that module transplantation brings the largest gain in recovery rate. All the ablated models fail to uncover the Nguyen-12 equation (the most difficult case), where the coefficient 1/2 needs to be represented by mathematical operators and symbols, e.g., $x / ( x + x )$ , $y / ( y + y )$ , etc. + +# D FREE FALLING BALLS WITH AIR RESISTANCE + +This appendix section reveals more details on discovering the physical laws, in the context of relationships between height and time, for the cases of free-falling objects with air resistance based on multiple experimental ball-drop datasets (de Silva et al., 2020). The datasets contain the records of 11 different types of balls, as shown in Figure D.1, dropped from a bridge, collected at a $3 0 \mathrm { H z }$ sampling rate. The time between dropping and landing varies in each case due to the fact that air resistance has different effects on these free-dropping balls and induces divergent physical laws. Consequently, we consider each ball as an individual experiment and discover the physical law for each of them. The measurement dataset of a free-dropping ball is split into a training set (records from the first 2 seconds, 60 measurements) and a testing set (records after 2 seconds). + +The physical laws of the three baseline models and distilled by the SPL machine from training data are exhibited in Table D.1. These discovered physical laws are then applied to forecast the height of the balls at the time slots in the testing dataset. These predictions, in comparison with the ground truth trajectory recorded, are shown in Figure D.2. The prediction error is shown in the Table 3 of the main text. + +# E NONLINEAR DYNAMICS DISCOVERY + +This appendix section elaborates more details on the two governing equation discovery experiments presented in the main text, including the datasets and full results. + +Table C.1: Summary of the ablation study results. + +
BenchmarkExpressionSPLModel AModel BModel CModel D
Nguyen-1+ +x 2 2100%100%100%14%12%
Nguyen-2x4 + +x100%100%100%0%0%
Nguyen-3+x4 +x² +x 25100%100%100%0%0%
Nguyen-4+x+x4+x²+x²+x 2699%96%92%0%0%
Nguyen-5sin(𝑥²)cos(x)-195%95%92%92%88%
Nguyen-6sin(x²)+sin(x+x²)100%100%96%100%100%
Nguyen-7ln(x +1) +ln(x² + 1)100%100%100%100%100%
Nguyen-8√x100%100%100%100%100%
Nguyen-9sin(x)+sin(y²)100%100%100%100%100%
Nguyen-102 sin(x) cos(y)100%100%100%0%0%
Nguyen-11x100%96%96%92%87%
Nguyen-12x4 2 + 1 -y28%0%0%0%0%
Average93.5%90.58%89.67%49.83%48.92%
+ +Table D.1: Uncovered physics from the motions of free-falling balls by the SPL machine and three baseline models. Note that these formulas are the raw equations produced by SRL. Further simplification helps better parsimony of the formulas. + +
TypeModelExpression
baseballSPL Model-1 Model-2H(t)= 47.8042 + 0.6253t- 4.5383t² H(t)= 47.682+ 1.456t- 5.629t² + 0.376t3 H(t)= 45.089- 8.156t + 5.448 exp(0t)
blue basketballModel-3 SPL Model-1 Model-2H(t)= 48.051- 183.467 log(cosh(0.217t)) H(t)= 46.4726-5.105t² + t³ -0.251t4 H(t)= 46.513-0.493t-3.912t² +0.03t H(t)= 43.522- 7.963t + 5.306 exp(0t)
green basketballModel-3 SPL Model-1H(t)= 46.402-84.791log(cosh(0.319t)) H(t)= 45.9087- 4.1465t² +log(cosh(1)) H(t)= 46.438-0.34t- 3.882t² - 0.055t3 H(t)= 43.512- 8.043t + 5.346 exp(0t)
volleyballModel-2 Model-3 SPL Model-1 Model-2H(t)= 46.391-124.424 log(cosh(0.263t)) H(t)= 48.0744-3.7772t² H(t)= 48.046+ 0.362t-4.352t² +0.218t
bowling ballModel-3 SPL Model-1H(t)= 45.32- 7.317t + 5.037exp(0t) H(t)= 48.124-107.816 log(cosh(0.27t)) H(t)= 46.1329-3.8173t² -0.2846t+4.14× 10-5 exp(20.7385t²)exp(-12.4538t3) H(t)= 46.139-0.091t-3.504t² -0.431t3
golf ballModel-2 Model-3 SPL Model-1H(t)= 43.336-8.525t + 5.676 exp(0t) H(t)= 46.342- 247.571log(cosh(0.189t)) H(t)= 49.5087-4.9633t² +log(cosh(t)) H(t)= 49.413+0.532t-5.061t²+0.102t3
tennis ballModel-2 Model-3 SPL Model-1H(t)= 46.356 -8.918t+ 5.964exp(0t) H(t)= 49.585-178.47log(cosh(0.23t)) H(t)= 47.8577-4.0574t² +log(cosh(0.121t))
whiffle ballModel-2 Model-3 SPL Model-1H(t)= 47.738+0.658t-4.901t²+0.325t H(t)= 45.016-7.717t+ 5.212 exp(0t) H(t)= 47.874- 114.19 log(cosh(0.269t)) H(t)= 4.1563t² -t + 47.0133exp(-0.1511t²) H(t)= 46.969+0.574t-4.505t²+0.522t
1 whiffle ballModel-2 Model-3 SPL Model-1H(t)= 44.259- 6.373t+ 4.689 exp(0t) H(t)= 47.062- 34.083log(cosh(0.462t)) H(t)= -18.6063 + 65.8583 exp(-0.0577t²) H(t)= 47.215+0.296t-4.379t²+0.421t3
2 yellow whiffleModel-2 Model-3 SPLH(t)= 44.443-6.744t + 4.813 exp(0t) H(t)= 47.255-38.29 log(cosh(0.447t)) H(t)= 148.9911/(log(cosh(t)) +3.065)-14.5828t²/(log(cosh(t)) +3.065)
ballModel-1 Model-2 Model-3+48.6092log(cosh(t))/(log(cosh(t)) + 3.065) H(t)= 48.613-0.047t-4.936t² +0.826t3 H(t)= 45.443- 6.789t+ 4.973exp(0t) H(t)= 48.594-12.49 log(cosh(0.86t))
orange whiffle ballSPL Model-1 Model-2 Model-3H(t)= -1.6626t + 47.8622 exp(-0.06815t²) H(t)= 47.836-1.397t-3.822t² +0.422t³ H(t)= 44.389-7.358t+ 5.152 exp(0t)
+ +![](images/e9b06f6ee845288af4c2619e93eb8e80a77b5c720bdb62d770ef47dca03524d9.jpg) +Figure D.1: The experimental balls that were dropped from the bridge (de Silva et al., 2020). From left to right: golf ball, tennis ball, whiffle ball 1, whiffle ball 2, baseball, yellow whiffle ball, orange whiffle ball, green basketball, and blue basketball. Volleyball is not shown here. + +![](images/c0c5774af163407ed9d73424cb4fb4fcf79658a7b1a4ed5734b9f71a03ee7860.jpg) +Figure D.2: Trajectories after 2 seconds predicted by uncovered physical laws. + +# E.1 LORENZ SYSTEM + +The 3-dimensional Lorenz system is governed by + +$$ +\begin{array} { l } { \dot { x } = \sigma ( y - x ) } \\ { \dot { y } = x ( \rho - z ) - y } \\ { \dot { z } = x y - \beta z } \end{array} +$$ + +with parameters $\sigma = 1 0$ , $\beta = 8 / 3$ , and $\rho = 2 8$ , under which the Lorenz attractor has two lobes and the system, starting from anywhere, makes cycles around one lobe before switching to the other and iterates repeatedly. The measurement data of the Lorenz system states in this experiment contains a clean signal with $5 \%$ Gaussian white noise. The derivatives of Lorenz’s state variables are unmeasured but numerically estimated and smoothed by the Savitzky–Golay filter. The noisy synthetic measurement data and numerically obtained derivatives are shown in Figure E.1. + +![](images/a41908088bcc678ad1a4d19638a3ca80f4aab863564157948f73cef957389e0f.jpg) +Figure E.1: Lorenz system for the experiment. Noisy measurement data and numerically estimated derivatives smoothed by Savitzky–Golay filter. + +Table E.1: Discovered governing equations for the Lorenz system. + +
ModelDiscovered Governing Equations
Eureqax= -0.56-9.02x +9.01y y= -0.047+18.79x+1.86y-0.046xy-0.74xz = -3.04-2.23z +0.88xy
pySINDyx = -0.46-9.18x +9.17y y= 22.32x+0.15y-0.85xz z = 6.04-2.83z +0.15x² + 0.81xy
NGGPx = 0.0047 -10.02x + 10.01y - 0.007x² + 0.007xy - 0.37x/z - 0.00074x²y +0.00063x +0.00018x²z +0.00011xy² -6.59e-5xyz -0.00011y² y = 26.36x-1.5y-0.83xz -7.20x/z + 13.08y/z + 4.52e-5x³ -0.0038xz² -44.25x/z²+0.00028x/z -0.00017xz + 5.98e 6x32 = -0.64- 0.036y － 2.64z + 1.038xy+ 0.00021xz + 0.0011yz － 0.00021x/z
SPL-8.04e-8y/x + 0.00022y/z - 8.04e-8z² - 0.00021xyz +0.00021xy/z -0.001y²z-0.00021y²/z+1.17y/z²-3.89e-7yz²/x+6.66e-9z²/x x = -9.966x + 9.964y y = 27.764x - 0.942y- 0.994xz
+ +Table E.1 presents the distilled governing equations for the Lorenz system by the SPL machine compared with 3 baseline models. It is observed that the SPL machine uncovers the explicit form of equations accurately in the context of active terms, whereas Eureqa, pySINDy and NGGP yield several false-positive terms in the underlying governing equations. The predicted system responses (starting from a different initial condition) simulated from these uncovered equations are shown in Figure E.2. Although it is hard to reproduce the most accurate coefficients due to the tremendous errors induced by numerical differentiation of noisy measurement data as depicted in Figure E.1, the SPL machine is still capable of distilling the most concise symbolic combination of operators and variables to correctly formulate the parsimonious mathematical expressions that govern the Lorenz system dynamics. The predicted responses for the governing equations unearthed by the SPL machine simulate the system in a decent manner. + +![](images/2bc7320d817370625240d80bb22bb51cdf8338752d89ab78274cb0f16c9e4e2e.jpg) +Figure E.2: Response prediction for 5 seconds by identified governing equations (dashed plots) under a different validation IC of Lorenz system, in comparison with the ground truth trajectory (grey). + +![](images/796fdf0e7e82ded4d13c43f8fb46e8d2be605444c2c5124087a627fcbb91c6c7.jpg) +Figure E.3: Double Pendulum system experiment and measurement data (Asseman et al., 2018): A. experiment setup. B. displacements of the two moving masses. C. model the system with $\theta _ { 1 }$ and $\theta _ { 2 }$ . D. angles of two masses transformed from displacements. + +# E.2 MOUNTED DOUBLE PENDULUM SYSTEM + +The second nonlinear dynamics discovery experiment is a chaotic double pendulum system (Asseman et al., 2018). The measured data, in form of videos, represents the chaotic motion of a double pendulum on the device shown in Figure E.3A filmed with a high-speed camera. The positional data is converted into angular form based on the geometry information (see the model shown in Figure E.3C). The governing equations can be derived using the Euler–Lagrange method, given by + +$$ +\begin{array} { r } { ( m _ { 1 } + m _ { 2 } ) l _ { 1 } \ddot { \theta } _ { 1 } + m _ { 2 } l _ { 2 } \ddot { \theta } _ { 2 } \cos ( \theta _ { 1 } - \theta _ { 2 } ) + m _ { 2 } l _ { 2 } \omega _ { 2 } ^ { 2 } \sin ( \theta _ { 1 } - \theta _ { 2 } ) + ( m _ { 1 } + m _ { 2 } ) g \sin ( \theta _ { 1 } ) = F _ { 1 } , } \\ { m _ { 2 } l _ { 2 } \ddot { \theta } _ { 2 } + m _ { 2 } l _ { 1 } \ddot { \theta } _ { 1 } \cos ( \theta _ { 1 } - \theta _ { 2 } ) - m _ { 2 } l _ { 1 } \omega _ { 1 } ^ { 2 } \sin ( \theta _ { 1 } - \theta _ { 2 } ) + m _ { 2 } g \sin ( \theta _ { 2 } ) = F _ { 2 } , } \end{array} +$$ + +which, by denoting $\omega$ as the velocity, can be converted to the following state-space form: + +$$ +\begin{array} { r l } & { \dot { \theta } _ { 1 } = \omega _ { 1 } , } \\ & { \dot { \theta } _ { 2 } = \omega _ { 2 } , } \\ & { \dot { \omega } _ { 1 } = c _ { 1 } \dot { \omega } _ { 2 } \cos ( \Delta \theta ) + c _ { 2 } \omega _ { 2 } ^ { 2 } \sin ( \Delta \theta ) + c _ { 3 } \sin ( \theta _ { 1 } ) + \mathcal { R } _ { 1 } ( \theta _ { 1 } , \theta _ { 2 } , \dot { \theta } _ { 1 } , \dot { \theta } _ { 2 } ) , } \\ & { \dot { \omega } _ { 2 } = c _ { 1 } \dot { \omega } _ { 1 } \cos ( \Delta \theta ) + c _ { 2 } \omega _ { 1 } ^ { 2 } \sin ( \Delta \theta ) + c _ { 3 } \sin ( \theta _ { 2 } ) + \mathcal { R } _ { 2 } ( \theta _ { 1 } , \theta _ { 2 } , \dot { \theta } _ { 1 } , \dot { \theta } _ { 2 } ) } \end{array} +$$ + +where $\Delta \theta = \theta _ { 1 } - \theta _ { 2 }$ , and $\mathcal { R } _ { 1 } ( \theta _ { 1 } , \theta _ { 2 } , \dot { \theta } _ { 1 } , \dot { \theta } _ { 2 } )$ and $\mathcal { R } _ { 2 } ( \theta _ { 1 } , \theta _ { 2 } , \dot { \theta } _ { 1 } , \dot { \theta } _ { 2 } )$ denote the damping terms for the last two differential equations. + +The data source contains multiple video datasets. For this discovery, 5,000 denoised random subsamples from 5 datasets are used for training purposes, and 2,000 random sub-samples from another 2 datasets for validation, and 1 dataset for testing. Some prior knowledge guiding this discovery includes: + +Table E.2: Discovered governing equations of the mounted double pendulum by the SPL model. + +
PhaseExpression
u1-0.0991ω2 cos(△0) - 0.103ω2 sin(△0) - 69.274 sin(0i) + 0.515 cos(01)
2-1.368ω1 c0s(△0) + 1.363ω² sin(△0) - 92.913 sin(02) + 0.032w1
+ +1. the two terms $\dot { \omega } _ { 2 } \cos ( \Delta \theta )$ for $\dot { \omega } _ { 1 }$ and $\dot { \omega } _ { 1 } \cos ( \Delta \theta )$ for $\dot { \omega } _ { 2 }$ , which can be easily derived based on our prior knowledge on the system, are assumed known in the two governing equations. However, their coefficients are unknown and need to be estimated. + +2. the remaining of the formulas are potentially comprised of the free combination of $\dot { \omega } _ { 1 } , \dot { \omega } _ { 2 }$ , $\omega _ { 1 } , \omega _ { 2 }$ , as well as the angles $( \theta _ { 1 } , \theta _ { 2 } , \Delta \theta )$ under the trigonometric functions $\cos ( \cdot )$ and $\sin ( \cdot )$ . + +3. velocities and relative velocities of two masses, as well as their directions (sign function) might contribute to the damping. + +Based on the above information, the candidate production rules for $\dot { \omega } _ { 1 }$ and $\dot { \omega } _ { 2 }$ equations are shown below, where non-terminal nodes are $V = \{ A , W , T , S \}$ and $C$ denotes the placeholder symbol for the constant coefficient values. + +$$ +\begin{array} { r l } & { A A + A , A A \times A , A C , A A + A , A W , } \\ & { W W \times W , W \omega _ { 1 } , W \omega _ { 2 } , W \dot { \omega } _ { 1 } , W \dot { \omega } _ { 2 } , } \\ & { A \cos ( T ) , A \sin ( T ) , T T + T , T T - T , T \theta _ { 1 } , T \theta _ { 2 } , } \\ & { A s i g n ( S ) , S S + S , S S - S , S \omega _ { 1 } , S \omega _ { 2 } , S \dot { \omega } _ { 1 } , S \dot { \omega } _ { 2 } , } \\ & { A \dot { \omega } _ { 1 } \cos ( \theta _ { 1 } - \theta _ { 2 } ) , A \dot { \omega } _ { 2 } \cos ( \theta _ { 1 } - \theta _ { 2 } ) . } \end{array} +$$ + +Note that our prior knowledge can be easily incorporated in the proposed SPL machine to improve the discovery performance, rather than relying on the free combination of mathematical operators and symbols. The hyperparameters are set as $\eta = 1$ , $t _ { m a x } = 2 0$ , and 40,000 episodes of training are regarded as one trail. 5 independent trials are performed and the equations with the highest validation scores are selected as the final result. The uncovered equations are shown in Table E.2. They are validated through interpolation on the testing set and compared with the smoothed derivatives, as shown in Figure E.4. The solution appears felicitous as the governing equations of the testing responses. + +# F DISCUSSION AND FUTURE DIRECTIONS + +The effectiveness of the proposed SPL machine is empowered by the following elements: (1) The use of MCTS enables the flexible representation of search space with customized computational grammars, composed of a finite set of mathematical operators and symbols, to guide the search tree expansion. (2) The exploration-exploitation trade-off nature of MCTS is remarkably useful for searching the optimal mathematical expression tree. (3) The key adjustments, including the greedy search, the adaptive-scaled rewarding, the reward regularizer, and the expression tree module transplantation, make it possible to efficiently uncover the best path to formulate complex equations. (4) The SPL machine straightforwardly accepts our prior or domain knowledge, or any sort of constraints of the tasks in the grammar design while leveraging great flexibility in expression formulation. + +While SPL shows huge potential in both symbolic regression and governing equation discovery tasks, there are still some imperfections to be improved. In this appendix section, a few bottlenecks and their potential solutions are presented: + +1. Computational cost. Computational cost for this framework is one of the major issues, especially when constant coefficient estimation is required. Evaluating the solution in the simulation phase happens very frequently for the MCTS algorithm where the policy selection relies heavily on a large number of historical rewards. However, the constant coefficient value estimation, which requires repeated calls for an optimization process and can be slow, is needed for evaluation purposes. In particular, the constant coefficient value is estimated via concurrently solving an optimization problem, e.g., by Powell’s conjugate direction method (Powell, 1964). For example, when the tree structure changes or is updated, the optimization of the constant coefficients should be re-performed simultaneously. The SPL machine is not the only symbolic regressor suffering from the computational cost in constant coefficient value estimation. In fact, the state-of-the-art symbolic regression model, the neural-guided GP (NGGP) (Mundhenk et al., 2021), becomes much slower in the Nguyen’s benchmark variant tasks (see Table E.3). The current implementation of the SPL machine tries to empirically avoid this issue by limiting the number of placeholders in a discovered expression and simplifying the expression before evaluation, but still cannot reach great efficiency. This bottleneck might be mitigated if parallel computing is introduced to the MCTS simulation phase. + +![](images/0f32e8e7d96b3ef5abada570cc03b00c86b665715bdd0a9076b3f8cd01758eb2.jpg) +Figure E.4: Discovered governing equations of the mounted double pendulum system on a different dataset in different time sections: in 2-8 seconds the two masses are in chaotic motions while in 30-36 seconds the masses tend to move periodically due to accumulative damping. $\ddot { \theta } _ { 1 }$ and ${ \ddot { \theta } } _ { 2 }$ are obtained through smoothed numerical differentiation and predicted from the discovered governing equations. + +Table E.3: Average training time (in seconds) of SPL and NGGP in the Nguyen’s benchmark problems + +
BenchmarkSPL [s]NGGP[s]
Nguyen-18.7762.734
Nguyen-27.2963.296
Nguyen-381.2873.945
Nguyen-4567.0615.764
Nguyen-5431.22877.627
Nguyen-664.651104.588
Nguyen-714.9953.024
Nguyen-85.592.896
Nguyen-95.74313.229
Nguyen-1053.24586.497
Nguyen-1110.16344.399
Nguyen-12187.9334.757
Nguyen-1c452.734362.075
Nguyen-2c295.7691188.215
Nguyen-5c2178.8911365.777
Nguyen-8c77.892129.349
Nguyen-9c2001.4023066.41
+ +2. Graph modularity underexamined. The current design of the SPL training scheme does not leverage the full graph modularity: modules are reached by transforming a complete parse tree into a grammar. However, in some cases, there might be some influential modules appearing frequently as part of the tree. This type of graph modularity is described in the AIFeynman method (Udrescu et al., 2020). Deploying a more comprehensive graph modularity into the SPL machine will boost its efficacy in the complex equation and nonlinear dynamics discovery tasks. + +3. Robustness against extreme data noise and scarcity. Although it is observed that this reinforcement learning-based method is able to unearth the parsimonious solution to the governing equations from synthetic or measurement data with a moderate level of noise and scarcity. It is not effective when the data condition is extreme, or if there are missing values that make it challenging to numerically calculate the state derivatives. It is reasonable to investigate the integration between the SPL framework with a differentiable surrogate model built upon neural networks (Long et al., 2018; Chen et al., 2021) or spline learning (Sun et al., 2021) for further robustness in nonlinear dynamics discovery tasks. \ No newline at end of file diff --git a/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z_content_list.json b/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..417c4425225f2b1123deb73808d34b163bdd15f0 --- /dev/null +++ b/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z_content_list.json @@ -0,0 +1,2177 @@ +[ + { + "type": "text", + "text": "SYMBOLIC PHYSICS LEARNER: DISCOVERING GOVERNING EQUATIONS VIA MONTE CARLO TREE SEARCH ", + "text_level": 1, + "bbox": [ + 176, + 98, + 826, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Fangzheng $\\mathbf { S u n ^ { 1 } }$ , Yang $\\mathbf { L i u ^ { 2 } }$ , Jian-Xun Wang3, Hao Sun4,∗ ", + "bbox": [ + 183, + 171, + 589, + 188 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1Northeastern University, Boston, MA, USA; 2University of Chinese Academy of Sciences, Beijing, China; 3University of Notre Dame, Notre Dame, IN, USA; 4Renmin University of China, Beijing, China. Emails: sun.fa@northeastern.edu; liuyang22@ucas.ac.cn; jwang33@nd.edu; haosun@ruc.edu.cn ", + "bbox": [ + 184, + 189, + 813, + 234 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 263, + 544, + 279 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nonlinear dynamics is ubiquitous in nature and commonly seen in various science and engineering disciplines. Distilling analytical expressions that govern nonlinear dynamics from limited data remains vital but challenging. To tackle this fundamental issue, we propose a novel Symbolic Physics Learner (SPL) machine to discover the mathematical structure of nonlinear dynamics. The key concept is to interpret mathematical operations and system state variables by computational rules and symbols, establish symbolic reasoning of mathematical formulas via expression trees, and employ a Monte Carlo tree search (MCTS) agent to explore optimal expression trees based on measurement data. The MCTS agent obtains an optimistic selection policy through the traversal of expression trees, featuring the one that maps to the arithmetic expression of underlying physics. Salient features of the proposed framework include search flexibility and enforcement of parsimony for discovered equations. The efficacy and superiority of the SPL machine are demonstrated by numerical examples, compared with state-of-the-art baselines. ", + "bbox": [ + 233, + 292, + 766, + 486 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 511, + 336, + 526 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We usually learn the behavior of a nonlinear dynamical system through its nonlinear governing differential equations. These equations can be formulated as $\\dot { \\mathbf { y } } ( t ) = d \\mathbf { y } / d t = \\mathcal { F } ( \\mathbf { y } ( t ) )$ , where $\\mathbf { y } ( t ) ~ = ~ \\{ y _ { 1 } ( \\hat { t } ) , y _ { 2 } ( t ) , . . . , y _ { n } ( t ) \\} ^ { \\hat { } } \\in ~ \\mathbb { R } ^ { 1 \\times n _ { s } }$ denotes the system state at time $t$ , $\\mathcal F ( \\cdot )$ a nonlinear function set defining the state motions and $n _ { s }$ the system dimension. The explicit form of $\\mathcal F ( \\cdot )$ for some nonlinear dynamics remains underexplored. For example, in a mounted double pendulum system, the mathematical description of the underlying physics might be unclear due to unknown viscous and frictional damping forms. These uncertainties yield critical demands for the discovery of nonlinear dynamics given observational data. Nevertheless, distilling the analytical form of governing equations from limited noisy data, commonly seen in practice, is an intractable challenge. ", + "bbox": [ + 174, + 541, + 825, + 666 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ever since the early work on the data-driven discovery of nonlinear dynamics (Džeroski & Todorovski, 1993; Dzeroski & Todorovski, 1995), many scientists have stepped into this field of study. During the recent decade, the escalating advances in machine learning, data science, and computing power have enabled several milestone efforts of unearthing the governing equations for nonlinear dynamical systems. Notably, a breakthrough model named SINDy (Sparse Identification of Nonlinear Dynamics) (Brunton et al., 2016) has shed light on tackling this achallenge. SINDy was invented to determine the sparse solution among a pre-defined basis function library recursively through a sequential threshold ridge regression (STRidge) algorithm. SINDy quickly became one of the state-of-art methods and kindled significant enthusiasm in this field of study (Rudy et al., 2017; Long et al., 2018; Champion et al., 2019; Chen et al., 2021; Sun et al., 2021; Rao et al., 2022). However, the success of this sparsity-promoting approach relies on a properly defined candidate function library that requires good prior knowledge of the system. It is also restricted by the fact that a linear combination of candidate functions might be insufficient to recover complicated mathematical expressions. Moreover, when the library size is massive, it empirically fails to hold the sparsity constraint. ", + "bbox": [ + 174, + 674, + 825, + 867 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "At the same time, attempts have been made to tackle the nonlinear dynamics discovery problems by introducing neural networks with activation functions replaced by commonly seen mathematical operators (Martius & Lampert, 2017; Sahoo et al., 2018; Kim et al., 2019; Long et al., 2019). The intricate formulas are obtained via symbolic expansion of the well-trained network. This interpretation of physical laws results in larger candidate pools compared with the library-based representation of physics employed by SINDy. Nevertheless, since the sparsity of discovered expressions is primarily achieved by empirical pruning of the network weights, this framework exhibits sensitivity to userdefined thresholds and may fall short to produce parsimonious equations for noisy and scarce data. ", + "bbox": [ + 176, + 875, + 823, + 904 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Alternatively, another inspiring work (Bongard & Lipson, 2007; Schmidt & Lipson, 2009) reenvisioned the data-driven nonlinear dynamics discovery tasks by casting them into symbolic regression problems which have been profoundly resolved by the genetic programming (GP) approach (Koza & Koza, 1992; Billard & Diday, 2003). Under this framework, a symbolic regressor is established to identify the governing equations that best describe the underlying physics through free combination of mathematical operators and symbols, leading to great flexibility in model selection. One essential weakness of this early methodology is that, driven exclusively by the goal of empirically seeking the best-fitting expression (e.g. minimizing the mean-square error) in a genetic expansion process, the GP-based model usually over-fits the target system with numerous false-positive terms under data noise, even sometimes at a subtle level, causing huge instability and uncertainty. However, this ingenious idea has inspired a series of subsequent endeavors (Cornforth & Lipson, 2012; Gaucel et al., 2014; Ly & Lipson, 2012; Quade et al., 2016; Vaddireddy et al., 2020). In a more recent work, Deep Symbolic Regression (DSR) (Petersen et al., 2021; Mundhenk et al., 2021), a reinforcement learning-based model was established and generally outperformed the GP based models including the commercial Eureqa software (Langdon & Gustafson, 2010). Additionally, the AI-Feynman methods (Udrescu & Tegmark, 2020; Udrescu et al., 2020; Udrescu & Tegmark, 2021) ameliorated symbolic regression for distilling physics laws from data by combining neural network fitting with a suite of physics-inspired techniques. This approach is also highlighted by a recursive decomposition of a complicated mathematical expression into different parts on a tree-based graph, which disentangles the original problem and speeds up the discovery. It outperformed Eureqa in the uncovering Feynman physics equations (Feynman et al., 1965). However, this approach is built upon ad-hoc steps and, to some extent, lacks flexible automation in equation discovery. ", + "bbox": [ + 173, + 193, + 826, + 500 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The popularity of adopting the tree-based symbolic reasoning of mathematical formulas (Lample & Charton, 2019) has been rising recently to discover unknown mathematical expressions with a reinforcement learning agent (Kubalík et al., 2019; Petersen et al., 2021; Mundhenk et al., 2021). However, some former work attempting to apply the Monte Carlo tree search (MCTS) algorithm as an alternative to GP for symbolic regression (Cazenave, 2013; White et al., 2015; Islam et al., 2018; Lu et al., 2021) failed to leverage the full flexibility of this algorithm, resulting in the similar shortage that GP-based symbolic regressors possess as discussed earlier. Despite these outcomes, we are conscious of the strengths of the MCTS algorithm in equation discovery: it enables the flexible representation of search space with customized computational grammars to guide the search tree expansion. A sound mathematical underpinning for the trade-off between exploration and exploitation is remarkably advantageous as well. These features make it possible to inform the MCTS agent by our prior physics knowledge in nonlinear dynamics discovery rather than randomly searching in large spaces. ", + "bbox": [ + 174, + 506, + 825, + 672 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contribution. We propose a promising model named Symbolic Physics Learner (SPL) machine, empowered by MCTS, for discovery of nonlinear dynamics. This architecture relies on a grammar composed of (i) computational rules and symbols to guide the search tree spanning and (ii) a composite objective rewarding function to simultaneously evaluate the generated equations with observational data and enforce the sparsity of the expression. Moreover, we design multiple adjustments to the conventional MCTS by: (1) replacing the expected reward in UCT score with maximum reward to better fit the equation discovery objective, (2) employing an adaptive scaling in policy evaluation which would eliminate the uncertainty of the reward value range owing to the unknown error of the system state derivatives, and (3) transplanting modules with high returns to the subsequent search as a single leaf node. With these adjustments, the SPL machine is capable of efficiently uncovering the best path to formulate the complex governing equations of the target dynamical system. ", + "bbox": [ + 174, + 684, + 825, + 837 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 861, + 326, + 877 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we expand and explain the background concepts brought up in the introduction to the SPL architecture, including the expression tree (parse tree) and the MCTS algorithm. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Expression tree. Any mathematical expression can be represented by a combinatorial set of symbols and mathematical operations, and further expressed by a parse tree structure (Hopcroft et al., 2006; Kusner et al., 2017) empowered by a context-free grammar (CFG). A CFG is a formal grammar characterized by a tuple comprised of 4 elements, namely, $\\mathcal { G } = ( V , \\Sigma , R , S )$ , where $V$ denotes a finite set of non-terminal nodes, $\\Sigma$ a finite set of terminal nodes, $R$ a finite set of production rules, each interpreted as a mapping from a single non-terminal symbol in $V$ to one or multiple terminal/non-terminal node(s) in $( V \\cup \\Sigma ) ^ { * }$ where $^ *$ represents the Kleene star operation, and $S$ a single non-terminal node standing for a start symbol. In our work, equations are symbolized into parse trees: we define the start node as equation symbol $f$ , terminal symbols (leaf nodes) corresponding to the independent variables formulating the equation (e.g, $x , y )$ , and a placeholder symbol $C$ for identifiable constant coefficients that stick to specific production rules. The non-terminal nodes between root and leaf nodes are represented by some symbols distinct from the start and terminal nodes (i.e., $M _ { ☉ }$ ). The production rules denote the commonly seen mathematical operators: unary rules (one non-terminal node mapping to one node) for operators like $\\cos ( \\cdot ) , \\exp ( \\cdot ) , \\bar { \\log ( | \\cdot | ) }$ , and binary rules (one non-terminal node mapping to two nodes) for operators such $\\mathrm { ~ \\imath s ~ } + , - , \\times , \\dot { \\mathrm { ~ \\cdot ~ } }$ . A parse tree is then generated via a pre-order traversal of production rules rooted at $f$ and terminates when all leaf nodes are entirely filled with terminal symbols. Each mathematical expression can be represented by such a traversal set of production rules. ", + "bbox": [ + 174, + 104, + 825, + 353 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Monte Carlo tree search. Monte Carlo tree search (MCTS) (Coulom, 2006) is an algorithm for searching optimal decisions in large combinatorial spaces represented by search trees. This technique complies with the best-first search principle based on the evaluations of stochastic simulations. It has already been widely employed in and proved the spectacular success by various gaming artificial intelligence systems, including the famous AlphaGo and AlphaZero (Silver et al., 2017) for computer Go game. A basic MCTS algorithm is composed of an iterative process with four steps: ", + "bbox": [ + 174, + 359, + 825, + 444 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. Selection. The MCTS agent, starting from the root node, moves through the visited nodes of the search tree and selects the next node according to a given selection policy until it reaches an expandable node or a leaf node. \n2. Expansion. At an expandable node, the MCTS agent expands the search tree by selecting one of its unvisited children. \n3. Simulation. After expansion, if the current node is non-terminal, the agent performs one or multiple independent simulations starting from the current node until reaching the terminal state. In this process, actions are randomly selected. \n4. Backpropagation. Statistics of nodes along the path from the current node to the root are updated with respect to search results (scores evaluated from the terminate states reached). ", + "bbox": [ + 210, + 455, + 825, + 612 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To maintain a proper balance between the less-tested paths and the best policy identified so far, the MCTS agent sticks to a trade-off between exploration and exploitation by taking action that maximizes the Upper Confidence Bounds applied for Trees (UCT), formulated as (Kocsis & Szepesvári, 2006): ", + "bbox": [ + 176, + 625, + 825, + 666 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b873fd134844f8973176b8e8dde3cf6d2e7009e599d08f7ae19b941e6903ae79.jpg", + "text": "$$\nU C T ( s , a ) = Q ( s , a ) + c \\sqrt { \\ln [ N ( s ) ] / N ( s , a ) }\n$$", + "text_format": "latex", + "bbox": [ + 344, + 674, + 655, + 694 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $Q ( s , a )$ is the average result/reward of playing action $a$ in state $s$ in the simulations performed in the history, encouraging the exploitation of current best child node; $N ( s )$ is number of times state $s$ visited, $N ( s , a )$ the number of times action $a$ has been selected at state $s$ , and $\\sqrt { \\ln [ N ( s ) ] / N ( s , a ) }$ consequently encourages exploration of less-visited child nodes. Constant $c$ controls the balance between exploration and exploitation, empirically defined upon the specific problem. Theoretical analysis of UCT-based MCTS (e.g., convergence, guarantees) is referred to Shah et al. (2019). ", + "bbox": [ + 174, + 700, + 825, + 787 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHODS ", + "text_level": 1, + "bbox": [ + 174, + 808, + 290, + 824 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Existing studies show that the MCTS agent continuously gains knowledge of specified tasks via the expansion of the search tree and, based on the backpropagation of evaluation results (i.e., rewards and number of visits), render a proper selection policy on visited states to guide the upcoming searching (Silver et al., 2017). In the proposed SPL machine, such a process is integrated with the symbolic reasoning of mathematical expressions to reproduce and evaluate valid mathematical expressions of the physical laws in nonlinear dynamics step-by-step, and then obtain a favorable selection policy pointing to the best solution. This algorithm is depicted in Figure 1 with an illustrative example and its overall training scheme is shown in Algorithm 1. Discussion of the hyperparameter setting for this algorithm is given in Appendix Section A. ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/a42dd81e12d9c668a474f4a45fa9cef2e188d64eac5f65dd6b946b67828ae5ab.jpg", + "image_caption": [ + "Figure 1: Schematic architecture of the SPL machine for nonlinear dynamics discovery. The graph explains the 4 MCTS phases of one learning episode with an illustrative example. " + ], + "image_footnote": [], + "bbox": [ + 217, + 99, + 782, + 393 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 463, + 825, + 506 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Rewarding. To evaluate the mathematical expression $\\tilde { f }$ projected from a parse tree, we define a numerical reward $r \\in \\mathcal { R } \\subset \\mathbb { R }$ based on this expression and input data ${ \\mathcal { D } } = \\{ { \\bf Y } ; \\dot { Y } _ { i } \\}$ , serving as the search result of the current expansion or simulation. It is formulated as ", + "bbox": [ + 174, + 512, + 825, + 558 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2a6cd658db7281014801e4731610a9b24e515be52d4164fb32811ac7cccf8cb9.jpg", + "text": "$$\nr = \\frac { \\eta ^ { n } } { 1 + \\sqrt { \\frac { 1 } { N } \\left. \\dot { Y } _ { i } - \\tilde { f } ( \\mathbf { Y } ) \\right. _ { 2 } ^ { 2 } } }\n$$", + "text_format": "latex", + "bbox": [ + 400, + 560, + 598, + 604 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbf { Y } = \\{ \\mathbf { y } _ { 1 } , \\mathbf { y } _ { 2 } , . . . , \\mathbf { y } _ { m } \\} \\in \\mathbb { R } ^ { m \\times N }$ is the $m$ dimensional state variables of a dynamical system, $\\dot { Y } _ { i } \\in \\mathbb { R } ^ { 1 \\times N }$ the numerically estimated state derivative for ith dimension, and $N$ the number of measurement data points. $\\eta$ denotes a discount factor, assigned slightly smaller than 1; $n$ is empirically defined as the total number of production rules in the parse tree. This numerator arrangement is designated to penalize non-parsimonious solutions. This rewarding formulation outputs a reasonable assessment to the distilled equations and encourages parsimonious solution by discounting the reward of a non-parsimonious one. The rooted mean square error (RMSE) in denominator evaluates the goodness-of-fit of the discovered equation w.r.t. the measurement data. ", + "bbox": [ + 173, + 608, + 825, + 723 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Training scheme. A grammar $\\mathcal { G } = ( V , \\Sigma , R , S )$ is defined with appropriate nodes and production rules to cover all possible forms of equations. To keep track of non-terminal nodes of the parsing tree, we apply a last-in-first-out (LIFO) strategy and denote the non-terminal node placed last on the stack $N T$ as the current node. We define the action space $A = R$ and the state space $s$ as all possible traversals of complete/incomplete parse trees (i.e., production rules selected) in ordered sequences. At the current state $s _ { t } = [ a _ { 1 } , a _ { 2 } , . . . a _ { t } ]$ where $t \\in \\mathbb N$ is the discrete traversal step-index of the upcoming production rule, the MCTS agent masks out the invalid production rules for current non-terminal node and on that basis selects a valid rule as action $a _ { t + 1 }$ (i.e, the left-hand side of a valid production rule is the current non-terminal symbol). Consequently, the parse tree gains a new terminal/non-terminal branch in accordance with $a _ { t + 1 }$ , meanwhile the agent finds itself in a new state $s _ { t + 1 } = [ a _ { 1 } , a _ { 2 } , . . . a _ { t } , a _ { t + 1 } ]$ . The agent subsequently pops off the current non-terminal symbol from $N T$ and pushes the non-terminal nodes, if there are any, on the right-hand side of the selected rule onto the stack. Once the agent attains an unvisited node, a certain amount of simulations are performed, where the agent starts to randomly select the next node until the parse tree is completed. ", + "bbox": [ + 173, + 728, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1 Input: Grammar $G = ( V , \\Sigma , R , S )$ , measurement data ${ \\mathcal { D } } = \\{ \\mathbf { Y } ; { \\dot { Y } } _ { i } \\}$ ; \n2 Parameters: discount/regularization factor $\\eta$ , exploration rate $c$ , $t _ { m a x }$ ; # $\\eta$ controls equation parsimony; \n3 Output: Optimal governing equation $\\tilde { f } ^ { \\star }$ ; \n4 for each episode do \n5 Selection: Initialize $s _ { 0 } = \\emptyset , t = 0 , N T = [ S ]$ ; \n6 while $s _ { t }$ expandable and $t < t _ { m a x }$ do \n7 Choose $a _ { t + 1 } = \\arg \\operatorname* { m a x } _ { \\mathcal { A } } U C T ( s _ { t } , a )$ ; \n8 Take action $a _ { t + 1 }$ , observe $s ^ { \\prime } , N T$ ; \n9 $s _ { t + 1 } \\gets s ^ { \\prime }$ note as visited, $t \\gets t + 1$ ; \n10 end \n11 Expansion: Randomly take an unvisited path with action $a$ , observe $s ^ { \\prime } , N T$ ; \n12 $s _ { t + 1 } \\gets s ^ { \\prime }$ note as visited, $t \\gets t + 1$ ; \n13 if $N T = \\emptyset$ then \n14 Project $\\tilde { f }$ , Backpropagate $r _ { t + 1 }$ and visited count and finish the episode; \n15 end \n16 Simulation: Fix the starting point $s _ { t } , N T$ ; \n17 for each simulation do \n18 while $s _ { t }$ non-terminal and $t < t _ { m a x }$ do \n19 Randomly take an action $a$ , observe $s ^ { \\prime } , N T$ ; \n20 $s _ { t + 1 } \\gets s ^ { \\prime } , t \\gets t + 1$ ; \n21 end \n22 if $N T = \\emptyset$ then \n23 Project $\\tilde { f }$ and calculate $r _ { t + 1 }$ ; \n24 end \n25 end \n26 Backpropagate simulation results; \n27 end ", + "bbox": [ + 155, + 127, + 818, + 492 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The reward is calculated or the maximal size is exceeded, resulting in a zero reward. The best result from the attempts counts as the reward of the current simulation phase and backpropagates from the current unvisited node all the way to the root node. ", + "bbox": [ + 173, + 523, + 825, + 565 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Greedy search. Different from the MCTS-based gaming AIs where the agents are inclined to pick the action with a high expected reward (average returns), the SPL machine seeks the unique optimal solution. In the proposed training framework, we apply a greedy search heuristic to encourage the agent to explore the branch which yields the best solution in the past: $Q ( s , a )$ is defined as the maximum reward of the state-action pair, and its value is backpropagated from the highest reward in the simulations upon the selection of the pair. Meanwhile, to overcome the local minima problems due to this greedy approach in policy search, we enforce a certain level of randomness by empirically adopting the $\\epsilon$ -greedy algorithm, a commonly seen approach in reinforcement learning models. ", + "bbox": [ + 173, + 571, + 825, + 685 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Adaptive-scaled rewarding. Owing to the unknown level of error from the numerically estimated state derivatives, the range of the RMSE in the SPL reward function is unpredictable. This uncertainty affects the scale of rewarding values thus the balance between exploration and exploitation is presented in Eq. (1). Besides adding “1” to the denominator of Eq. (2) to avoid dramatically large numerical rewards, we also apply an adaptive scale of the reward, given by ", + "bbox": [ + 173, + 690, + 825, + 761 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d8935421f0b8f51731a9f8f664f181390e861adbb61ba90a95b4821d4ad5a341.jpg", + "text": "$$\nQ ( s , a ) = { \\frac { r ^ { * } ( s , a ) } { \\operatorname* { m a x } _ { s ^ { \\prime } \\in S , a ^ { \\prime } \\in A } Q ( s ^ { \\prime } , a ^ { \\prime } ) } }\n$$", + "text_format": "latex", + "bbox": [ + 382, + 768, + 616, + 803 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $r ^ { * }$ denotes the maximum reward of the state-action pair. It is scaled by the current maximum reward among all states $s$ and actions $\\mathcal { A }$ to reach an equilibrium that the $Q$ -values are stretched to the scale $[ 0 , 1 ]$ at any time. This self-adaptive fashion yields a well-scaled calculation of UCT under different value ranges of rewards throughout the training. ", + "bbox": [ + 174, + 809, + 825, + 866 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Module transplantation. A function can be decomposed into smaller modules where each is simpler than the original one (Udrescu et al., 2020). This modularity feature, as shown in Figure 2, helps us develop a divide-and-conquer heuristic for distilling some complicated functions: after every certain amount of MCTS iterations, the discovered parse trees with high rewards are picked out and thereupon ", + "bbox": [ + 173, + 872, + 825, + 929 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "reckoned as individual production rules and appended to the set of production rules $R$ ; accordingly, these trees are “transplanted” to the future ones as their modules (i.e, the leaves). To avoid early overfitting problem, we incrementally enlarge the sizes of such modules from a baseline length ", + "bbox": [ + 173, + 104, + 356, + 256 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/5a25f743cfc4fd3f1a08cb871a3cdb3f18582c0ca7d6fe587e43f81249f2f211.jpg", + "image_caption": [ + "Figure 2: A module transplantation process: A complete parse serves as a single production rule and is appended to the grammar pool. " + ], + "image_footnote": [], + "bbox": [ + 369, + 104, + 820, + 215 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "to the maximum allowed size of the parse tree throughout the iterations. The augmentation part of $R$ is refreshed whenever new production rules are created, keeping only the ones engendering high rewards. This approach accelerates the policy search by capturing and locking some modules that likely contribute to, or appear as part of the optimal solution, especially in the cases of the mathematical expression containing “deep” operations (e.g., high-order polynomials) whose structures are difficult for the MCTS agent to repeatedly obtain during the selection and expansion. ", + "bbox": [ + 174, + 256, + 825, + 339 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 SYMBOLIC REGRESSION: FINDING MATHEMATICAL FORMULAS ", + "text_level": 1, + "bbox": [ + 174, + 362, + 738, + 377 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 DATA NOISE & SCARCITY ", + "text_level": 1, + "bbox": [ + 176, + 393, + 397, + 407 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Data scarcity and noise are commonly seen in measurement data and become one of the bottleneck issues for discovering the governing equations of nonlinear dynamics. Tackling the challenges in high-level data scarcity and noise situations is traditionally regarded as an essential robustness indicator for a nonlinear dynamics discovery model. To this end, we present an examination of the proposed SPL machine by an equation discovery task in the presence of multiple levels of data noise and ", + "bbox": [ + 176, + 421, + 472, + 573 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/47b9fbce3bf68ac13532690055be70536c76e905d4ef341af2b871f31de9beea.jpg", + "image_caption": [ + "Figure 3: The effect of data noise/scarcity on recovery rate. The heatmaps demonstrate the recovery rate of GP and the SPL machine under different data conditions, summarized over 100 independent trials. " + ], + "image_footnote": [], + "bbox": [ + 485, + 386, + 818, + 508 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "volume, comparing with a GP-based symbolic regressor (implemented with gplearn python package)1. The target equation is $f ( x ) = 0 . 3 x ^ { 3 } + 0 . 5 x ^ { 2 } + 2 x$ , and the independent variable $X$ is uniformly sampled in the given range $[ - 1 0 , 1 0 ]$ . Gaussian white noise is added to the dependent variable $Y$ with the noise level defined as the root-mean-square ratio between the noise and the exact values. For discovery, the two models are fed with equivalent search space: $\\{ + , - , \\times , \\div , c o s t , x \\}$ as candidate mathematical operations and symbols. The hyperparameters of the SPL machine are set as $\\eta = 0 . 9 9$ , $t _ { m a x } = 5 0$ , and 10,000 episodes of training is regarded as one trail. For the GP-based symbolic regressor, the population of programs is set as 2,000, the number of generations as 20. The range of constant coefficient values is $[ - 1 0 , 1 0 ]$ . For 16 different data noise and scarcity levels, each model was performed 100 independent trails. The recovery rates are displayed as a $4 \\times 4$ mesh grid w.r.t. different noise/scarcity levels in Figure 3. It is observed that the SPL machine outperforms the GP-based symbolic regressor in all the cases. A T-test also proves that the recovery rate of the SPL machine is significantly higher than that of GP (e.g., $p$ -value $= 1 . 0 6 \\times 1 0 ^ { - 7 }$ ). ", + "bbox": [ + 173, + 574, + 826, + 753 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 NGUYEN’S SYMBOLIC REGRESSION BENCHMARK ", + "text_level": 1, + "bbox": [ + 174, + 772, + 562, + 786 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Nguyen’s symbolic regression benchmark task (Uy et al., 2011) is widely used to test the model’s robustness in symbolic regression problems. Given a set of allowed operators, a target equation, and data generated by the specified equation (see Table 1 for example), the tested model is supposed to distill the mathematical expression that is identical to the target equation, or equivalent to it (e.g., Nguyen-7 equation can be recovered as $\\log ( x ^ { 3 } + x ^ { 2 } + x + 1 )$ , Nguyen-10 equation can be recovered as $\\sin ( x + y )$ , and Nguyen-11 equation can be recovered as $\\exp ( y \\log ( x ) ) )$ . Some variants of Nguyen’s benchmark equations are also considered in this experiment. Their discoveries require numerical estimation of the constant coefficient values. Each equation generates two datasets: one for training and another for testing. The discovered equation that perfectly fits the testing data is regarded as a successful discovery (i.e., the discovered equation should be identical or equivalent to the target one). The recovery rate is calculated based on 100 independent tests for each task. In these benchmark tasks, three algorithms are tested: GP-based symbolic regressor, the neural-guided GP (NGGP) (Mundhenk et al., 2021), and the SPL machine. Note that NGGP is an improved approach over DSR (Petersen et al., 2021). They are given the same set of candidate operations: √ $\\{ + , - , \\times , \\div , \\exp ( \\cdot ) , \\cos ( \\cdot ) , \\sin ( \\cdot ) \\}$ for all benchmarks and $\\{ { \\sqrt { \\cdot } } , \\ln ( \\cdot ) \\}$ are added to the 7, 8, 11 benchmarks. The hyperparameters of the GP-based symbolic regressor are the same as those mentioned in Section 4.1; configurations of the NGGP models are obtained from its source code2; detailed setting of the benchmark tasks and the SPL model is described in Appendix Section B. The success rates are shown in Table 1. It is observed that the SPL machine and the NGGP model both produce reliable results in Nguyen’s benchmark problems and the SPL machine slightly outperforms NGGP. This experiment betokens the capacity of the SPL machine in discovery of equations with divergent forms. ", + "bbox": [ + 174, + 799, + 825, + 897 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/e0759a006d1825e8846f66cd9b75f7bdde5b0af2da31d83d89bea998f47920b9.jpg", + "table_caption": [ + "Table 1: Recovery rate of three algorithms in Nguyen’s benchmark symbolic regression problems. The SPL machine outperforms the other two models in average recovery rate. " + ], + "table_footnote": [], + "table_body": "
BenchmarkExpressionSPLNGGPGP
Nguyen-1x²+x²+x100%100%99%
Nguyen-2x²+x²+x²+x100%100%90%
Nguyen-3+x4+x²+x²+x 25100%100%34%
Nguyen-4x+x+x4+x²+x²+x99%100%54%
Nguyen-5sin(x²) cos(𝑥) -195%80%12%
Nguyen-6sin(x²)+sin(x+x²)100%100%11%
Nguyen-7ln(𝑥 +1) + ln(x²+ 1)100%100%17%
Nguyen-8√x100%100%100%
Nguyen-9sin(x) + sin(y²)100%100%76%
Nguyen-102 sin(x) cos(y)100%100%86%
Nguyen-11xy100%100%13%
Nguyen-12x4-x²+¹y²-y28%4%0%
Nguyen-1c3.39x +2.12x² +1.78x100%100%0%
Nguyen-2c0.48x4 +3.39x+2.12x² +1.78x94%100%0%
Nguyen-5csin(x²) cos(x) -0.7595%98%1%
Nguyen-8c√1.23x100%100%56%
Nguyen-9csin(1.5x) + sin(0.5y²)96%90%0%
Average94.5%92.4%38.2%
", + "bbox": [ + 240, + 147, + 754, + 429 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 448, + 825, + 643 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Ablation Study: We consider four ablation studies by removing: (a) the adaptive scaling in reward calculation, (b) the discount factor $\\eta ^ { n }$ that drives equation parsimony in Eq. (2), (c) module transplantation in tree generation, and (d) all of the above. The four models were tested on the first 12 Nguyen equations (see Appendix Section C). Results show the average recovery rates for these models are all smaller than that produced by SPL (see Appendix Table C.1), where the module transplantation brings the largest effect. Hence, these modules are critical to guarantee the proposed model efficacy. ", + "bbox": [ + 174, + 650, + 825, + 733 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 PHYSICAL LAW DISCOVERY: FREE FALLING BALLS WITH AIR RESISTANCE ", + "bbox": [ + 173, + 768, + 820, + 786 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "It is well known that, in 1589–1592, Galileo dropped two objects of unequal mass from the Leaning Tower of Pisa and drew a conclusion that their velocities were not affected by the mass. This has been well recognized globally as the “textbook” physical law for the vertical motion of a free-falling object: ", + "bbox": [ + 174, + 801, + 465, + 898 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/490486d65f41be7269fa6dd26d1e4f6d524cf70f185edfa282ea8b1ee14afd94.jpg", + "table_caption": [ + "Table 2: Baseline models ( $\\dot { c } _ { i }$ : unknown constants). " + ], + "table_footnote": [], + "table_body": "
Physics ModelDerived model expression
Model-1H(t)= co+cit + c2t²+c3t3
Model-2H(t)= Co +Cit + C2eCt
Model-3H(t)= co + c1 log(cosh(c2t))
", + "bbox": [ + 488, + 821, + 812, + 891 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "the height of the object is formulated as $\\begin{array} { r } { H ( t ) = h _ { 0 } + v _ { 0 } t - \\frac { 1 } { 2 } g t ^ { 2 } } \\end{array}$ , where $h _ { 0 }$ denotes initial height, $v _ { 0 }$ the initial velocity, and $g$ the gravitational acceleration. However, this ideal situation is rarely reached in our daily life because air resistance serves as a significant damping factor that prevents the above physical law from occurring in real-life cases. ", + "bbox": [ + 174, + 102, + 825, + 159 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Many efforts have been made to uncover the effect of the air resistance and derive mathematical models to describe the free-falling objects with air resistance (Clancy, 1975; Lindemuth, 1971; Greenwood et al., 1986). This section provides data-driven discovery of the physical laws of relationships between height and time in the cases of free-falling objects with air resistance based on multiple experimental ball-drop datasets (de Silva et al., 2020), which contain the records of 11 different types of balls dropped from a ", + "bbox": [ + 173, + 166, + 400, + 388 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/761b3c943ce74c563a97c5018151ec457e6d985ca4238773b55ddb0c5adc14bb.jpg", + "table_caption": [ + "Table 3: Mean square error (MSE) between ball motion prediction with the measurements in the test set. The SPL machine reaches the best prediction results in most (9 out of 11) cases. " + ], + "table_footnote": [], + "table_body": "
TypeSPLModel-1Model-2Model-3
baseball0.32.79894.5893.507
blue basketball0.4570.51369.2092.227
green basketball0.0880.185.4351.604
volleyball0.1110.57480.9650.76
bowling ball0.0030.3387.023.167
golf ball0.0090.21486.0931.684
tennis ball0.0910.24672.2780.161
whiffle ball 11.581.61965.4260.21
whiffle ball 20.0990.62858.5330.966
yellow whiffle ball0.42817.34144.9842.57
orange whiffle ball0.7450.37936.7653.257
", + "bbox": [ + 416, + 215, + 818, + 383 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "bridge (see Appendix Figure D.1). For discovery, each dataset is split into a training set (records from the first 2 seconds) and a testing set (records after 2 seconds). Three mathematically derived physics models are selected from the literature as baseline models3,4,5 for this experiment (see Table 2), and the unknown constant coefficient values are estimated by Powell’s conjugate direction method (Powell, 1964). Based on our prior knowledge of the physical law that may appear in this case, we use $\\{ + , - , \\times , \\div , \\exp ( \\cdot ) , \\cosh ( \\cdot ) , \\log ( \\cdot ) \\}$ as the candidate grammars for the SPL discovery, with terminal nodes $\\{ t , c o n s t \\}$ . The hyperparameters are set as $\\eta = 0 . 9 9 9 9$ , $t _ { m a x } = 2 0$ , and one single discovery is built upon 6,000 episodes of training. The physical laws distilled by SPL from training data are applied to the test data and compared with the ground truth. Their prediction errors, in terms of MSE, are presented in Table 3 (the SPL-discovered equations are shown in Appendix Table D.1). The full results can be found in Appendix Section D. It can be concluded that the data-driven discovery of physical laws leads to a better approximation of the free-falling objects with air resistance. ", + "bbox": [ + 173, + 388, + 825, + 554 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CHAOTIC DYNAMICS DISCOVERY: THE LORENZ SYSTEM", + "text_level": 1, + "bbox": [ + 176, + 568, + 679, + 584 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The first nonlinear dynamics discovery example is a 3-dimensional Lorenz system (Lorenz, 1963) whose dynamical behavior $( x , y , z )$ is governed by $\\dot { x } = \\sigma ( y -$ $x ) , \\dot { y } = x ( \\rho - z ) - y , \\dot { z } = x y - \\beta z$ with parameters $\\sigma = 1 0$ , $\\beta = 8 / 3$ , and $\\rho = 2 8$ . The Lorenz attractor has two lobes and the system, starting from anywhere, makes cycles around one lobe before switching to the other and iterates repeatedly, exhibiting strong chaos. The synthetic system states $( x , y , z )$ are generated by solving the nonlinear differential equations using the Matlab ode113 (Shampine, 1975; Shampine & Reichelt, 1997) function. $5 \\%$ Gaussian white noise is added to the clean data to generate noisy measurement. ", + "bbox": [ + 174, + 599, + 529, + 779 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/39ef080190b639c69c315bb191062a2b2d59be81ed334877eec499826590ef23.jpg", + "table_caption": [ + "Table 4: Summary of the discovered governing equations for Lonrez system. Each cell concludes if target physics terms are distilled (if yes, number of false positive terms in uncovered expression). " + ], + "table_footnote": [ + "The derivatives of the system states $( \\dot { x } , \\dot { y } , \\dot { z } )$ are unmeasured but estimated by central difference and smoothed by the Savitzky–Golay filter (Savitzky & Golay, 1964) in order to reduce the noise effect. " + ], + "table_body": "
Model宝y之
EureqaYes (1)Yes (3)Yes (1)
pySINDyYes (1)No (N/A)Yes (2)
NGGPYes (10)Yes (8)Yes (16)
SPLYes (0)Yes (0)Yes (0)
", + "bbox": [ + 540, + 678, + 834, + 777 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this experiment, the proposed SPL machine is compared with three benchmark methods: Eureqa, pySINDy and NGGP. For Eureqa, NGGP, and the SPL machine, $\\{ + , - , \\times , \\div \\}$ are used as candidate operations; the upper bound of complexity is set to be 50; for pySINDy, the candidate function library includes all polynomial basis of $( x , y , z )$ from degree 1 to degree 4. Appendix Table E.1 presents the distilled governing equations by each approach and Table 4 summarizes these results: the SPL machine uncovers the explicit form of equations accurately in the context of active terms, whereas Eureqa, pySINDy and NGGP yield several false-positive terms in the governing equations. In particular, although Eureqa and NGGP are capable of uncovering the correct terms, their performance is very sensitive to the measurement noise as indicated by the redundant terms (despite with small coefficients) shown in Appendix Table E.1. Overall, the baseline methods fail to handle the large noise effect, essentially limiting their applicability in nonlinear dynamics discovery. It is evident that the SPL machine is capable of distilling the concise symbolic combination of operators and variables to correctly formulate parsimonious mathematical expressions that govern the Lorenz system, outperforming the baseline methods of Eureqa, pySINDy and NGGP. ", + "bbox": [ + 174, + 815, + 825, + 871 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 243 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 EXPERIMENTAL DYNAMICS DISCOVERY: DOUBLE PENDULUM ", + "text_level": 1, + "bbox": [ + 174, + 261, + 723, + 279 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This section shows SPL-based discovery of a chaotic double pendulum system with experimental data (Asseman et al., 2018) as shown in Appendix Figure E.3. The governing equations are given by: ", + "bbox": [ + 171, + 291, + 825, + 320 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/4277b4b26af472757877edfbfe2b1ed201a2ca2fb690ffae1bb359f3358633ea.jpg", + "text": "$$\n\\begin{array} { r l } & { \\dot { \\omega } _ { 1 } = c _ { 1 } \\dot { \\omega } _ { 2 } \\cos ( \\Delta \\theta ) + c _ { 2 } \\omega _ { 2 } ^ { 2 } \\sin ( \\Delta \\theta ) + c _ { 3 } \\sin ( \\theta _ { 1 } ) + \\mathcal { R } _ { 1 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } ) , } \\\\ & { \\dot { \\omega } _ { 2 } = c _ { 1 } \\dot { \\omega } _ { 1 } \\cos ( \\Delta \\theta ) + c _ { 2 } \\omega _ { 1 } ^ { 2 } \\sin ( \\Delta \\theta ) + c _ { 3 } \\sin ( \\theta _ { 2 } ) + \\mathcal { R } _ { 2 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 263, + 321, + 730, + 363 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where $\\theta _ { 1 } , \\theta _ { 2 }$ denote the angular displacements; $\\omega _ { 1 } = \\dot { \\theta } _ { 1 }$ , $\\omega _ { 2 } ~ = ~ { \\dot { \\theta } } _ { 2 }$ the velocities; $\\dot { \\omega } _ { 1 } , \\dot { \\omega } _ { 2 }$ the accelerations; $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ denote the unknown damping forces. Note that $\\Delta \\theta = \\theta _ { 1 } - \\theta _ { 2 }$ . ", + "bbox": [ + 176, + 364, + 823, + 395 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The data source contains multiple camera-sensed datasets. Here, 5,000 denoised random sub-samples from 5 datasets are used for training, 2,000 random sub-samples from another 2 datasets for validation, and 1 dataset for testing. The derivatives of the system states are numerically estimated by the same approach discussed in the Lorenz case. Some prior physics knowledge is employed to guide the discovery: (1) the terms $\\dot { \\omega } _ { 2 } \\cos ( \\Delta \\theta )$ for $\\dot { \\omega } _ { 1 }$ and $\\dot { \\omega } _ { 1 } \\cos ( \\Delta \\theta )$ for $\\dot { \\omega } _ { 2 }$ are assumed to be part of the governing equations based on the Lagrange derivation; (2) the angles $( \\theta _ { 1 } , \\theta _ { 2 } , \\Delta \\theta )$ are under the trigonometric functions $\\cos ( \\cdot )$ and $\\sin ( \\cdot )$ ; (3) directions of velocities/relative velocities may appear in damping. Production rules fulfilling the above prior knowledge are exhibited in Appendix Section E.2. The hyperparameters are set as $\\eta = 1$ , $t _ { m a x } = 2 0$ , and 40,000 episodes of training are regarded as one trail. 5 independent trials are performed and the equations with the highest validation scores are selected as the final results. The uncovered equations are given as follows: ", + "bbox": [ + 173, + 400, + 825, + 554 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/d9367ad4b4dbfa6afdea2867b710bf63aee402b6d965aea4ca8a5966167e73a5.jpg", + "text": "$$\n\\dot { \\omega } _ { 1 } = - 0 . 0 9 9 1 \\dot { \\omega } _ { 2 } \\cos ( \\Delta \\theta ) - 0 . 1 0 3 \\omega _ { 2 } ^ { 2 } \\sin ( \\Delta \\theta ) - 6 9 . 2 7 4 \\sin ( \\theta _ { 1 } ) + 0 . 5 1 5 \\cos ( \\theta _ { 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 227, + 556, + 766, + 573 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/fe1921586e2af2fa27f9f0a3317709a0eeca10ce27b6d67a275851c0c2ec108c.jpg", + "text": "$$\n\\begin{array} { r } { \\dot { \\omega } _ { 2 } = - 1 . 3 6 8 \\dot { \\omega } _ { 1 } \\cos ( \\Delta \\theta ) + 1 . 3 6 3 \\omega _ { 1 } ^ { 2 } \\sin ( \\Delta \\theta ) - 9 2 . 9 1 3 \\sin ( \\theta _ { 2 } ) + 0 . 0 3 2 \\omega _ { 1 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 577, + 723, + 594 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where the explicit expression of physics in an ideal double pendulum system, as displayed in Eq. (5), are successfully distilled and damping terms are estimated. This set of equation is validated through interpolation on the testing set and compared with the smoothed derivatives, as shown in Appendix Figure E.4. The solution appears felicitous as the governing equations of the testing responses. ", + "bbox": [ + 174, + 597, + 825, + 652 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 CONCLUSION AND DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 671, + 470, + 688 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This paper introduces a Symbolic Physics Learner (SPL) machine to tackle the challenge of distilling the mathematical structure of equations for physical systems (e.g., nonlinear dynamics) with scarce/noisy data. This framework is built upon the expression tree interpretation of mathematical operations and variables and an MCTS agent that searches for the optimistic policy to reconstruct the target mathematical formula. With some remarkable adjustments to the MCTS algorithms, the SPL model straightforwardly accepts our prior or domain knowledge, or any sort of constraints of the tasks in the grammar design while leveraging great flexibility in expression formulation. The robustness of the proposed SPL machine for complex target expression discovery within a large search space is indicated in the Nguyen’s symbolic regression benchmark problems, where the SPL machine outperforms state-of-the-art symbolic regression methods. Moreover, encouraging results are obtained in the tasks of discovering physical laws and nonlinear dynamics, based on synthetic or experimental datasets. While the proposed SPL machine shows huge potential in both symbolic regression and physical law discovery tasks, there are still some imperfections that can be improved: (i) the computational cost is high for constant coefficient value estimation due to repeated calls for an optimization process, (ii) graph modularity is underexamined, and (iii) robustness against extreme data noise and scarcity is not optimal. These limitations are further explained in Appendix Section F. ", + "bbox": [ + 173, + 702, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 328, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The work is supported by the National Natural Science Foundation of China (No. 92270118) and the Beijing Outstanding Young Scientist Program (No. BJJWZYJH012019100020098). 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", + "bbox": [ + 174, + 507, + 825, + 551 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Nguyen Quang Uy, Nguyen Xuan Hoai, Michael O’Neill, Robert I McKay, and Edgar GalvánLópez. Semantically-based crossover in genetic programming: application to real-valued symbolic regression. Genetic Programming and Evolvable Machines, 12(2):91–119, 2011. ", + "bbox": [ + 173, + 559, + 823, + 603 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Harsha Vaddireddy, Adil Rasheed, Anne E Staples, and Omer San. Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data. Physics of Fluids, 32(1):015113, 2020. ", + "bbox": [ + 171, + 611, + 823, + 655 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "David R White, Shin Yoo, and Jeremy Singer. The programming game: evaluating mcts as an alternative to gp for symbolic regression. In Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation, pp. 1521–1522, 2015. ", + "bbox": [ + 171, + 662, + 823, + 707 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 263, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A HYPERPARAMETER SETTING ", + "text_level": 1, + "bbox": [ + 178, + 136, + 449, + 152 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We perform a parametric study on the value of discount factor $\\eta$ based on Nguyen’s benchmark problems without measurement noise. Empirically, setting $\\eta = 0 . 9 9 9 9$ ensures the scores of ground truth equations stand out and successfully enforces the sparsity in all experiments. For the discovery of very chaotic dynamical systems based on measurement data, we expect some physics terms from the governing equations to have a weak impact on the state variables (e.g., in the chaotic double pendulum system experiments, the effects from physics terms $\\sin ( \\theta _ { 1 } )$ and $\\mathrm { s i n } ( \\theta _ { 2 } )$ are hard to be captured), and the effect of data noise is unknown. Hence, we set $\\eta = 1$ to leverage the full strength of data fitting to enable the detection of physics terms that are offset or overwhelmed by data noise but are pivotal to the systems. Nevertheless, we must acknowledge that this selection process is empirical, which depends on our desire for the degree of parsimony of the target equation(s). ", + "bbox": [ + 174, + 167, + 825, + 308 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "As for the hyperparameters in the training schema (i.e., maximum module transplantation, episodes, maximum tree size, maximum augmented grammars), we have conducted parametric convergence tests for each experiment to ensure the learning curves (i.e., maximum scores in the history) converge. For example, as discussed in Section B, Table B.2 shows the setting of these hyperparameters. ", + "bbox": [ + 174, + 314, + 826, + 369 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B NGUYEN’S BENCHMARK PROBLEMS ", + "text_level": 1, + "bbox": [ + 176, + 392, + 514, + 409 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "This section provides more detailed experiment settings for the Nguyen’s benchmark tasks that are described in Section 4.2 of the main text, where the training hyperparameters for the SPL machine in these equation discovery experiments are also listed. Table B.1 presents the candidate mathematical operations allowed for three tested models and Table B.2 displays training hyperparameters for the SPL machine in the Nguyen’s benchmark tasks. ", + "bbox": [ + 174, + 424, + 823, + 493 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Moreover, the utilization of CFG in the SPL machine facilitates the flexibility of applying some prior knowledge including the universal mathematical rules and constraints. This feature empirically turns out to be an accessible and scalable approach for avoiding meaningless mathematical expressions. In the Nguyen’s benchmark tasks, one or multiple mathematical constraints are given to the SPL machine. These constraints include ", + "bbox": [ + 174, + 501, + 825, + 570 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1. Only variables and constant values are allowed in trigonometric functions. ", + "bbox": [ + 212, + 583, + 715, + 598 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/c977a0116b88c9d45bd82c616594fda31c9b4695a6f5a0ef5949d9aa48170324.jpg", + "table_caption": [ + "Table B.1: Candidate operators for each Nguyen’s benchmark task. const denotes constant values. " + ], + "table_footnote": [], + "table_body": "
Benchmark Candidate Operations
Nguyen-1 +,-,×,÷,cos(-),sin(-),exp(·),sin(+,-,×,÷,cos(-),sin(-),exp(-),log(),√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√+,-,×,÷,cos(-),sin(-),exp(-),log()，√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√,const
Nguyen-2 +,-,×,÷,cos(·),sin(.),exp()
Nguyen-3×,÷,cos(·,sin(·
Nguyen-4X，÷,cos(,sin(·
Nguyen-5×，÷,cos(·,sin(·
Nguyen-6X,÷,cos(·,sin(·
Nguyen-7×÷,cos(·,sin(·,exp(·
Nguyen-8×÷,cos(·,sin(·,exp(·
Nguyen-9×÷cos(·,sin(·,exp(·
Nguyen-10×÷Cos,sin(·,exp(.
Nguyen-11×÷Cos,sin(·,exp(.
Nguyen-12×÷Cos,sin(·,exp(.
Nguyen-1c×÷Cos,sin(·,exp(.
Nguyen-2c×÷cos(.,sin(·,exp(.
Nguyen-5c十，一×÷Cos,sin(·,exp(·
Nguyen-8c十，一×÷cos,sin(·,exp(.
Nguyen-9c×÷cos(·,sin(·,exp(·
", + "bbox": [ + 266, + 654, + 718, + 921 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/416ad4ab18fc2d9291ba37bc67a1725b1c6cc83c6f8e51a41dca5b624f8319ec.jpg", + "table_caption": [ + "Table B.2: Training Hyperparameter settings for the SPL model in Nguyen’s benchmark tasks. " + ], + "table_footnote": [], + "table_body": "
BenchmarkMaximum Module TransplantationEpisodes Between Module TransplantationMaximum Tree SizeMaximum Augmented Grammars
Nguyen-12010,000505
Nguyen-22010,000505
Nguyen-320100,000505
Nguyen-420100,000505
Nguyen-520100,000505
Nguyen-62010,000505
Nguyen-7205,000505
Nguyen-8205,000505
Nguyen-92010,000505
Nguyen-102010,000505
Nguyen-112010,000505
Nguyen-1220100,000505
Nguyen-1c202,000505
Nguyen-2c2010,000505
Nguyen-5c2010.000505
Nguyen-8c202,000505
Nguyen-9c201,000505
", + "bbox": [ + 222, + 132, + 766, + 412 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/03feee0ecc0cb69c29a7745189ecc54a262e04f556486cf0765f7ea4931903e3.jpg", + "table_caption": [ + "Table B.3: Mathematical constraints for the SPL machine in each Nguyen’s benchmark problem. " + ], + "table_footnote": [], + "table_body": "
BenchmarkConstraints Utilization
Nguyen-1{1,3}
Nguyen-2{1,3}
Nguyen-3{1,3}
Nguyen-4{1,3}
Nguyen-5{2,3}
Nguyen-6{2,3}
Nguyen-7{2,3}
Nguyen-80
Nguyen-9{2,3,4}
Nguyen-10{2,3,4}
Nguyen-11{2,3}
Nguyen-12{1,3,4}
Nguyen-1c{1,3}
Nguyen-2c{1,3}
Nguyen-5c{2,3}
Nguyen-8c0
Nguyen-9c{2,3,4}
", + "bbox": [ + 352, + 474, + 637, + 741 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2. Variables in trigonometric functions, logarithms and roots are up to the polynomial of 3. \n3. Trigonometric functions, logarithms and roots are not allowed to form unreasonable composite functions with each other, such as $\\sin ( \\cos ( . . . ) )$ . \n4. Some small integers (e.g. 1, 2) are used directly as leaves. ", + "bbox": [ + 207, + 773, + 826, + 838 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "They can be easily implemented into the SPL machine by defining or adjusting non-terminal nodes or production rules in the customized CFG. Constraints adopted by each task are shown in Table B.3. ", + "bbox": [ + 174, + 849, + 825, + 878 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C RESULTS OF ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 174, + 898, + 467, + 915 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We consider four ablation studies by removing the following: ", + "bbox": [ + 173, + 103, + 575, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "(a) the adaptive scaling in reward calculation, (b) the discount factor $\\eta ^ { n }$ that drives equation parsimony in Eq. (2), (c) module transplantation in tree generation, (d) all of the above three. ", + "bbox": [ + 205, + 130, + 655, + 195 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The resulting models are denoted with Model A, Model B, Model C, and Model D (note that Model $\\mathbf { D }$ is equivalent to the vanilla MCTS). The ablation study is performed on the 12 classic Nguyen’s benchmark problems, where the recovery rate of each model is calculated. The results of the ablation study are summarized in Table C.1. It is clear that module transplantation brings the largest gain in recovery rate. All the ablated models fail to uncover the Nguyen-12 equation (the most difficult case), where the coefficient 1/2 needs to be represented by mathematical operators and symbols, e.g., $x / ( x + x )$ , $y / ( y + y )$ , etc. ", + "bbox": [ + 174, + 208, + 826, + 306 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D FREE FALLING BALLS WITH AIR RESISTANCE ", + "text_level": 1, + "bbox": [ + 174, + 327, + 593, + 343 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "This appendix section reveals more details on discovering the physical laws, in the context of relationships between height and time, for the cases of free-falling objects with air resistance based on multiple experimental ball-drop datasets (de Silva et al., 2020). The datasets contain the records of 11 different types of balls, as shown in Figure D.1, dropped from a bridge, collected at a $3 0 \\mathrm { H z }$ sampling rate. The time between dropping and landing varies in each case due to the fact that air resistance has different effects on these free-dropping balls and induces divergent physical laws. Consequently, we consider each ball as an individual experiment and discover the physical law for each of them. The measurement dataset of a free-dropping ball is split into a training set (records from the first 2 seconds, 60 measurements) and a testing set (records after 2 seconds). ", + "bbox": [ + 173, + 358, + 825, + 484 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The physical laws of the three baseline models and distilled by the SPL machine from training data are exhibited in Table D.1. These discovered physical laws are then applied to forecast the height of the balls at the time slots in the testing dataset. These predictions, in comparison with the ground truth trajectory recorded, are shown in Figure D.2. The prediction error is shown in the Table 3 of the main text. ", + "bbox": [ + 174, + 491, + 825, + 560 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "E NONLINEAR DYNAMICS DISCOVERY ", + "text_level": 1, + "bbox": [ + 174, + 582, + 511, + 598 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "This appendix section elaborates more details on the two governing equation discovery experiments presented in the main text, including the datasets and full results. ", + "bbox": [ + 173, + 613, + 823, + 642 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/569676fb9b326d807eec17772684ab7f6941c4e98a1f2834431b6fce545b954a.jpg", + "table_caption": [ + "Table C.1: Summary of the ablation study results. " + ], + "table_footnote": [], + "table_body": "
BenchmarkExpressionSPLModel AModel BModel CModel D
Nguyen-1+ +x 2 2100%100%100%14%12%
Nguyen-2x4 + +x100%100%100%0%0%
Nguyen-3+x4 +x² +x 25100%100%100%0%0%
Nguyen-4+x+x4+x²+x²+x 2699%96%92%0%0%
Nguyen-5sin(𝑥²)cos(x)-195%95%92%92%88%
Nguyen-6sin(x²)+sin(x+x²)100%100%96%100%100%
Nguyen-7ln(x +1) +ln(x² + 1)100%100%100%100%100%
Nguyen-8√x100%100%100%100%100%
Nguyen-9sin(x)+sin(y²)100%100%100%100%100%
Nguyen-102 sin(x) cos(y)100%100%100%0%0%
Nguyen-11x100%96%96%92%87%
Nguyen-12x4 2 + 1 -y28%0%0%0%0%
Average93.5%90.58%89.67%49.83%48.92%
", + "bbox": [ + 173, + 698, + 821, + 906 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/5a12a0d0ad2bb8b23eb0affa2dee2b2b2c7c2cc587809ad8e1291c37f1149cdf.jpg", + "table_caption": [ + "Table D.1: Uncovered physics from the motions of free-falling balls by the SPL machine and three baseline models. Note that these formulas are the raw equations produced by SRL. Further simplification helps better parsimony of the formulas. " + ], + "table_footnote": [], + "table_body": "
TypeModelExpression
baseballSPL Model-1 Model-2H(t)= 47.8042 + 0.6253t- 4.5383t² H(t)= 47.682+ 1.456t- 5.629t² + 0.376t3 H(t)= 45.089- 8.156t + 5.448 exp(0t)
blue basketballModel-3 SPL Model-1 Model-2H(t)= 48.051- 183.467 log(cosh(0.217t)) H(t)= 46.4726-5.105t² + t³ -0.251t4 H(t)= 46.513-0.493t-3.912t² +0.03t H(t)= 43.522- 7.963t + 5.306 exp(0t)
green basketballModel-3 SPL Model-1H(t)= 46.402-84.791log(cosh(0.319t)) H(t)= 45.9087- 4.1465t² +log(cosh(1)) H(t)= 46.438-0.34t- 3.882t² - 0.055t3 H(t)= 43.512- 8.043t + 5.346 exp(0t)
volleyballModel-2 Model-3 SPL Model-1 Model-2H(t)= 46.391-124.424 log(cosh(0.263t)) H(t)= 48.0744-3.7772t² H(t)= 48.046+ 0.362t-4.352t² +0.218t
bowling ballModel-3 SPL Model-1H(t)= 45.32- 7.317t + 5.037exp(0t) H(t)= 48.124-107.816 log(cosh(0.27t)) H(t)= 46.1329-3.8173t² -0.2846t+4.14× 10-5 exp(20.7385t²)exp(-12.4538t3) H(t)= 46.139-0.091t-3.504t² -0.431t3
golf ballModel-2 Model-3 SPL Model-1H(t)= 43.336-8.525t + 5.676 exp(0t) H(t)= 46.342- 247.571log(cosh(0.189t)) H(t)= 49.5087-4.9633t² +log(cosh(t)) H(t)= 49.413+0.532t-5.061t²+0.102t3
tennis ballModel-2 Model-3 SPL Model-1H(t)= 46.356 -8.918t+ 5.964exp(0t) H(t)= 49.585-178.47log(cosh(0.23t)) H(t)= 47.8577-4.0574t² +log(cosh(0.121t))
whiffle ballModel-2 Model-3 SPL Model-1H(t)= 47.738+0.658t-4.901t²+0.325t H(t)= 45.016-7.717t+ 5.212 exp(0t) H(t)= 47.874- 114.19 log(cosh(0.269t)) H(t)= 4.1563t² -t + 47.0133exp(-0.1511t²) H(t)= 46.969+0.574t-4.505t²+0.522t
1 whiffle ballModel-2 Model-3 SPL Model-1H(t)= 44.259- 6.373t+ 4.689 exp(0t) H(t)= 47.062- 34.083log(cosh(0.462t)) H(t)= -18.6063 + 65.8583 exp(-0.0577t²) H(t)= 47.215+0.296t-4.379t²+0.421t3
2 yellow whiffleModel-2 Model-3 SPLH(t)= 44.443-6.744t + 4.813 exp(0t) H(t)= 47.255-38.29 log(cosh(0.447t)) H(t)= 148.9911/(log(cosh(t)) +3.065)-14.5828t²/(log(cosh(t)) +3.065)
ballModel-1 Model-2 Model-3+48.6092log(cosh(t))/(log(cosh(t)) + 3.065) H(t)= 48.613-0.047t-4.936t² +0.826t3 H(t)= 45.443- 6.789t+ 4.973exp(0t) H(t)= 48.594-12.49 log(cosh(0.86t))
orange whiffle ballSPL Model-1 Model-2 Model-3H(t)= -1.6626t + 47.8622 exp(-0.06815t²) H(t)= 47.836-1.397t-3.822t² +0.422t³ H(t)= 44.389-7.358t+ 5.152 exp(0t)
", + "bbox": [ + 173, + 239, + 826, + 840 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/e9b06f6ee845288af4c2619e93eb8e80a77b5c720bdb62d770ef47dca03524d9.jpg", + "image_caption": [ + "Figure D.1: The experimental balls that were dropped from the bridge (de Silva et al., 2020). From left to right: golf ball, tennis ball, whiffle ball 1, whiffle ball 2, baseball, yellow whiffle ball, orange whiffle ball, green basketball, and blue basketball. Volleyball is not shown here. " + ], + "image_footnote": [], + "bbox": [ + 191, + 102, + 807, + 186 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/c0c5774af163407ed9d73424cb4fb4fcf79658a7b1a4ed5734b9f71a03ee7860.jpg", + "image_caption": [ + "Figure D.2: Trajectories after 2 seconds predicted by uncovered physical laws. " + ], + "image_footnote": [], + "bbox": [ + 179, + 253, + 808, + 742 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E.1 LORENZ SYSTEM", + "text_level": 1, + "bbox": [ + 174, + 792, + 338, + 808 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The 3-dimensional Lorenz system is governed by ", + "bbox": [ + 173, + 819, + 498, + 834 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5a6c104208a2aa5909aae8249740fb8fc45dc5fe4a1a04654dedc5193af2c777.jpg", + "text": "$$\n\\begin{array} { l } { \\dot { x } = \\sigma ( y - x ) } \\\\ { \\dot { y } = x ( \\rho - z ) - y } \\\\ { \\dot { z } = x y - \\beta z } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 437, + 837, + 560, + 892 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "with parameters $\\sigma = 1 0$ , $\\beta = 8 / 3$ , and $\\rho = 2 8$ , under which the Lorenz attractor has two lobes and the system, starting from anywhere, makes cycles around one lobe before switching to the other and iterates repeatedly. The measurement data of the Lorenz system states in this experiment contains a clean signal with $5 \\%$ Gaussian white noise. The derivatives of Lorenz’s state variables are unmeasured but numerically estimated and smoothed by the Savitzky–Golay filter. The noisy synthetic measurement data and numerically obtained derivatives are shown in Figure E.1. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/a41908088bcc678ad1a4d19638a3ca80f4aab863564157948f73cef957389e0f.jpg", + "image_caption": [ + "Figure E.1: Lorenz system for the experiment. Noisy measurement data and numerically estimated derivatives smoothed by Savitzky–Golay filter. " + ], + "image_footnote": [], + "bbox": [ + 228, + 104, + 767, + 280 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/50464c9ef51077e4e559cca455475503a1b7b4a715709832969a2dd56e514596.jpg", + "table_caption": [ + "Table E.1: Discovered governing equations for the Lorenz system. " + ], + "table_footnote": [], + "table_body": "
ModelDiscovered Governing Equations
Eureqax= -0.56-9.02x +9.01y y= -0.047+18.79x+1.86y-0.046xy-0.74xz = -3.04-2.23z +0.88xy
pySINDyx = -0.46-9.18x +9.17y y= 22.32x+0.15y-0.85xz z = 6.04-2.83z +0.15x² + 0.81xy
NGGPx = 0.0047 -10.02x + 10.01y - 0.007x² + 0.007xy - 0.37x/z - 0.00074x²y +0.00063x +0.00018x²z +0.00011xy² -6.59e-5xyz -0.00011y² y = 26.36x-1.5y-0.83xz -7.20x/z + 13.08y/z + 4.52e-5x³ -0.0038xz² -44.25x/z²+0.00028x/z -0.00017xz + 5.98e 6x32 = -0.64- 0.036y － 2.64z + 1.038xy+ 0.00021xz + 0.0011yz － 0.00021x/z
SPL-8.04e-8y/x + 0.00022y/z - 8.04e-8z² - 0.00021xyz +0.00021xy/z -0.001y²z-0.00021y²/z+1.17y/z²-3.89e-7yz²/x+6.66e-9z²/x x = -9.966x + 9.964y y = 27.764x - 0.942y- 0.994xz
", + "bbox": [ + 194, + 371, + 795, + 672 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 708, + 823, + 763 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Table E.1 presents the distilled governing equations for the Lorenz system by the SPL machine compared with 3 baseline models. It is observed that the SPL machine uncovers the explicit form of equations accurately in the context of active terms, whereas Eureqa, pySINDy and NGGP yield several false-positive terms in the underlying governing equations. The predicted system responses (starting from a different initial condition) simulated from these uncovered equations are shown in Figure E.2. Although it is hard to reproduce the most accurate coefficients due to the tremendous errors induced by numerical differentiation of noisy measurement data as depicted in Figure E.1, the SPL machine is still capable of distilling the most concise symbolic combination of operators and variables to correctly formulate the parsimonious mathematical expressions that govern the Lorenz system dynamics. The predicted responses for the governing equations unearthed by the SPL machine simulate the system in a decent manner. ", + "bbox": [ + 173, + 770, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/2bc7320d817370625240d80bb22bb51cdf8338752d89ab78274cb0f16c9e4e2e.jpg", + "image_caption": [ + "Figure E.2: Response prediction for 5 seconds by identified governing equations (dashed plots) under a different validation IC of Lorenz system, in comparison with the ground truth trajectory (grey). " + ], + "image_footnote": [], + "bbox": [ + 258, + 103, + 741, + 243 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/796fdf0e7e82ded4d13c43f8fb46e8d2be605444c2c5124087a627fcbb91c6c7.jpg", + "image_caption": [ + "Figure E.3: Double Pendulum system experiment and measurement data (Asseman et al., 2018): A. experiment setup. B. displacements of the two moving masses. C. model the system with $\\theta _ { 1 }$ and $\\theta _ { 2 }$ . D. angles of two masses transformed from displacements. " + ], + "image_footnote": [], + "bbox": [ + 179, + 305, + 818, + 472 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.2 MOUNTED DOUBLE PENDULUM SYSTEM ", + "text_level": 1, + "bbox": [ + 173, + 556, + 501, + 571 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The second nonlinear dynamics discovery experiment is a chaotic double pendulum system (Asseman et al., 2018). The measured data, in form of videos, represents the chaotic motion of a double pendulum on the device shown in Figure E.3A filmed with a high-speed camera. The positional data is converted into angular form based on the geometry information (see the model shown in Figure E.3C). The governing equations can be derived using the Euler–Lagrange method, given by ", + "bbox": [ + 174, + 582, + 825, + 654 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/0aa9469031e8512c1adeb7ea2154809ba7efdb2a1b97a35b75f750a2e2d87a4d.jpg", + "text": "$$\n\\begin{array} { r } { ( m _ { 1 } + m _ { 2 } ) l _ { 1 } \\ddot { \\theta } _ { 1 } + m _ { 2 } l _ { 2 } \\ddot { \\theta } _ { 2 } \\cos ( \\theta _ { 1 } - \\theta _ { 2 } ) + m _ { 2 } l _ { 2 } \\omega _ { 2 } ^ { 2 } \\sin ( \\theta _ { 1 } - \\theta _ { 2 } ) + ( m _ { 1 } + m _ { 2 } ) g \\sin ( \\theta _ { 1 } ) = F _ { 1 } , } \\\\ { m _ { 2 } l _ { 2 } \\ddot { \\theta } _ { 2 } + m _ { 2 } l _ { 1 } \\ddot { \\theta } _ { 1 } \\cos ( \\theta _ { 1 } - \\theta _ { 2 } ) - m _ { 2 } l _ { 1 } \\omega _ { 1 } ^ { 2 } \\sin ( \\theta _ { 1 } - \\theta _ { 2 } ) + m _ { 2 } g \\sin ( \\theta _ { 2 } ) = F _ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 657, + 807, + 702 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "which, by denoting $\\omega$ as the velocity, can be converted to the following state-space form: ", + "bbox": [ + 171, + 719, + 754, + 736 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/7825dba022f8430042074524b761f319102d6d78d907c0cfa29118d84bb72956.jpg", + "text": "$$\n\\begin{array} { r l } & { \\dot { \\theta } _ { 1 } = \\omega _ { 1 } , } \\\\ & { \\dot { \\theta } _ { 2 } = \\omega _ { 2 } , } \\\\ & { \\dot { \\omega } _ { 1 } = c _ { 1 } \\dot { \\omega } _ { 2 } \\cos ( \\Delta \\theta ) + c _ { 2 } \\omega _ { 2 } ^ { 2 } \\sin ( \\Delta \\theta ) + c _ { 3 } \\sin ( \\theta _ { 1 } ) + \\mathcal { R } _ { 1 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } ) , } \\\\ & { \\dot { \\omega } _ { 2 } = c _ { 1 } \\dot { \\omega } _ { 1 } \\cos ( \\Delta \\theta ) + c _ { 2 } \\omega _ { 1 } ^ { 2 } \\sin ( \\Delta \\theta ) + c _ { 3 } \\sin ( \\theta _ { 2 } ) + \\mathcal { R } _ { 2 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 739, + 732, + 825 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $\\Delta \\theta = \\theta _ { 1 } - \\theta _ { 2 }$ , and $\\mathcal { R } _ { 1 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } )$ and $\\mathcal { R } _ { 2 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } )$ denote the damping terms for the last two differential equations. ", + "bbox": [ + 176, + 830, + 821, + 861 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The data source contains multiple video datasets. For this discovery, 5,000 denoised random subsamples from 5 datasets are used for training purposes, and 2,000 random sub-samples from another 2 datasets for validation, and 1 dataset for testing. Some prior knowledge guiding this discovery includes: ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/09d9d9086e2db27bb035a32a65924c6e5b6c6bf009e6d7604b0b6de5c33c5619.jpg", + "table_caption": [ + "Table E.2: Discovered governing equations of the mounted double pendulum by the SPL model. " + ], + "table_footnote": [], + "table_body": "
PhaseExpression
u1-0.0991ω2 cos(△0) - 0.103ω2 sin(△0) - 69.274 sin(0i) + 0.515 cos(01)
2-1.368ω1 c0s(△0) + 1.363ω² sin(△0) - 92.913 sin(02) + 0.032w1
", + "bbox": [ + 209, + 132, + 784, + 193 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1. the two terms $\\dot { \\omega } _ { 2 } \\cos ( \\Delta \\theta )$ for $\\dot { \\omega } _ { 1 }$ and $\\dot { \\omega } _ { 1 } \\cos ( \\Delta \\theta )$ for $\\dot { \\omega } _ { 2 }$ , which can be easily derived based on our prior knowledge on the system, are assumed known in the two governing equations. However, their coefficients are unknown and need to be estimated. ", + "bbox": [ + 214, + 217, + 825, + 260 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "2. the remaining of the formulas are potentially comprised of the free combination of $\\dot { \\omega } _ { 1 } , \\dot { \\omega } _ { 2 }$ , $\\omega _ { 1 } , \\omega _ { 2 }$ , as well as the angles $( \\theta _ { 1 } , \\theta _ { 2 } , \\Delta \\theta )$ under the trigonometric functions $\\cos ( \\cdot )$ and $\\sin ( \\cdot )$ . ", + "bbox": [ + 212, + 263, + 825, + 306 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "3. velocities and relative velocities of two masses, as well as their directions (sign function) might contribute to the damping. ", + "bbox": [ + 209, + 310, + 823, + 339 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Based on the above information, the candidate production rules for $\\dot { \\omega } _ { 1 }$ and $\\dot { \\omega } _ { 2 }$ equations are shown below, where non-terminal nodes are $V = \\{ A , W , T , S \\}$ and $C$ denotes the placeholder symbol for the constant coefficient values. ", + "bbox": [ + 178, + 351, + 826, + 392 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/05457b967d967912683654bfc300655a9b6cc56001e9c06f9bb923c91936bb73.jpg", + "text": "$$\n\\begin{array} { r l } & { A A + A , A A \\times A , A C , A A + A , A W , } \\\\ & { W W \\times W , W \\omega _ { 1 } , W \\omega _ { 2 } , W \\dot { \\omega } _ { 1 } , W \\dot { \\omega } _ { 2 } , } \\\\ & { A \\cos ( T ) , A \\sin ( T ) , T T + T , T T - T , T \\theta _ { 1 } , T \\theta _ { 2 } , } \\\\ & { A s i g n ( S ) , S S + S , S S - S , S \\omega _ { 1 } , S \\omega _ { 2 } , S \\dot { \\omega } _ { 1 } , S \\dot { \\omega } _ { 2 } , } \\\\ & { A \\dot { \\omega } _ { 1 } \\cos ( \\theta _ { 1 } - \\theta _ { 2 } ) , A \\dot { \\omega } _ { 2 } \\cos ( \\theta _ { 1 } - \\theta _ { 2 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 401, + 750, + 497 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Note that our prior knowledge can be easily incorporated in the proposed SPL machine to improve the discovery performance, rather than relying on the free combination of mathematical operators and symbols. The hyperparameters are set as $\\eta = 1$ , $t _ { m a x } = 2 0$ , and 40,000 episodes of training are regarded as one trail. 5 independent trials are performed and the equations with the highest validation scores are selected as the final result. The uncovered equations are shown in Table E.2. They are validated through interpolation on the testing set and compared with the smoothed derivatives, as shown in Figure E.4. The solution appears felicitous as the governing equations of the testing responses. ", + "bbox": [ + 173, + 505, + 826, + 617 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "F DISCUSSION AND FUTURE DIRECTIONS ", + "text_level": 1, + "bbox": [ + 174, + 636, + 537, + 652 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The effectiveness of the proposed SPL machine is empowered by the following elements: (1) The use of MCTS enables the flexible representation of search space with customized computational grammars, composed of a finite set of mathematical operators and symbols, to guide the search tree expansion. (2) The exploration-exploitation trade-off nature of MCTS is remarkably useful for searching the optimal mathematical expression tree. (3) The key adjustments, including the greedy search, the adaptive-scaled rewarding, the reward regularizer, and the expression tree module transplantation, make it possible to efficiently uncover the best path to formulate complex equations. (4) The SPL machine straightforwardly accepts our prior or domain knowledge, or any sort of constraints of the tasks in the grammar design while leveraging great flexibility in expression formulation. ", + "bbox": [ + 173, + 667, + 826, + 794 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "While SPL shows huge potential in both symbolic regression and governing equation discovery tasks, there are still some imperfections to be improved. In this appendix section, a few bottlenecks and their potential solutions are presented: ", + "bbox": [ + 176, + 800, + 825, + 842 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1. Computational cost. Computational cost for this framework is one of the major issues, especially when constant coefficient estimation is required. Evaluating the solution in the simulation phase happens very frequently for the MCTS algorithm where the policy selection relies heavily on a large number of historical rewards. However, the constant coefficient value estimation, which requires repeated calls for an optimization process and can be slow, is needed for evaluation purposes. In particular, the constant coefficient value is estimated via concurrently solving an optimization problem, e.g., by Powell’s conjugate direction method (Powell, 1964). For example, when the tree structure changes or is updated, the optimization of the constant coefficients should be re-performed simultaneously. The SPL machine is not the only symbolic regressor suffering from the computational cost in constant coefficient value estimation. In fact, the state-of-the-art symbolic regression model, the neural-guided GP (NGGP) (Mundhenk et al., 2021), becomes much slower in the Nguyen’s benchmark variant tasks (see Table E.3). The current implementation of the SPL machine tries to empirically avoid this issue by limiting the number of placeholders in a discovered expression and simplifying the expression before evaluation, but still cannot reach great efficiency. This bottleneck might be mitigated if parallel computing is introduced to the MCTS simulation phase. ", + "bbox": [ + 215, + 854, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/0f32e8e7d96b3ef5abada570cc03b00c86b665715bdd0a9076b3f8cd01758eb2.jpg", + "image_caption": [ + "Figure E.4: Discovered governing equations of the mounted double pendulum system on a different dataset in different time sections: in 2-8 seconds the two masses are in chaotic motions while in 30-36 seconds the masses tend to move periodically due to accumulative damping. $\\ddot { \\theta } _ { 1 }$ and ${ \\ddot { \\theta } } _ { 2 }$ are obtained through smoothed numerical differentiation and predicted from the discovered governing equations. " + ], + "image_footnote": [], + "bbox": [ + 200, + 99, + 790, + 497 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/67ceec60ca2e657eed15890f24b5ca1710c5a77846bf0a9957d6781aaaae57b3.jpg", + "table_caption": [ + "Table E.3: Average training time (in seconds) of SPL and NGGP in the Nguyen’s benchmark problems " + ], + "table_footnote": [], + "table_body": "
BenchmarkSPL [s]NGGP[s]
Nguyen-18.7762.734
Nguyen-27.2963.296
Nguyen-381.2873.945
Nguyen-4567.0615.764
Nguyen-5431.22877.627
Nguyen-664.651104.588
Nguyen-714.9953.024
Nguyen-85.592.896
Nguyen-95.74313.229
Nguyen-1053.24586.497
Nguyen-1110.16344.399
Nguyen-12187.9334.757
Nguyen-1c452.734362.075
Nguyen-2c295.7691188.215
Nguyen-5c2178.8911365.777
Nguyen-8c77.892129.349
Nguyen-9c2001.4023066.41
", + "bbox": [ + 356, + 611, + 630, + 877 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 232, + 104, + 825, + 270 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "2. Graph modularity underexamined. The current design of the SPL training scheme does not leverage the full graph modularity: modules are reached by transforming a complete parse tree into a grammar. However, in some cases, there might be some influential modules appearing frequently as part of the tree. This type of graph modularity is described in the AIFeynman method (Udrescu et al., 2020). Deploying a more comprehensive graph modularity into the SPL machine will boost its efficacy in the complex equation and nonlinear dynamics discovery tasks. ", + "bbox": [ + 214, + 275, + 825, + 372 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "3. Robustness against extreme data noise and scarcity. Although it is observed that this reinforcement learning-based method is able to unearth the parsimonious solution to the governing equations from synthetic or measurement data with a moderate level of noise and scarcity. It is not effective when the data condition is extreme, or if there are missing values that make it challenging to numerically calculate the state derivatives. It is reasonable to investigate the integration between the SPL framework with a differentiable surrogate model built upon neural networks (Long et al., 2018; Chen et al., 2021) or spline learning (Sun et al., 2021) for further robustness in nonlinear dynamics discovery tasks. ", + "bbox": [ + 214, + 377, + 825, + 488 + ], + "page_idx": 21 + } +] \ No newline at end of file diff --git a/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z_middle.json b/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..fcb0c978ba899b985720282a973cb4233b0ebb7f --- /dev/null +++ b/parse/dev/ZTK3SefE8_Z/ZTK3SefE8_Z_middle.json @@ -0,0 +1,52921 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 506, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 508, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 508, + 97 + ], + "score": 1.0, + "content": "SYMBOLIC PHYSICS LEARNER: DISCOVERING GOV-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 507, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 507, + 118 + ], + "score": 1.0, + "content": "ERNING EQUATIONS VIA MONTE CARLO TREE SEARCH", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 361, + 149 + ], + "lines": [ + { + "bbox": [ + 110, + 136, + 363, + 150 + ], + "spans": [ + { + "bbox": [ + 110, + 136, + 160, + 150 + ], + "score": 1.0, + "content": "Fangzheng", + "type": "text" + }, + { + "bbox": [ + 161, + 137, + 183, + 149 + ], + "score": 0.81, + "content": "\\mathbf { S u n ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 136, + 211, + 150 + ], + "score": 1.0, + "content": ", Yang", + "type": "text" + }, + { + "bbox": [ + 211, + 137, + 232, + 149 + ], + "score": 0.49, + "content": "\\mathbf { L i u ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 136, + 363, + 150 + ], + "score": 1.0, + "content": ", Jian-Xun Wang3, Hao Sun4,∗", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 113, + 150, + 498, + 186 + ], + "lines": [ + { + "bbox": [ + 112, + 149, + 500, + 164 + ], + "spans": [ + { + "bbox": [ + 112, + 149, + 500, + 164 + ], + "score": 1.0, + "content": "1Northeastern University, Boston, MA, USA; 2University of Chinese Academy of Sciences, Beijing, China;", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 111, + 160, + 466, + 174 + ], + "spans": [ + { + "bbox": [ + 111, + 160, + 466, + 174 + ], + "score": 1.0, + "content": "3University of Notre Dame, Notre Dame, IN, USA; 4Renmin University of China, Beijing, China.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 112, + 175, + 468, + 187 + ], + "spans": [ + { + "bbox": [ + 112, + 175, + 468, + 187 + ], + "score": 1.0, + "content": "Emails: sun.fa@northeastern.edu; liuyang22@ucas.ac.cn; jwang33@nd.edu; haosun@ruc.edu.cn", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 278, + 209, + 333, + 221 + ], + "lines": [ + { + "bbox": [ + 277, + 208, + 335, + 222 + ], + "spans": [ + { + "bbox": [ + 277, + 208, + 335, + 222 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 143, + 232, + 469, + 385 + ], + "lines": [ + { + "bbox": [ + 141, + 231, + 469, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 469, + 245 + ], + "score": 1.0, + "content": "Nonlinear dynamics is ubiquitous in nature and commonly seen in various science", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "score": 1.0, + "content": "and engineering disciplines. Distilling analytical expressions that govern nonlinear", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 470, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 470, + 266 + ], + "score": 1.0, + "content": "dynamics from limited data remains vital but challenging. To tackle this funda-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "score": 1.0, + "content": "mental issue, we propose a novel Symbolic Physics Learner (SPL) machine to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "score": 1.0, + "content": "discover the mathematical structure of nonlinear dynamics. The key concept is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "score": 1.0, + "content": "to interpret mathematical operations and system state variables by computational", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 310 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 310 + ], + "score": 1.0, + "content": "rules and symbols, establish symbolic reasoning of mathematical formulas via", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "score": 1.0, + "content": "expression trees, and employ a Monte Carlo tree search (MCTS) agent to explore", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 470, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 470, + 332 + ], + "score": 1.0, + "content": "optimal expression trees based on measurement data. The MCTS agent obtains an", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 469, + 344 + ], + "score": 1.0, + "content": "optimistic selection policy through the traversal of expression trees, featuring the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 469, + 353 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 469, + 353 + ], + "score": 1.0, + "content": "one that maps to the arithmetic expression of underlying physics. 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The efficacy and superiority of the SPL machine are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 375, + 461, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 461, + 387 + ], + "score": 1.0, + "content": "demonstrated by numerical examples, compared with state-of-the-art baselines.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 405, + 206, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 208, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 208, + 420 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 443 + ], + "score": 1.0, + "content": "We usually learn the behavior of a nonlinear dynamical system through its nonlinear governing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 362, + 453 + ], + "score": 1.0, + "content": "differential equations. 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2University of Chinese Academy of Sciences, Beijing, China;", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 111, + 160, + 466, + 174 + ], + "spans": [ + { + "bbox": [ + 111, + 160, + 466, + 174 + ], + "score": 1.0, + "content": "3University of Notre Dame, Notre Dame, IN, USA; 4Renmin University of China, Beijing, China.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 112, + 175, + 468, + 187 + ], + "spans": [ + { + "bbox": [ + 112, + 175, + 468, + 187 + ], + "score": 1.0, + "content": "Emails: sun.fa@northeastern.edu; liuyang22@ucas.ac.cn; jwang33@nd.edu; haosun@ruc.edu.cn", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 111, + 149, + 500, + 187 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 209, + 333, + 221 + ], + "lines": [ + { + "bbox": [ + 277, + 208, + 335, + 222 + ], + "spans": [ + { + "bbox": [ + 277, + 208, + 335, + 222 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 143, + 232, + 469, + 385 + ], + "lines": [ + { + "bbox": [ + 141, + 231, + 469, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 469, + 245 + ], + "score": 1.0, + "content": "Nonlinear dynamics is ubiquitous in nature and commonly seen in various science", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 469, + 256 + ], + "score": 1.0, + "content": "and engineering disciplines. Distilling analytical expressions that govern nonlinear", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 470, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 470, + 266 + ], + "score": 1.0, + "content": "dynamics from limited data remains vital but challenging. To tackle this funda-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "score": 1.0, + "content": "mental issue, we propose a novel Symbolic Physics Learner (SPL) machine to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "score": 1.0, + "content": "discover the mathematical structure of nonlinear dynamics. The key concept is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 299 + ], + "score": 1.0, + "content": "to interpret mathematical operations and system state variables by computational", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 310 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 310 + ], + "score": 1.0, + "content": "rules and symbols, establish symbolic reasoning of mathematical formulas via", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "score": 1.0, + "content": "expression trees, and employ a Monte Carlo tree search (MCTS) agent to explore", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 470, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 470, + 332 + ], + "score": 1.0, + "content": "optimal expression trees based on measurement data. 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Salient features", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 352, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 469, + 366 + ], + "score": 1.0, + "content": "of the proposed framework include search flexibility and enforcement of parsimony", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 469, + 376 + ], + "score": 1.0, + "content": "for discovered equations. 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The explicit form of", + "type": "text" + }, + { + "bbox": [ + 470, + 462, + 489, + 474 + ], + "score": 0.91, + "content": "\\mathcal F ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "some nonlinear dynamics remains underexplored. For example, in a mounted double pendulum", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "system, the mathematical description of the underlying physics might be unclear due to unknown", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "viscous and frictional damping forms. These uncertainties yield critical demands for the discovery of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 504, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 505, + 520 + ], + "score": 1.0, + "content": "nonlinear dynamics given observational data. Nevertheless, distilling the analytical form of governing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 517, + 466, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 466, + 530 + ], + "score": 1.0, + "content": "equations from limited noisy data, commonly seen in practice, is an intractable challenge.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 428, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "Ever since the early work on the data-driven discovery of nonlinear dynamics (Džeroski & Todorovski,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 559 + ], + "score": 1.0, + "content": "1993; Dzeroski & Todorovski, 1995), many scientists have stepped into this field of study. During", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "the recent decade, the escalating advances in machine learning, data science, and computing power", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 581 + ], + "score": 1.0, + "content": "have enabled several milestone efforts of unearthing the governing equations for nonlinear dynamical", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "systems. Notably, a breakthrough model named SINDy (Sparse Identification of Nonlinear Dynamics)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "(Brunton et al., 2016) has shed light on tackling this achallenge. SINDy was invented to determine the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "sparse solution among a pre-defined basis function library recursively through a sequential threshold", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "ridge regression (STRidge) algorithm. SINDy quickly became one of the state-of-art methods and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "kindled significant enthusiasm in this field of study (Rudy et al., 2017; Long et al., 2018; Champion", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 633, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 644 + ], + "score": 1.0, + "content": "et al., 2019; Chen et al., 2021; Sun et al., 2021; Rao et al., 2022). However, the success of this", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "sparsity-promoting approach relies on a properly defined candidate function library that requires good", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "prior knowledge of the system. It is also restricted by the fact that a linear combination of candidate", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "functions might be insufficient to recover complicated mathematical expressions. Moreover, when", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 412, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 412, + 689 + ], + "score": 1.0, + "content": "the library size is massive, it empirically fails to hold the sparsity constraint.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 534, + 506, + 689 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 693, + 504, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 505, + 705 + ], + "score": 1.0, + "content": "At the same time, attempts have been made to tackle the nonlinear dynamics discovery problems", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 704, + 505, + 717 + ], + "spans": [ + { + "bbox": [ + 106, + 704, + 505, + 717 + ], + "score": 1.0, + "content": "by introducing neural networks with activation functions replaced by commonly seen mathematical", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "operators (Martius & Lampert, 2017; Sahoo et al., 2018; Kim et al., 2019; Long et al., 2019). The", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "intricate formulas are obtained via symbolic expansion of the well-trained network. This interpretation", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "of physical laws results in larger candidate pools compared with the library-based representation of", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "physics employed by SINDy. Nevertheless, since the sparsity of discovered expressions is primarily", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "achieved by empirical pruning of the network weights, this framework exhibits sensitivity to user-", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 503, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 503, + 150 + ], + "score": 1.0, + "content": "defined thresholds and may fall short to produce parsimonious equations for noisy and scarce data.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 45.5, + "bbox_fs": [ + 106, + 692, + 505, + 717 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "operators (Martius & Lampert, 2017; Sahoo et al., 2018; Kim et al., 2019; Long et al., 2019). The", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "intricate formulas are obtained via symbolic expansion of the well-trained network. This interpretation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "of physical laws results in larger candidate pools compared with the library-based representation of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "physics employed by SINDy. Nevertheless, since the sparsity of discovered expressions is primarily", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "achieved by empirical pruning of the network weights, this framework exhibits sensitivity to user-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 503, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 503, + 150 + ], + "score": 1.0, + "content": "defined thresholds and may fall short to produce parsimonious equations for noisy and scarce data.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 153, + 506, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "Alternatively, another inspiring work (Bongard & Lipson, 2007; Schmidt & Lipson, 2009) re-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "envisioned the data-driven nonlinear dynamics discovery tasks by casting them into symbolic regres-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "sion problems which have been profoundly resolved by the genetic programming (GP) approach", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "(Koza & Koza, 1992; Billard & Diday, 2003). Under this framework, a symbolic regressor is es-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "tablished to identify the governing equations that best describe the underlying physics through free", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 506, + 221 + ], + "score": 1.0, + "content": "combination of mathematical operators and symbols, leading to great flexibility in model selection.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "One essential weakness of this early methodology is that, driven exclusively by the goal of empirically", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "seeking the best-fitting expression (e.g. minimizing the mean-square error) in a genetic expansion", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "process, the GP-based model usually over-fits the target system with numerous false-positive terms", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 507, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 507, + 267 + ], + "score": 1.0, + "content": "under data noise, even sometimes at a subtle level, causing huge instability and uncertainty. However,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "this ingenious idea has inspired a series of subsequent endeavors (Cornforth & Lipson, 2012; Gaucel", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "et al., 2014; Ly & Lipson, 2012; Quade et al., 2016; Vaddireddy et al., 2020). In a more recent work,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "Deep Symbolic Regression (DSR) (Petersen et al., 2021; Mundhenk et al., 2021), a reinforcement", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "learning-based model was established and generally outperformed the GP based models including the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "commercial Eureqa software (Langdon & Gustafson, 2010). Additionally, the AI-Feynman methods", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "(Udrescu & Tegmark, 2020; Udrescu et al., 2020; Udrescu & Tegmark, 2021) ameliorated symbolic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "regression for distilling physics laws from data by combining neural network fitting with a suite of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "physics-inspired techniques. This approach is also highlighted by a recursive decomposition of a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "complicated mathematical expression into different parts on a tree-based graph, which disentangles", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "the original problem and speeds up the discovery. It outperformed Eureqa in the uncovering Feynman", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "physics equations (Feynman et al., 1965). However, this approach is built upon ad-hoc steps and, to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 351, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 351, + 398 + ], + "score": 1.0, + "content": "some extent, lacks flexible automation in equation discovery.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "The popularity of adopting the tree-based symbolic reasoning of mathematical formulas (Lample", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "& Charton, 2019) has been rising recently to discover unknown mathematical expressions with a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "reinforcement learning agent (Kubalík et al., 2019; Petersen et al., 2021; Mundhenk et al., 2021).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "However, some former work attempting to apply the Monte Carlo tree search (MCTS) algorithm as an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "alternative to GP for symbolic regression (Cazenave, 2013; White et al., 2015; Islam et al., 2018; Lu", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "et al., 2021) failed to leverage the full flexibility of this algorithm, resulting in the similar shortage that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "GP-based symbolic regressors possess as discussed earlier. Despite these outcomes, we are conscious", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "of the strengths of the MCTS algorithm in equation discovery: it enables the flexible representation of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "search space with customized computational grammars to guide the search tree expansion. A sound", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "mathematical underpinning for the trade-off between exploration and exploitation is remarkably", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "advantageous as well. These features make it possible to inform the MCTS agent by our prior physics", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 522, + 474, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 474, + 534 + ], + "score": 1.0, + "content": "knowledge in nonlinear dynamics discovery rather than randomly searching in large spaces.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "Contribution. We propose a promising model named Symbolic Physics Learner (SPL) machine,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "empowered by MCTS, for discovery of nonlinear dynamics. This architecture relies on a grammar", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "composed of (i) computational rules and symbols to guide the search tree spanning and (ii) a composite", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "objective rewarding function to simultaneously evaluate the generated equations with observational", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "data and enforce the sparsity of the expression. 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Under this framework, a symbolic regressor is es-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "tablished to identify the governing equations that best describe the underlying physics through free", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 506, + 221 + ], + "score": 1.0, + "content": "combination of mathematical operators and symbols, leading to great flexibility in model selection.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "One essential weakness of this early methodology is that, driven exclusively by the goal of empirically", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "seeking the best-fitting expression (e.g. minimizing the mean-square error) in a genetic expansion", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "process, the GP-based model usually over-fits the target system with numerous false-positive terms", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 507, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 507, + 267 + ], + "score": 1.0, + "content": "under data noise, even sometimes at a subtle level, causing huge instability and uncertainty. However,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "this ingenious idea has inspired a series of subsequent endeavors (Cornforth & Lipson, 2012; Gaucel", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "et al., 2014; Ly & Lipson, 2012; Quade et al., 2016; Vaddireddy et al., 2020). In a more recent work,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "Deep Symbolic Regression (DSR) (Petersen et al., 2021; Mundhenk et al., 2021), a reinforcement", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "learning-based model was established and generally outperformed the GP based models including the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "commercial Eureqa software (Langdon & Gustafson, 2010). Additionally, the AI-Feynman methods", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "(Udrescu & Tegmark, 2020; Udrescu et al., 2020; Udrescu & Tegmark, 2021) ameliorated symbolic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "regression for distilling physics laws from data by combining neural network fitting with a suite of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "physics-inspired techniques. This approach is also highlighted by a recursive decomposition of a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "complicated mathematical expression into different parts on a tree-based graph, which disentangles", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "the original problem and speeds up the discovery. It outperformed Eureqa in the uncovering Feynman", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "physics equations (Feynman et al., 1965). However, this approach is built upon ad-hoc steps and, to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 351, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 351, + 398 + ], + "score": 1.0, + "content": "some extent, lacks flexible automation in equation discovery.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 154, + 507, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "The popularity of adopting the tree-based symbolic reasoning of mathematical formulas (Lample", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "& Charton, 2019) has been rising recently to discover unknown mathematical expressions with a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "reinforcement learning agent (Kubalík et al., 2019; Petersen et al., 2021; Mundhenk et al., 2021).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "However, some former work attempting to apply the Monte Carlo tree search (MCTS) algorithm as an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "alternative to GP for symbolic regression (Cazenave, 2013; White et al., 2015; Islam et al., 2018; Lu", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "et al., 2021) failed to leverage the full flexibility of this algorithm, resulting in the similar shortage that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "GP-based symbolic regressors possess as discussed earlier. 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A sound", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "mathematical underpinning for the trade-off between exploration and exploitation is remarkably", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "advantageous as well. 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Any mathematical expression can be represented by a combinatorial set of symbols", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "and mathematical operations, and further expressed by a parse tree structure (Hopcroft et al., 2006;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "Kusner et al., 2017) empowered by a context-free grammar (CFG). A CFG is a formal grammar", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 353, + 128 + ], + "score": 1.0, + "content": "characterized by a tuple comprised of 4 elements, namely,", + "type": "text" + }, + { + "bbox": [ + 354, + 115, + 426, + 127 + ], + "score": 0.92, + "content": "\\mathcal { G } = ( V , \\Sigma , R , S )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 115, + 460, + 128 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 460, + 115, + 469, + 126 + ], + "score": 0.75, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "denotes", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 251, + 138 + ], + "score": 1.0, + "content": "a finite set of non-terminal nodes,", + "type": "text" + }, + { + "bbox": [ + 252, + 127, + 260, + 137 + ], + "score": 0.75, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 127, + 388, + 138 + ], + "score": 1.0, + "content": "a finite set of terminal nodes,", + "type": "text" + }, + { + "bbox": [ + 388, + 127, + 398, + 137 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "a finite set of production", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 504, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 416, + 149 + ], + "score": 1.0, + "content": "rules, each interpreted as a mapping from a single non-terminal symbol in", + "type": "text" + }, + { + "bbox": [ + 417, + 138, + 426, + 147 + ], + "score": 0.72, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 137, + 504, + 149 + ], + "score": 1.0, + "content": "to one or multiple", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 243, + 162 + ], + "score": 1.0, + "content": "terminal/non-terminal node(s) in", + "type": "text" + }, + { + "bbox": [ + 243, + 149, + 282, + 160 + ], + "score": 0.92, + "content": "( V \\cup \\Sigma ) ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 147, + 311, + 162 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 311, + 150, + 318, + 158 + ], + "score": 0.79, + "content": "^ *", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 147, + 488, + 162 + ], + "score": 1.0, + "content": "represents the Kleene star operation, and", + "type": "text" + }, + { + "bbox": [ + 489, + 149, + 497, + 159 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "single non-terminal node standing for a start symbol. 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The non-terminal nodes", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "between root and leaf nodes are represented by some symbols distinct from the start and terminal", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 151, + 226 + ], + "score": 1.0, + "content": "nodes (i.e.,", + "type": "text" + }, + { + "bbox": [ + 152, + 214, + 164, + 225 + ], + "score": 0.62, + "content": "M _ { ☉ }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "). The production rules denote the commonly seen mathematical operators: unary rules", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 363, + 239 + ], + "score": 1.0, + "content": "(one non-terminal node mapping to one node) for operators like", + "type": "text" + }, + { + "bbox": [ + 363, + 225, + 456, + 237 + ], + "score": 0.4, + "content": "\\cos ( \\cdot ) , \\exp ( \\cdot ) , \\bar { \\log ( | \\cdot | ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 224, + 505, + 239 + ], + "score": 1.0, + "content": ", and binary", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 395, + 249 + ], + "score": 1.0, + "content": "rules (one non-terminal node mapping to two nodes) for operators such", + "type": "text" + }, + { + "bbox": [ + 396, + 237, + 450, + 247 + ], + "score": 0.61, + "content": "\\mathrm { ~ \\imath s ~ } + , - , \\times , \\dot { \\mathrm { ~ \\cdot ~ } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 236, + 506, + 249 + ], + "score": 1.0, + "content": ". A parse tree", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 384, + 260 + ], + "score": 1.0, + "content": "is then generated via a pre-order traversal of production rules rooted at", + "type": "text" + }, + { + "bbox": [ + 384, + 248, + 392, + 258 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "and terminates when all leaf", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "nodes are entirely filled with terminal symbols. Each mathematical expression can be represented by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 270, + 264, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 264, + 281 + ], + "score": 1.0, + "content": "such a traversal set of production rules.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "Monte Carlo tree search. Monte Carlo tree search (MCTS) (Coulom, 2006) is an algorithm for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "searching optimal decisions in large combinatorial spaces represented by search trees. This technique", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "complies with the best-first search principle based on the evaluations of stochastic simulations. It", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "has already been widely employed in and proved the spectacular success by various gaming artificial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "intelligence systems, including the famous AlphaGo and AlphaZero (Silver et al., 2017) for computer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 458, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 458, + 355 + ], + "score": 1.0, + "content": "Go game. A basic MCTS algorithm is composed of an iterative process with four steps:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 129, + 361, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 130, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 130, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "1. Selection. 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After expansion, if the current node is non-terminal, the agent performs one or", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 142, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "multiple independent simulations starting from the current node until reaching the terminal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 448, + 351, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 351, + 460 + ], + "score": 1.0, + "content": "state. In this process, actions are randomly selected.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 128, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 128, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "4. Backpropagation. Statistics of nodes along the path from the current node to the root are", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 141, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "updated with respect to search results (scores evaluated from the terminate states reached).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 108, + 495, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "To maintain a proper balance between the less-tested paths and the best policy identified so far, the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "MCTS agent sticks to a trade-off between exploration and exploitation by taking action that maximizes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 517, + 507, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 507, + 530 + ], + "score": 1.0, + "content": "the Upper Confidence Bounds applied for Trees (UCT), formulated as (Kocsis & Szepesvári, 2006):", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 534, + 401, + 550 + ], + "lines": [ + { + "bbox": [ + 211, + 534, + 401, + 550 + ], + "spans": [ + { + "bbox": [ + 211, + 534, + 401, + 550 + ], + "score": 0.93, + "content": "U C T ( s , a ) = Q ( s , a ) + c \\sqrt { \\ln [ N ( s ) ] / N ( s , a ) }", + "type": "interline_equation", + "image_path": "b873fd134844f8973176b8e8dde3cf6d2e7009e599d08f7ae19b941e6903ae79.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 211, + 534, + 401, + 550 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 132, + 568 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 555, + 164, + 568 + ], + "score": 0.93, + "content": "Q ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 556, + 344, + 568 + ], + "score": 1.0, + "content": "is the average result/reward of playing action", + "type": "text" + }, + { + "bbox": [ + 344, + 558, + 351, + 565 + ], + "score": 0.76, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 556, + 382, + 568 + ], + "score": 1.0, + "content": "in state", + "type": "text" + }, + { + "bbox": [ + 383, + 558, + 388, + 565 + ], + "score": 0.75, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "in the simulations performed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 384, + 579 + ], + "score": 1.0, + "content": "in the history, encouraging the exploitation of current best child node;", + "type": "text" + }, + { + "bbox": [ + 385, + 567, + 407, + 579 + ], + "score": 0.9, + "content": "N ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "is number of times state", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 578, + 504, + 593 + ], + "spans": [ + { + "bbox": [ + 107, + 583, + 112, + 589 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 578, + 145, + 593 + ], + "score": 1.0, + "content": "visited,", + "type": "text" + }, + { + "bbox": [ + 145, + 579, + 177, + 591 + ], + "score": 0.93, + "content": "N ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 578, + 286, + 593 + ], + "score": 1.0, + "content": "the number of times action", + "type": "text" + }, + { + "bbox": [ + 286, + 582, + 293, + 589 + ], + "score": 0.7, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 578, + 395, + 593 + ], + "score": 1.0, + "content": "has been selected at state", + "type": "text" + }, + { + "bbox": [ + 396, + 582, + 401, + 589 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 578, + 422, + 593 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 423, + 579, + 504, + 592 + ], + "score": 0.91, + "content": "\\sqrt { \\ln [ N ( s ) ] / N ( s , a ) }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 414, + 603 + ], + "score": 1.0, + "content": "consequently encourages exploration of less-visited child nodes. Constant", + "type": "text" + }, + { + "bbox": [ + 414, + 595, + 419, + 600 + ], + "score": 0.69, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "controls the balance", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "between exploration and exploitation, empirically defined upon the specific problem. Theoretical", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 484, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 484, + 626 + ], + "score": 1.0, + "content": "analysis of UCT-based MCTS (e.g., convergence, guarantees) is referred to Shah et al. (2019).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 178, + 653 + ], + "lines": [ + { + "bbox": [ + 104, + 638, + 181, + 656 + ], + "spans": [ + { + "bbox": [ + 104, + 638, + 181, + 656 + ], + "score": 1.0, + "content": "3 METHODS", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Existing studies show that the MCTS agent continuously gains knowledge of specified tasks via the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "expansion of the search tree and, based on the backpropagation of evaluation results (i.e., rewards and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "number of visits), render a proper selection policy on visited states to guide the upcoming searching", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "(Silver et al., 2017). 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Any mathematical expression can be represented by a combinatorial set of symbols", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "and mathematical operations, and further expressed by a parse tree structure (Hopcroft et al., 2006;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "Kusner et al., 2017) empowered by a context-free grammar (CFG). 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The non-terminal nodes", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "between root and leaf nodes are represented by some symbols distinct from the start and terminal", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 151, + 226 + ], + "score": 1.0, + "content": "nodes (i.e.,", + "type": "text" + }, + { + "bbox": [ + 152, + 214, + 164, + 225 + ], + "score": 0.62, + "content": "M _ { ☉ }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "). 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This technique", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "complies with the best-first search principle based on the evaluations of stochastic simulations. It", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "has already been widely employed in and proved the spectacular success by various gaming artificial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "intelligence systems, including the famous AlphaGo and AlphaZero (Silver et al., 2017) for computer", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 458, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 458, + 355 + ], + "score": 1.0, + "content": "Go game. 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Constant", + "type": "text" + }, + { + "bbox": [ + 414, + 595, + 419, + 600 + ], + "score": 0.69, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "controls the balance", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "between exploration and exploitation, empirically defined upon the specific problem. Theoretical", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 484, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 484, + 626 + ], + "score": 1.0, + "content": "analysis of UCT-based MCTS (e.g., convergence, guarantees) is referred to Shah et al. 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The graph", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 336, + 434, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 434, + 351 + ], + "score": 1.0, + "content": "explains the 4 MCTS phases of one learning episode with an illustrative example.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "pointing to the best solution. This algorithm is depicted in Figure 1 with an illustrative example and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 378, + 504, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 504, + 390 + ], + "score": 1.0, + "content": "its overall training scheme is shown in Algorithm 1. 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This numerator arrangement is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 529, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 504, + 540 + ], + "score": 1.0, + "content": "designated to penalize non-parsimonious solutions. This rewarding formulation outputs a reasonable", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "assessment to the distilled equations and encourages parsimonious solution by discounting the reward", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "of a non-parsimonious one. The rooted mean square error (RMSE) in denominator evaluates the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 561, + 391, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 391, + 573 + ], + "score": 1.0, + "content": "goodness-of-fit of the discovered equation w.r.t. the measurement data.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 232, + 592 + ], + "score": 1.0, + "content": "Training scheme. A grammar", + "type": "text" + }, + { + "bbox": [ + 233, + 578, + 302, + 590 + ], + "score": 0.92, + "content": "\\mathcal { G } = ( V , \\Sigma , R , S )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 577, + 506, + 592 + ], + "score": 1.0, + "content": "is defined with appropriate nodes and production", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 587, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 506, + 603 + ], + "score": 1.0, + "content": "rules to cover all possible forms of equations. To keep track of non-terminal nodes of the parsing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "score": 1.0, + "content": "tree, we apply a last-in-first-out (LIFO) strategy and denote the non-terminal node placed last on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 145, + 624 + ], + "score": 1.0, + "content": "the stack", + "type": "text" + }, + { + "bbox": [ + 145, + 612, + 163, + 621 + ], + "score": 0.74, + "content": "N T", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 611, + 360, + 624 + ], + "score": 1.0, + "content": "as the current node. We define the action space", + "type": "text" + }, + { + "bbox": [ + 360, + 611, + 391, + 621 + ], + "score": 0.91, + "content": "A = R", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 611, + 471, + 624 + ], + "score": 1.0, + "content": "and the state space", + "type": "text" + }, + { + "bbox": [ + 471, + 612, + 480, + 621 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "as all", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "possible traversals of complete/incomplete parse trees (i.e., production rules selected) in ordered", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 230, + 646 + ], + "score": 1.0, + "content": "sequences. 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The graph", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 336, + 434, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 434, + 351 + ], + "score": 1.0, + "content": "explains the 4 MCTS phases of one learning episode with an illustrative example.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 401 + ], + "lines": [], + "index": 6, + "bbox_fs": [ + 105, + 367, + 505, + 402 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 330, + 420 + ], + "score": 1.0, + "content": "Rewarding. To evaluate the mathematical expression", + "type": "text" + }, + { + "bbox": [ + 330, + 406, + 338, + 419 + ], + "score": 0.86, + "content": "\\tilde { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "projected from a parse tree, we define a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 179, + 432 + ], + "score": 1.0, + "content": "numerical reward", + "type": "text" + }, + { + "bbox": [ + 180, + 420, + 227, + 430 + ], + "score": 0.91, + "content": "r \\in \\mathcal { R } \\subset \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 419, + 388, + 432 + ], + "score": 1.0, + "content": "based on this expression and input data", + "type": "text" + }, + { + "bbox": [ + 389, + 419, + 443, + 432 + ], + "score": 0.94, + "content": "{ \\mathcal { D } } = \\{ { \\bf Y } ; \\dot { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 419, + 505, + 432 + ], + "score": 1.0, + "content": ", serving as the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 431, + 391, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 391, + 442 + ], + "score": 1.0, + "content": "search result of the current expansion or simulation. 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This numerator arrangement is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 529, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 504, + 540 + ], + "score": 1.0, + "content": "designated to penalize non-parsimonious solutions. This rewarding formulation outputs a reasonable", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "assessment to the distilled equations and encourages parsimonious solution by discounting the reward", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "of a non-parsimonious one. The rooted mean square error (RMSE) in denominator evaluates the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 561, + 391, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 391, + 573 + ], + "score": 1.0, + "content": "goodness-of-fit of the discovered equation w.r.t. the measurement data.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 480, + 507, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 232, + 592 + ], + "score": 1.0, + "content": "Training scheme. A grammar", + "type": "text" + }, + { + "bbox": [ + 233, + 578, + 302, + 590 + ], + "score": 0.92, + "content": "\\mathcal { G } = ( V , \\Sigma , R , S )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 577, + 506, + 592 + ], + "score": 1.0, + "content": "is defined with appropriate nodes and production", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 587, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 104, + 587, + 506, + 603 + ], + "score": 1.0, + "content": "rules to cover all possible forms of equations. To keep track of non-terminal nodes of the parsing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "score": 1.0, + "content": "tree, we apply a last-in-first-out (LIFO) strategy and denote the non-terminal node placed last on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 145, + 624 + ], + "score": 1.0, + "content": "the stack", + "type": "text" + }, + { + "bbox": [ + 145, + 612, + 163, + 621 + ], + "score": 0.74, + "content": "N T", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 611, + 360, + 624 + ], + "score": 1.0, + "content": "as the current node. 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At the current state", + "type": "text" + }, + { + "bbox": [ + 231, + 633, + 304, + 645 + ], + "score": 0.93, + "content": "s _ { t } = [ a _ { 1 } , a _ { 2 } , . . . a _ { t } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 633, + 331, + 646 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 332, + 633, + 357, + 643 + ], + "score": 0.91, + "content": "t \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "is the discrete traversal step-index of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the upcoming production rule, the MCTS agent masks out the invalid production rules for current", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 372, + 668 + ], + "score": 1.0, + "content": "non-terminal node and on that basis selects a valid rule as action", + "type": "text" + }, + { + "bbox": [ + 373, + 656, + 393, + 667 + ], + "score": 0.89, + "content": "a _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "(i.e, the left-hand side of a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "valid production rule is the current non-terminal symbol). 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The range of", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 225, + 554 + ], + "score": 1.0, + "content": "constant coefficient values is", + "type": "text" + }, + { + "bbox": [ + 225, + 542, + 263, + 554 + ], + "score": 0.83, + "content": "[ - 1 0 , 1 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 542, + 505, + 554 + ], + "score": 1.0, + "content": ". For 16 different data noise and scarcity levels, each model", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 416, + 566 + ], + "score": 1.0, + "content": "was performed 100 independent trails. The recovery rates are displayed as a", + "type": "text" + }, + { + "bbox": [ + 417, + 553, + 441, + 564 + ], + "score": 0.89, + "content": "4 \\times 4", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "mesh grid w.r.t.", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "different noise/scarcity levels in Figure 3. It is observed that the SPL machine outperforms the", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "GP-based symbolic regressor in all the cases. 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Some variants of Nguyen’s", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "benchmark equations are also considered in this experiment. Their discoveries require numerical", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "estimation of the constant coefficient values. Each equation generates two datasets: one for training", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "and another for testing. The discovered equation that perfectly fits the testing data is regarded as a", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 378, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 506, + 389 + ], + "score": 1.0, + "content": "successful discovery (i.e., the discovered equation should be identical or equivalent to the target one).", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 388, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 506, + 400 + ], + "score": 1.0, + "content": "The recovery rate is calculated based on 100 independent tests for each task. 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The hyperparameters of the", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "GP-based symbolic regressor are the same as those mentioned in Section 4.1; configurations of the", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "NGGP models are obtained from its source code2; detailed setting of the benchmark tasks and the", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "SPL model is described in Appendix Section B. The success rates are shown in Table 1. 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BenchmarkExpressionSPLNGGPGP
Nguyen-1x²+x²+x100%100%99%
Nguyen-2x²+x²+x²+x100%100%90%
Nguyen-3+x4+x²+x²+x 25100%100%34%
Nguyen-4x+x+x4+x²+x²+x99%100%54%
Nguyen-5sin(x²) cos(𝑥) -195%80%12%
Nguyen-6sin(x²)+sin(x+x²)100%100%11%
Nguyen-7ln(𝑥 +1) + ln(x²+ 1)100%100%17%
Nguyen-8√x100%100%100%
Nguyen-9sin(x) + sin(y²)100%100%76%
Nguyen-102 sin(x) cos(y)100%100%86%
Nguyen-11xy100%100%13%
Nguyen-12x4-x²+¹y²-y28%4%0%
Nguyen-1c3.39x +2.12x² +1.78x100%100%0%
Nguyen-2c0.48x4 +3.39x+2.12x² +1.78x94%100%0%
Nguyen-5csin(x²) cos(x) -0.7595%98%1%
Nguyen-8c√1.23x100%100%56%
Nguyen-9csin(1.5x) + sin(0.5y²)96%90%0%
Average94.5%92.4%38.2%
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It is observed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "that the SPL machine and the NGGP model both produce reliable results in Nguyen’s benchmark", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "problems and the SPL machine slightly outperforms NGGP. This experiment betokens the capacity", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 497, + 379, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 379, + 511 + ], + "score": 1.0, + "content": "of the SPL machine in discovery of equations with divergent forms.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "Ablation Study: We consider four ablation studies by removing: (a) the adaptive scaling in reward", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 243, + 538 + ], + "score": 1.0, + "content": "calculation, (b) the discount factor", + "type": "text" + }, + { + "bbox": [ + 243, + 526, + 254, + 537 + ], + "score": 0.89, + "content": "\\eta ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "that drives equation parsimony in Eq. 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Physics ModelDerived model expression
Model-1H(t)= co+cit + c2t²+c3t3
Model-2H(t)= Co +Cit + C2eCt
Model-3H(t)= co + c1 log(cosh(c2t))
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BenchmarkExpressionSPLNGGPGP
Nguyen-1x²+x²+x100%100%99%
Nguyen-2x²+x²+x²+x100%100%90%
Nguyen-3+x4+x²+x²+x 25100%100%34%
Nguyen-4x+x+x4+x²+x²+x99%100%54%
Nguyen-5sin(x²) cos(𝑥) -195%80%12%
Nguyen-6sin(x²)+sin(x+x²)100%100%11%
Nguyen-7ln(𝑥 +1) + ln(x²+ 1)100%100%17%
Nguyen-8√x100%100%100%
Nguyen-9sin(x) + sin(y²)100%100%76%
Nguyen-102 sin(x) cos(y)100%100%86%
Nguyen-11xy100%100%13%
Nguyen-12x4-x²+¹y²-y28%4%0%
Nguyen-1c3.39x +2.12x² +1.78x100%100%0%
Nguyen-2c0.48x4 +3.39x+2.12x² +1.78x94%100%0%
Nguyen-5csin(x²) cos(x) -0.7595%98%1%
Nguyen-8c√1.23x100%100%56%
Nguyen-9csin(1.5x) + sin(0.5y²)96%90%0%
Average94.5%92.4%38.2%
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Physics ModelDerived model expression
Model-1H(t)= co+cit + c2t²+c3t3
Model-2H(t)= Co +Cit + C2eCt
Model-3H(t)= co + c1 log(cosh(c2t))
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baseball0.32.79894.5893.507
blue basketball0.4570.51369.2092.227
green basketball0.0880.185.4351.604
volleyball0.1110.57480.9650.76
bowling ball0.0030.3387.023.167
golf ball0.0090.21486.0931.684
tennis ball0.0910.24672.2780.161
whiffle ball 11.581.61965.4260.21
whiffle ball 20.0990.62858.5330.966
yellow whiffle ball0.42817.34144.9842.57
orange whiffle ball0.7450.37936.7653.257
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Model宝y之
EureqaYes (1)Yes (3)Yes (1)
pySINDyYes (1)No (N/A)Yes (2)
NGGPYes (10)Yes (8)Yes (16)
SPLYes (0)Yes (0)Yes (0)
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yellow whiffle ball0.42817.34144.9842.57
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Model宝y之
EureqaYes (1)Yes (3)Yes (1)
pySINDyYes (1)No (N/A)Yes (2)
NGGPYes (10)Yes (8)Yes (16)
SPLYes (0)Yes (0)Yes (0)
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It is evident", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "that the SPL machine is capable of distilling the concise symbolic combination of operators and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "variables to correctly formulate parsimonious mathematical expressions that govern the Lorenz", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 418, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 418, + 194 + ], + "score": 1.0, + "content": "system, outperforming the baseline methods of Eureqa, pySINDy and NGGP.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 207, + 443, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 445, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 445, + 222 + ], + "score": 1.0, + "content": "7 EXPERIMENTAL DYNAMICS DISCOVERY: DOUBLE PENDULUM", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 105, + 231, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "This section shows SPL-based discovery of a chaotic double pendulum system with experimental", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 256 + ], + "score": 1.0, + "content": "data (Asseman et al., 2018) as shown in Appendix Figure E.3. The governing equations are given by:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 255, + 447, + 288 + ], + "lines": [ + { + "bbox": [ + 161, + 255, + 446, + 288 + ], + "spans": [ + { + "bbox": [ + 161, + 255, + 446, + 288 + ], + "score": 0.68, + "content": "\\begin{array} { r l } & { \\dot { \\omega } _ { 1 } = c _ { 1 } \\dot { \\omega } _ { 2 } \\cos ( \\Delta \\theta ) + c _ { 2 } \\omega _ { 2 } ^ { 2 } \\sin ( \\Delta \\theta ) + c _ { 3 } \\sin ( \\theta _ { 1 } ) + \\mathcal { R } _ { 1 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } ) , } \\\\ & { \\dot { \\omega } _ { 2 } = c _ { 1 } \\dot { \\omega } _ { 1 } \\cos ( \\Delta \\theta ) + c _ { 2 } \\omega _ { 1 } ^ { 2 } \\sin ( \\Delta \\theta ) + c _ { 3 } \\sin ( \\theta _ { 2 } ) + \\mathcal { R } _ { 2 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "4277b4b26af472757877edfbfe2b1ed201a2ca2fb690ffae1bb359f3358633ea.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 161, + 255, + 447, + 266.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 161, + 266.0, + 447, + 277.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 161, + 277.0, + 447, + 288.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 289, + 504, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 134, + 302 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 290, + 161, + 302 + ], + "score": 0.32, + "content": "\\theta _ { 1 } , \\theta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 289, + 309, + 302 + ], + "score": 1.0, + "content": "denote the angular displacements;", + "type": "text" + }, + { + "bbox": [ + 309, + 288, + 349, + 301 + ], + "score": 0.84, + "content": "\\omega _ { 1 } = \\dot { \\theta } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 289, + 354, + 302 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 354, + 288, + 393, + 301 + ], + "score": 0.86, + "content": "\\omega _ { 2 } ~ = ~ { \\dot { \\theta } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 289, + 458, + 302 + ], + "score": 1.0, + "content": "the velocities;", + "type": "text" + }, + { + "bbox": [ + 458, + 290, + 487, + 301 + ], + "score": 0.68, + "content": "\\dot { \\omega } _ { 1 } , \\dot { \\omega } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 464, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 164, + 314 + ], + "score": 1.0, + "content": "accelerations;", + "type": "text" + }, + { + "bbox": [ + 164, + 302, + 178, + 312 + ], + "score": 0.89, + "content": "\\mathcal { R } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 300, + 196, + 314 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 196, + 302, + 210, + 312 + ], + "score": 0.9, + "content": "\\mathcal { R } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 300, + 401, + 314 + ], + "score": 1.0, + "content": "denote the unknown damping forces. Note that", + "type": "text" + }, + { + "bbox": [ + 401, + 301, + 459, + 312 + ], + "score": 0.93, + "content": "\\Delta \\theta = \\theta _ { 1 } - \\theta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 300, + 464, + 314 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "The data source contains multiple camera-sensed datasets. 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Production rules fulfilling the above prior knowledge are exhibited in Appendix Section", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 250, + 420 + ], + "score": 1.0, + "content": "E.2. 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(5),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "are successfully distilled and damping terms are estimated. This set of equation is validated through", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "interpolation on the testing set and compared with the smoothed derivatives, as shown in Appendix", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 506, + 485, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 485, + 518 + ], + "score": 1.0, + "content": "Figure E.4. The solution appears felicitous as the governing equations of the testing responses.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 108, + 532, + 288, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 290, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 290, + 547 + ], + "score": 1.0, + "content": "8 CONCLUSION AND DISCUSSION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "This paper introduces a Symbolic Physics Learner (SPL) machine to tackle the challenge of distill-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "ing the mathematical structure of equations for physical systems (e.g., nonlinear dynamics) with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "scarce/noisy data. This framework is built upon the expression tree interpretation of mathematical", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "operations and variables and an MCTS agent that searches for the optimistic policy to reconstruct", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "score": 1.0, + "content": "the target mathematical formula. With some remarkable adjustments to the MCTS algorithms, the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "SPL model straightforwardly accepts our prior or domain knowledge, or any sort of constraints of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "the tasks in the grammar design while leveraging great flexibility in expression formulation. The", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "score": 1.0, + "content": "robustness of the proposed SPL machine for complex target expression discovery within a large", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "search space is indicated in the Nguyen’s symbolic regression benchmark problems, where the SPL", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "machine outperforms state-of-the-art symbolic regression methods. Moreover, encouraging results", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "are obtained in the tasks of discovering physical laws and nonlinear dynamics, based on synthetic", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "or experimental datasets. While the proposed SPL machine shows huge potential in both symbolic", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "regression and physical law discovery tasks, there are still some imperfections that can be improved:", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "(i) the computational cost is high for constant coefficient value estimation due to repeated calls for an", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "optimization process, (ii) graph modularity is underexamined, and (iii) robustness against extreme", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 507, + 733 + ], + "score": 1.0, + "content": "data noise and scarcity is not optimal. These limitations are further explained in Appendix Section F.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 43.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 193 + ], + "lines": [], + "index": 4.5, + "bbox_fs": [ + 105, + 82, + 506, + 194 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 207, + 443, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 445, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 445, + 222 + ], + "score": 1.0, + "content": "7 EXPERIMENTAL DYNAMICS DISCOVERY: DOUBLE PENDULUM", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 105, + 231, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "This section shows SPL-based discovery of a chaotic double pendulum system with experimental", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 256 + ], + "score": 1.0, + "content": "data (Asseman et al., 2018) as shown in Appendix Figure E.3. The governing equations are given by:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 231, + 505, + 256 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 255, + 447, + 288 + ], + "lines": [ + { + "bbox": [ + 161, + 255, + 446, + 288 + ], + "spans": [ + { + "bbox": [ + 161, + 255, + 446, + 288 + ], + "score": 0.68, + "content": "\\begin{array} { r l } & { \\dot { \\omega } _ { 1 } = c _ { 1 } \\dot { \\omega } _ { 2 } \\cos ( \\Delta \\theta ) + c _ { 2 } \\omega _ { 2 } ^ { 2 } \\sin ( \\Delta \\theta ) + c _ { 3 } \\sin ( \\theta _ { 1 } ) + \\mathcal { R } _ { 1 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } ) , } \\\\ & { \\dot { \\omega } _ { 2 } = c _ { 1 } \\dot { \\omega } _ { 1 } \\cos ( \\Delta \\theta ) + c _ { 2 } \\omega _ { 1 } ^ { 2 } \\sin ( \\Delta \\theta ) + c _ { 3 } \\sin ( \\theta _ { 2 } ) + \\mathcal { R } _ { 2 } ( \\theta _ { 1 } , \\theta _ { 2 } , \\dot { \\theta } _ { 1 } , \\dot { \\theta } _ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "4277b4b26af472757877edfbfe2b1ed201a2ca2fb690ffae1bb359f3358633ea.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 161, + 255, + 447, + 266.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 161, + 266.0, + 447, + 277.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 161, + 277.0, + 447, + 288.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 289, + 504, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 134, + 302 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 290, + 161, + 302 + ], + "score": 0.32, + "content": "\\theta _ { 1 } , \\theta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 289, + 309, + 302 + ], + "score": 1.0, + "content": "denote the angular displacements;", + "type": "text" + }, + { + "bbox": [ + 309, + 288, + 349, + 301 + ], + "score": 0.84, + "content": "\\omega _ { 1 } = \\dot { \\theta } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 289, + 354, + 302 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 354, + 288, + 393, + 301 + ], + "score": 0.86, + "content": "\\omega _ { 2 } ~ = ~ { \\dot { \\theta } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 289, + 458, + 302 + ], + "score": 1.0, + "content": "the velocities;", + "type": "text" + }, + { + "bbox": [ + 458, + 290, + 487, + 301 + ], + "score": 0.68, + "content": "\\dot { \\omega } _ { 1 } , \\dot { \\omega } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 464, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 164, + 314 + ], + "score": 1.0, + "content": "accelerations;", + "type": "text" + }, + { + "bbox": [ + 164, + 302, + 178, + 312 + ], + "score": 0.89, + "content": "\\mathcal { R } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 300, + 196, + 314 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 196, + 302, + 210, + 312 + ], + "score": 0.9, + "content": "\\mathcal { R } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 300, + 401, + 314 + ], + "score": 1.0, + "content": "denote the unknown damping forces. 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Production rules fulfilling the above prior knowledge are exhibited in Appendix Section", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 250, + 420 + ], + "score": 1.0, + "content": "E.2. 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(5),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "are successfully distilled and damping terms are estimated. This set of equation is validated through", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "interpolation on the testing set and compared with the smoothed derivatives, as shown in Appendix", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 506, + 485, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 485, + 518 + ], + "score": 1.0, + "content": "Figure E.4. The solution appears felicitous as the governing equations of the testing responses.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 106, + 472, + 506, + 518 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 532, + 288, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 290, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 290, + 547 + ], + "score": 1.0, + "content": "8 CONCLUSION AND DISCUSSION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "This paper introduces a Symbolic Physics Learner (SPL) machine to tackle the challenge of distill-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "ing the mathematical structure of equations for physical systems (e.g., nonlinear dynamics) with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "scarce/noisy data. This framework is built upon the expression tree interpretation of mathematical", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "operations and variables and an MCTS agent that searches for the optimistic policy to reconstruct", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "score": 1.0, + "content": "the target mathematical formula. With some remarkable adjustments to the MCTS algorithms, the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "SPL model straightforwardly accepts our prior or domain knowledge, or any sort of constraints of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "the tasks in the grammar design while leveraging great flexibility in expression formulation. The", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 647 + ], + "score": 1.0, + "content": "robustness of the proposed SPL machine for complex target expression discovery within a large", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "search space is indicated in the Nguyen’s symbolic regression benchmark problems, where the SPL", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "machine outperforms state-of-the-art symbolic regression methods. 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Feature engineering and symbolic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 115, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "regression methods for detecting hidden physics from sparse sensor observation data. Physics of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 507, + 230, + 518 + ], + "spans": [ + { + "bbox": [ + 115, + 507, + 230, + 518 + ], + "score": 1.0, + "content": "Fluids, 32(1):015113, 2020.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 484, + 506, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 525, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "David R White, Shin Yoo, and Jeremy Singer. The programming game: evaluating mcts as an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 116, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 116, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "alternative to gp for symbolic regression. 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Empirically, setting", + "type": "text" + }, + { + "bbox": [ + 341, + 146, + 388, + 157 + ], + "score": 0.9, + "content": "\\eta = 0 . 9 9 9 9", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "ensures the scores of ground", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 154, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 170 + ], + "score": 1.0, + "content": "truth equations stand out and successfully enforces the sparsity in all experiments. For the discovery", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "of very chaotic dynamical systems based on measurement data, we expect some physics terms from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "the governing equations to have a weak impact on the state variables (e.g., in the chaotic double", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 363, + 201 + ], + "score": 1.0, + "content": "pendulum system experiments, the effects from physics terms", + "type": "text" + }, + { + "bbox": [ + 363, + 189, + 393, + 200 + ], + "score": 0.88, + "content": "\\sin ( \\theta _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 189, + 413, + 201 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 413, + 189, + 443, + 200 + ], + "score": 0.9, + "content": "\\mathrm { s i n } ( \\theta _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "are hard to be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 367, + 213 + ], + "score": 1.0, + "content": "captured), and the effect of data noise is unknown. Hence, we set", + "type": "text" + }, + { + "bbox": [ + 368, + 201, + 393, + 211 + ], + "score": 0.91, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 199, + 505, + 213 + ], + "score": 1.0, + "content": "to leverage the full strength", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "of data fitting to enable the detection of physics terms that are offset or overwhelmed by data noise", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 234 + ], + "score": 1.0, + "content": "but are pivotal to the systems. Nevertheless, we must acknowledge that this selection process is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 478, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 478, + 245 + ], + "score": 1.0, + "content": "empirical, which depends on our desire for the degree of parsimony of the target equation(s).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 506, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "As for the hyperparameters in the training schema (i.e., maximum module transplantation, episodes,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "score": 1.0, + "content": "maximum tree size, maximum augmented grammars), we have conducted parametric convergence", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 507, + 285 + ], + "score": 1.0, + "content": "tests for each experiment to ensure the learning curves (i.e., maximum scores in the history) converge.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 483, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 483, + 295 + ], + "score": 1.0, + "content": "For example, as discussed in Section B, Table B.2 shows the setting of these hyperparameters.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 311, + 315, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 316, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 316, + 325 + ], + "score": 1.0, + "content": "B NGUYEN’S BENCHMARK PROBLEMS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 504, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "This section provides more detailed experiment settings for the Nguyen’s benchmark tasks that are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "described in Section 4.2 of the main text, where the training hyperparameters for the SPL machine in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "these equation discovery experiments are also listed. Table B.1 presents the candidate mathematical", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "operations allowed for three tested models and Table B.2 displays training hyperparameters for the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 380, + 299, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 299, + 392 + ], + "score": 1.0, + "content": "SPL machine in the Nguyen’s benchmark tasks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "Moreover, the utilization of CFG in the SPL machine facilitates the flexibility of applying some prior", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "knowledge including the universal mathematical rules and constraints. This feature empirically turns", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "out to be an accessible and scalable approach for avoiding meaningless mathematical expressions.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "In the Nguyen’s benchmark tasks, one or multiple mathematical constraints are given to the SPL", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 249, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 249, + 453 + ], + "score": 1.0, + "content": "machine. 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Only variables and constant values are allowed in trigonometric functions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "table", + "bbox": [ + 163, + 518, + 440, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 104, + 501, + 500, + 513 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 500, + 502, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 502, + 514 + ], + "score": 1.0, + "content": "Table B.1: Candidate operators for each Nguyen’s benchmark task. const denotes constant values.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "table_body", + "bbox": [ + 163, + 518, + 440, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 163, + 518, + 440, + 730 + ], + "spans": [ + { + "bbox": [ + 163, + 518, + 440, + 730 + ], + "score": 0.866, + "html": "
Benchmark Candidate Operations
Nguyen-1 +,-,×,÷,cos(-),sin(-),exp(·),sin(+,-,×,÷,cos(-),sin(-),exp(-),log(),√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√+,-,×,÷,cos(-),sin(-),exp(-),log()，√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√,const
Nguyen-2 +,-,×,÷,cos(·),sin(.),exp()
Nguyen-3×,÷,cos(·,sin(·
Nguyen-4X，÷,cos(,sin(·
Nguyen-5×，÷,cos(·,sin(·
Nguyen-6X,÷,cos(·,sin(·
Nguyen-7×÷,cos(·,sin(·,exp(·
Nguyen-8×÷,cos(·,sin(·,exp(·
Nguyen-9×÷cos(·,sin(·,exp(·
Nguyen-10×÷Cos,sin(·,exp(.
Nguyen-11×÷Cos,sin(·,exp(.
Nguyen-12×÷Cos,sin(·,exp(.
Nguyen-1c×÷Cos,sin(·,exp(.
Nguyen-2c×÷cos(.,sin(·,exp(.
Nguyen-5c十，一×÷Cos,sin(·,exp(·
Nguyen-8c十，一×÷cos,sin(·,exp(.
Nguyen-9c×÷cos(·,sin(·,exp(·
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Empirically, setting", + "type": "text" + }, + { + "bbox": [ + 341, + 146, + 388, + 157 + ], + "score": 0.9, + "content": "\\eta = 0 . 9 9 9 9", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "ensures the scores of ground", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 154, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 170 + ], + "score": 1.0, + "content": "truth equations stand out and successfully enforces the sparsity in all experiments. For the discovery", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "of very chaotic dynamical systems based on measurement data, we expect some physics terms from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "the governing equations to have a weak impact on the state variables (e.g., in the chaotic double", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 363, + 201 + ], + "score": 1.0, + "content": "pendulum system experiments, the effects from physics terms", + "type": "text" + }, + { + "bbox": [ + 363, + 189, + 393, + 200 + ], + "score": 0.88, + "content": "\\sin ( \\theta _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 189, + 413, + 201 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 413, + 189, + 443, + 200 + ], + "score": 0.9, + "content": "\\mathrm { s i n } ( \\theta _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "are hard to be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 367, + 213 + ], + "score": 1.0, + "content": "captured), and the effect of data noise is unknown. Hence, we set", + "type": "text" + }, + { + "bbox": [ + 368, + 201, + 393, + 211 + ], + "score": 0.91, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 199, + 505, + 213 + ], + "score": 1.0, + "content": "to leverage the full strength", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "of data fitting to enable the detection of physics terms that are offset or overwhelmed by data noise", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 234 + ], + "score": 1.0, + "content": "but are pivotal to the systems. Nevertheless, we must acknowledge that this selection process is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 478, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 478, + 245 + ], + "score": 1.0, + "content": "empirical, which depends on our desire for the degree of parsimony of the target equation(s).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 134, + 506, + 245 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 506, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "As for the hyperparameters in the training schema (i.e., maximum module transplantation, episodes,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 274 + ], + "score": 1.0, + "content": "maximum tree size, maximum augmented grammars), we have conducted parametric convergence", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 507, + 285 + ], + "score": 1.0, + "content": "tests for each experiment to ensure the learning curves (i.e., maximum scores in the history) converge.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 282, + 483, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 483, + 295 + ], + "score": 1.0, + "content": "For example, as discussed in Section B, Table B.2 shows the setting of these hyperparameters.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 250, + 507, + 295 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 311, + 315, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 316, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 316, + 325 + ], + "score": 1.0, + "content": "B NGUYEN’S BENCHMARK PROBLEMS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 504, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "This section provides more detailed experiment settings for the Nguyen’s benchmark tasks that are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "described in Section 4.2 of the main text, where the training hyperparameters for the SPL machine in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "these equation discovery experiments are also listed. Table B.1 presents the candidate mathematical", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "operations allowed for three tested models and Table B.2 displays training hyperparameters for the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 380, + 299, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 299, + 392 + ], + "score": 1.0, + "content": "SPL machine in the Nguyen’s benchmark tasks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 335, + 506, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "Moreover, the utilization of CFG in the SPL machine facilitates the flexibility of applying some prior", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 421 + ], + "score": 1.0, + "content": "knowledge including the universal mathematical rules and constraints. This feature empirically turns", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "out to be an accessible and scalable approach for avoiding meaningless mathematical expressions.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "In the Nguyen’s benchmark tasks, one or multiple mathematical constraints are given to the SPL", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 249, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 249, + 453 + ], + "score": 1.0, + "content": "machine. These constraints include", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 397, + 506, + 453 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 462, + 438, + 474 + ], + "lines": [ + { + "bbox": [ + 130, + 462, + 441, + 476 + ], + "spans": [ + { + "bbox": [ + 130, + 462, + 441, + 476 + ], + "score": 1.0, + "content": "1. Only variables and constant values are allowed in trigonometric functions.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 130, + 462, + 441, + 476 + ] + }, + { + "type": "table", + "bbox": [ + 163, + 518, + 440, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 104, + 501, + 500, + 513 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 500, + 502, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 502, + 514 + ], + "score": 1.0, + "content": "Table B.1: Candidate operators for each Nguyen’s benchmark task. const denotes constant values.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "table_body", + "bbox": [ + 163, + 518, + 440, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 163, + 518, + 440, + 730 + ], + "spans": [ + { + "bbox": [ + 163, + 518, + 440, + 730 + ], + "score": 0.866, + "html": "
Benchmark Candidate Operations
Nguyen-1 +,-,×,÷,cos(-),sin(-),exp(·),sin(+,-,×,÷,cos(-),sin(-),exp(-),log(),√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√+,-,×,÷,cos(-),sin(-),exp(-),log()，√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√,const
Nguyen-2 +,-,×,÷,cos(·),sin(.),exp()
Nguyen-3×,÷,cos(·,sin(·
Nguyen-4X，÷,cos(,sin(·
Nguyen-5×，÷,cos(·,sin(·
Nguyen-6X,÷,cos(·,sin(·
Nguyen-7×÷,cos(·,sin(·,exp(·
Nguyen-8×÷,cos(·,sin(·,exp(·
Nguyen-9×÷cos(·,sin(·,exp(·
Nguyen-10×÷Cos,sin(·,exp(.
Nguyen-11×÷Cos,sin(·,exp(.
Nguyen-12×÷Cos,sin(·,exp(.
Nguyen-1c×÷Cos,sin(·,exp(.
Nguyen-2c×÷cos(.,sin(·,exp(.
Nguyen-5c十，一×÷Cos,sin(·,exp(·
Nguyen-8c十，一×÷cos,sin(·,exp(.
Nguyen-9c×÷cos(·,sin(·,exp(·
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BenchmarkMaximum Module TransplantationEpisodes Between Module TransplantationMaximum Tree SizeMaximum Augmented Grammars
Nguyen-12010,000505
Nguyen-22010,000505
Nguyen-320100,000505
Nguyen-420100,000505
Nguyen-520100,000505
Nguyen-62010,000505
Nguyen-7205,000505
Nguyen-8205,000505
Nguyen-92010,000505
Nguyen-102010,000505
Nguyen-112010,000505
Nguyen-1220100,000505
Nguyen-1c202,000505
Nguyen-2c2010,000505
Nguyen-5c2010.000505
Nguyen-8c202,000505
Nguyen-9c201,000505
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BenchmarkConstraints Utilization
Nguyen-1{1,3}
Nguyen-2{1,3}
Nguyen-3{1,3}
Nguyen-4{1,3}
Nguyen-5{2,3}
Nguyen-6{2,3}
Nguyen-7{2,3}
Nguyen-80
Nguyen-9{2,3,4}
Nguyen-10{2,3,4}
Nguyen-11{2,3}
Nguyen-12{1,3,4}
Nguyen-1c{1,3}
Nguyen-2c{1,3}
Nguyen-5c{2,3}
Nguyen-8c0
Nguyen-9c{2,3,4}
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Some small integers (e.g. 1, 2) are used directly as leaves.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 673, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 505, + 686 + ], + "score": 1.0, + "content": "They can be easily implemented into the SPL machine by defining or adjusting non-terminal nodes or", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 684, + 503, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 503, + 697 + ], + "score": 1.0, + "content": "production rules in the customized CFG. 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BenchmarkMaximum Module TransplantationEpisodes Between Module TransplantationMaximum Tree SizeMaximum Augmented Grammars
Nguyen-12010,000505
Nguyen-22010,000505
Nguyen-320100,000505
Nguyen-420100,000505
Nguyen-520100,000505
Nguyen-62010,000505
Nguyen-7205,000505
Nguyen-8205,000505
Nguyen-92010,000505
Nguyen-102010,000505
Nguyen-112010,000505
Nguyen-1220100,000505
Nguyen-1c202,000505
Nguyen-2c2010,000505
Nguyen-5c2010.000505
Nguyen-8c202,000505
Nguyen-9c201,000505
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BenchmarkConstraints Utilization
Nguyen-1{1,3}
Nguyen-2{1,3}
Nguyen-3{1,3}
Nguyen-4{1,3}
Nguyen-5{2,3}
Nguyen-6{2,3}
Nguyen-7{2,3}
Nguyen-80
Nguyen-9{2,3,4}
Nguyen-10{2,3,4}
Nguyen-11{2,3}
Nguyen-12{1,3,4}
Nguyen-1c{1,3}
Nguyen-2c{1,3}
Nguyen-5c{2,3}
Nguyen-8c0
Nguyen-9c{2,3,4}
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The ablation study is performed on the 12 classic Nguyen’s", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "benchmark problems, where the recovery rate of each model is calculated. The results of the ablation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "study are summarized in Table C.1. It is clear that module transplantation brings the largest gain", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "in recovery rate. All the ablated models fail to uncover the Nguyen-12 equation (the most difficult", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 507, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 507, + 234 + ], + "score": 1.0, + "content": "case), where the coefficient 1/2 needs to be represented by mathematical operators and symbols, e.g.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 216, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 149, + 243 + ], + "score": 0.68, + "content": "x / ( x + x )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 230, + 153, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 154, + 231, + 195, + 244 + ], + "score": 0.59, + "content": "y / ( y + y )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 230, + 216, + 245 + ], + "score": 1.0, + "content": ", etc.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 259, + 363, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 364, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 364, + 274 + ], + "score": 1.0, + "content": "D FREE FALLING BALLS WITH AIR RESISTANCE", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "This appendix section reveals more details on discovering the physical laws, in the context of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "relationships between height and time, for the cases of free-falling objects with air resistance based on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "multiple experimental ball-drop datasets (de Silva et al., 2020). The datasets contain the records of 11", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 440, + 330 + ], + "score": 1.0, + "content": "different types of balls, as shown in Figure D.1, dropped from a bridge, collected at a", + "type": "text" + }, + { + "bbox": [ + 441, + 318, + 466, + 328 + ], + "score": 0.69, + "content": "3 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "sampling", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "rate. 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These discovered physical laws are then applied to forecast the height of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "the balls at the time slots in the testing dataset. These predictions, in comparison with the ground", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "truth trajectory recorded, are shown in Figure D.2. The prediction error is shown in the Table 3 of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 149, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 149, + 446 + ], + "score": 1.0, + "content": "main text.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 461, + 313, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 315, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 315, + 476 + ], + "score": 1.0, + "content": "E NONLINEAR DYNAMICS DISCOVERY", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 504, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "This appendix section elaborates more details on the two governing equation discovery experiments", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 498, + 367, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 367, + 509 + ], + "score": 1.0, + "content": "presented in the main text, including the datasets and full results.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "table", + "bbox": [ + 106, + 553, + 503, + 718 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 205, + 537, + 405, + 549 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 204, + 536, + 406, + 550 + ], + "spans": [ + { + "bbox": [ + 204, + 536, + 406, + 550 + ], + "score": 1.0, + "content": "Table C.1: Summary of the ablation study results.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "table_body", + "bbox": [ + 106, + 553, + 503, + 718 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 553, + 503, + 718 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 503, + 718 + ], + "score": 0.985, + "html": "
BenchmarkExpressionSPLModel AModel BModel CModel D
Nguyen-1+ +x 2 2100%100%100%14%12%
Nguyen-2x4 + +x100%100%100%0%0%
Nguyen-3+x4 +x² +x 25100%100%100%0%0%
Nguyen-4+x+x4+x²+x²+x 2699%96%92%0%0%
Nguyen-5sin(𝑥²)cos(x)-195%95%92%92%88%
Nguyen-6sin(x²)+sin(x+x²)100%100%96%100%100%
Nguyen-7ln(x +1) +ln(x² + 1)100%100%100%100%100%
Nguyen-8√x100%100%100%100%100%
Nguyen-9sin(x)+sin(y²)100%100%100%100%100%
Nguyen-102 sin(x) cos(y)100%100%100%0%0%
Nguyen-11x100%96%96%92%87%
Nguyen-12x4 2 + 1 -y28%0%0%0%0%
Average93.5%90.58%89.67%49.83%48.92%
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(2),", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 125, + 131, + 311, + 144 + ], + "spans": [ + { + "bbox": [ + 125, + 131, + 311, + 144 + ], + "score": 1.0, + "content": "(c) module transplantation in tree generation,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 125, + 144, + 231, + 156 + ], + "spans": [ + { + "bbox": [ + 125, + 144, + 231, + 156 + ], + "score": 1.0, + "content": "(d) all of the above three.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 124, + 103, + 400, + 156 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 506, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "The resulting models are denoted with Model A, Model B, Model C, and Model D (note that Model", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 115, + 187 + ], + "score": 0.3, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "is equivalent to the vanilla MCTS). The ablation study is performed on the 12 classic Nguyen’s", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "benchmark problems, where the recovery rate of each model is calculated. The results of the ablation", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "study are summarized in Table C.1. It is clear that module transplantation brings the largest gain", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "in recovery rate. All the ablated models fail to uncover the Nguyen-12 equation (the most difficult", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 507, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 507, + 234 + ], + "score": 1.0, + "content": "case), where the coefficient 1/2 needs to be represented by mathematical operators and symbols, e.g.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 216, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 149, + 243 + ], + "score": 0.68, + "content": "x / ( x + x )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 230, + 153, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 154, + 231, + 195, + 244 + ], + "score": 0.59, + "content": "y / ( y + y )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 230, + 216, + 245 + ], + "score": 1.0, + "content": ", etc.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 165, + 507, + 245 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 259, + 363, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 364, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 364, + 274 + ], + "score": 1.0, + "content": "D FREE FALLING BALLS WITH AIR RESISTANCE", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "This appendix section reveals more details on discovering the physical laws, in the context of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "relationships between height and time, for the cases of free-falling objects with air resistance based on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "multiple experimental ball-drop datasets (de Silva et al., 2020). The datasets contain the records of 11", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 440, + 330 + ], + "score": 1.0, + "content": "different types of balls, as shown in Figure D.1, dropped from a bridge, collected at a", + "type": "text" + }, + { + "bbox": [ + 441, + 318, + 466, + 328 + ], + "score": 0.69, + "content": "3 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "sampling", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "rate. The time between dropping and landing varies in each case due to the fact that air resistance", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "has different effects on these free-dropping balls and induces divergent physical laws. 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The prediction error is shown in the Table 3 of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 149, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 149, + 446 + ], + "score": 1.0, + "content": "main text.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 389, + 506, + 446 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 461, + 313, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 315, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 315, + 476 + ], + "score": 1.0, + "content": "E NONLINEAR DYNAMICS DISCOVERY", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 504, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "This appendix section elaborates more details on the two governing equation discovery experiments", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 498, + 367, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 367, + 509 + ], + "score": 1.0, + "content": "presented in the main text, including the datasets and full results.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 486, + 505, + 509 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 553, + 503, + 718 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 205, + 537, + 405, + 549 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 204, + 536, + 406, + 550 + ], + "spans": [ + { + "bbox": [ + 204, + 536, + 406, + 550 + ], + "score": 1.0, + "content": "Table C.1: Summary of the ablation study results.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "table_body", + "bbox": [ + 106, + 553, + 503, + 718 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 553, + 503, + 718 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 503, + 718 + ], + "score": 0.985, + "html": "
BenchmarkExpressionSPLModel AModel BModel CModel D
Nguyen-1+ +x 2 2100%100%100%14%12%
Nguyen-2x4 + +x100%100%100%0%0%
Nguyen-3+x4 +x² +x 25100%100%100%0%0%
Nguyen-4+x+x4+x²+x²+x 2699%96%92%0%0%
Nguyen-5sin(𝑥²)cos(x)-195%95%92%92%88%
Nguyen-6sin(x²)+sin(x+x²)100%100%96%100%100%
Nguyen-7ln(x +1) +ln(x² + 1)100%100%100%100%100%
Nguyen-8√x100%100%100%100%100%
Nguyen-9sin(x)+sin(y²)100%100%100%100%100%
Nguyen-102 sin(x) cos(y)100%100%100%0%0%
Nguyen-11x100%96%96%92%87%
Nguyen-12x4 2 + 1 -y28%0%0%0%0%
Average93.5%90.58%89.67%49.83%48.92%
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TypeModelExpression
baseballSPL Model-1 Model-2H(t)= 47.8042 + 0.6253t- 4.5383t² H(t)= 47.682+ 1.456t- 5.629t² + 0.376t3 H(t)= 45.089- 8.156t + 5.448 exp(0t)
blue basketballModel-3 SPL Model-1 Model-2H(t)= 48.051- 183.467 log(cosh(0.217t)) H(t)= 46.4726-5.105t² + t³ -0.251t4 H(t)= 46.513-0.493t-3.912t² +0.03t H(t)= 43.522- 7.963t + 5.306 exp(0t)
green basketballModel-3 SPL Model-1H(t)= 46.402-84.791log(cosh(0.319t)) H(t)= 45.9087- 4.1465t² +log(cosh(1)) H(t)= 46.438-0.34t- 3.882t² - 0.055t3 H(t)= 43.512- 8.043t + 5.346 exp(0t)
volleyballModel-2 Model-3 SPL Model-1 Model-2H(t)= 46.391-124.424 log(cosh(0.263t)) H(t)= 48.0744-3.7772t² H(t)= 48.046+ 0.362t-4.352t² +0.218t
bowling ballModel-3 SPL Model-1H(t)= 45.32- 7.317t + 5.037exp(0t) H(t)= 48.124-107.816 log(cosh(0.27t)) H(t)= 46.1329-3.8173t² -0.2846t+4.14× 10-5 exp(20.7385t²)exp(-12.4538t3) H(t)= 46.139-0.091t-3.504t² -0.431t3
golf ballModel-2 Model-3 SPL Model-1H(t)= 43.336-8.525t + 5.676 exp(0t) H(t)= 46.342- 247.571log(cosh(0.189t)) H(t)= 49.5087-4.9633t² +log(cosh(t)) H(t)= 49.413+0.532t-5.061t²+0.102t3
tennis ballModel-2 Model-3 SPL Model-1H(t)= 46.356 -8.918t+ 5.964exp(0t) H(t)= 49.585-178.47log(cosh(0.23t)) H(t)= 47.8577-4.0574t² +log(cosh(0.121t))
whiffle ballModel-2 Model-3 SPL Model-1H(t)= 47.738+0.658t-4.901t²+0.325t H(t)= 45.016-7.717t+ 5.212 exp(0t) H(t)= 47.874- 114.19 log(cosh(0.269t)) H(t)= 4.1563t² -t + 47.0133exp(-0.1511t²) H(t)= 46.969+0.574t-4.505t²+0.522t
1 whiffle ballModel-2 Model-3 SPL Model-1H(t)= 44.259- 6.373t+ 4.689 exp(0t) H(t)= 47.062- 34.083log(cosh(0.462t)) H(t)= -18.6063 + 65.8583 exp(-0.0577t²) H(t)= 47.215+0.296t-4.379t²+0.421t3
2 yellow whiffleModel-2 Model-3 SPLH(t)= 44.443-6.744t + 4.813 exp(0t) H(t)= 47.255-38.29 log(cosh(0.447t)) H(t)= 148.9911/(log(cosh(t)) +3.065)-14.5828t²/(log(cosh(t)) +3.065)
ballModel-1 Model-2 Model-3+48.6092log(cosh(t))/(log(cosh(t)) + 3.065) H(t)= 48.613-0.047t-4.936t² +0.826t3 H(t)= 45.443- 6.789t+ 4.973exp(0t) H(t)= 48.594-12.49 log(cosh(0.86t))
orange whiffle ballSPL Model-1 Model-2 Model-3H(t)= -1.6626t + 47.8622 exp(-0.06815t²) H(t)= 47.836-1.397t-3.822t² +0.422t³ H(t)= 44.389-7.358t+ 5.152 exp(0t)
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TypeModelExpression
baseballSPL Model-1 Model-2H(t)= 47.8042 + 0.6253t- 4.5383t² H(t)= 47.682+ 1.456t- 5.629t² + 0.376t3 H(t)= 45.089- 8.156t + 5.448 exp(0t)
blue basketballModel-3 SPL Model-1 Model-2H(t)= 48.051- 183.467 log(cosh(0.217t)) H(t)= 46.4726-5.105t² + t³ -0.251t4 H(t)= 46.513-0.493t-3.912t² +0.03t H(t)= 43.522- 7.963t + 5.306 exp(0t)
green basketballModel-3 SPL Model-1H(t)= 46.402-84.791log(cosh(0.319t)) H(t)= 45.9087- 4.1465t² +log(cosh(1)) H(t)= 46.438-0.34t- 3.882t² - 0.055t3 H(t)= 43.512- 8.043t + 5.346 exp(0t)
volleyballModel-2 Model-3 SPL Model-1 Model-2H(t)= 46.391-124.424 log(cosh(0.263t)) H(t)= 48.0744-3.7772t² H(t)= 48.046+ 0.362t-4.352t² +0.218t
bowling ballModel-3 SPL Model-1H(t)= 45.32- 7.317t + 5.037exp(0t) H(t)= 48.124-107.816 log(cosh(0.27t)) H(t)= 46.1329-3.8173t² -0.2846t+4.14× 10-5 exp(20.7385t²)exp(-12.4538t3) H(t)= 46.139-0.091t-3.504t² -0.431t3
golf ballModel-2 Model-3 SPL Model-1H(t)= 43.336-8.525t + 5.676 exp(0t) H(t)= 46.342- 247.571log(cosh(0.189t)) H(t)= 49.5087-4.9633t² +log(cosh(t)) H(t)= 49.413+0.532t-5.061t²+0.102t3
tennis ballModel-2 Model-3 SPL Model-1H(t)= 46.356 -8.918t+ 5.964exp(0t) H(t)= 49.585-178.47log(cosh(0.23t)) H(t)= 47.8577-4.0574t² +log(cosh(0.121t))
whiffle ballModel-2 Model-3 SPL Model-1H(t)= 47.738+0.658t-4.901t²+0.325t H(t)= 45.016-7.717t+ 5.212 exp(0t) H(t)= 47.874- 114.19 log(cosh(0.269t)) H(t)= 4.1563t² -t + 47.0133exp(-0.1511t²) H(t)= 46.969+0.574t-4.505t²+0.522t
1 whiffle ballModel-2 Model-3 SPL Model-1H(t)= 44.259- 6.373t+ 4.689 exp(0t) H(t)= 47.062- 34.083log(cosh(0.462t)) H(t)= -18.6063 + 65.8583 exp(-0.0577t²) H(t)= 47.215+0.296t-4.379t²+0.421t3
2 yellow whiffleModel-2 Model-3 SPLH(t)= 44.443-6.744t + 4.813 exp(0t) H(t)= 47.255-38.29 log(cosh(0.447t)) H(t)= 148.9911/(log(cosh(t)) +3.065)-14.5828t²/(log(cosh(t)) +3.065)
ballModel-1 Model-2 Model-3+48.6092log(cosh(t))/(log(cosh(t)) + 3.065) H(t)= 48.613-0.047t-4.936t² +0.826t3 H(t)= 45.443- 6.789t+ 4.973exp(0t) H(t)= 48.594-12.49 log(cosh(0.86t))
orange whiffle ballSPL Model-1 Model-2 Model-3H(t)= -1.6626t + 47.8622 exp(-0.06815t²) H(t)= 47.836-1.397t-3.822t² +0.422t³ H(t)= 44.389-7.358t+ 5.152 exp(0t)
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ModelDiscovered Governing Equations
Eureqax= -0.56-9.02x +9.01y y= -0.047+18.79x+1.86y-0.046xy-0.74xz = -3.04-2.23z +0.88xy
pySINDyx = -0.46-9.18x +9.17y y= 22.32x+0.15y-0.85xz z = 6.04-2.83z +0.15x² + 0.81xy
NGGPx = 0.0047 -10.02x + 10.01y - 0.007x² + 0.007xy - 0.37x/z - 0.00074x²y +0.00063x +0.00018x²z +0.00011xy² -6.59e-5xyz -0.00011y² y = 26.36x-1.5y-0.83xz -7.20x/z + 13.08y/z + 4.52e-5x³ -0.0038xz² -44.25x/z²+0.00028x/z -0.00017xz + 5.98e 6x32 = -0.64- 0.036y － 2.64z + 1.038xy+ 0.00021xz + 0.0011yz － 0.00021x/z
SPL-8.04e-8y/x + 0.00022y/z - 8.04e-8z² - 0.00021xyz +0.00021xy/z -0.001y²z-0.00021y²/z+1.17y/z²-3.89e-7yz²/x+6.66e-9z²/x x = -9.966x + 9.964y y = 27.764x - 0.942y- 0.994xz
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ModelDiscovered Governing Equations
Eureqax= -0.56-9.02x +9.01y y= -0.047+18.79x+1.86y-0.046xy-0.74xz = -3.04-2.23z +0.88xy
pySINDyx = -0.46-9.18x +9.17y y= 22.32x+0.15y-0.85xz z = 6.04-2.83z +0.15x² + 0.81xy
NGGPx = 0.0047 -10.02x + 10.01y - 0.007x² + 0.007xy - 0.37x/z - 0.00074x²y +0.00063x +0.00018x²z +0.00011xy² -6.59e-5xyz -0.00011y² y = 26.36x-1.5y-0.83xz -7.20x/z + 13.08y/z + 4.52e-5x³ -0.0038xz² -44.25x/z²+0.00028x/z -0.00017xz + 5.98e 6x32 = -0.64- 0.036y － 2.64z + 1.038xy+ 0.00021xz + 0.0011yz － 0.00021x/z
SPL-8.04e-8y/x + 0.00022y/z - 8.04e-8z² - 0.00021xyz +0.00021xy/z -0.001y²z-0.00021y²/z+1.17y/z²-3.89e-7yz²/x+6.66e-9z²/x x = -9.966x + 9.964y y = 27.764x - 0.942y- 0.994xz
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The solution appears felicitous as the governing equations of the testing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 479, + 150, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 150, + 490 + ], + "score": 1.0, + "content": "responses.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 504, + 329, + 517 + ], + "lines": [ + { + "bbox": [ + 104, + 503, + 330, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 503, + 330, + 520 + ], + "score": 1.0, + "content": "F DISCUSSION AND FUTURE DIRECTIONS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 506, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 543 + ], + "score": 1.0, + "content": "The effectiveness of the proposed SPL machine is empowered by the following elements: (1) The use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 539, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 554 + ], + "score": 1.0, + "content": "of MCTS enables the flexible representation of search space with customized computational grammars,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "composed of a finite set of mathematical operators and symbols, to guide the search tree expansion.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "score": 1.0, + "content": "(2) The exploration-exploitation trade-off nature of MCTS is remarkably useful for searching the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "optimal mathematical expression tree. 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The hyperparameters are set as", + "type": "text" + }, + { + "bbox": [ + 287, + 423, + 312, + 434 + ], + "score": 0.84, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 422, + 315, + 436 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 315, + 423, + 360, + 434 + ], + "score": 0.77, + "content": "t _ { m a x } = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 422, + 506, + 436 + ], + "score": 1.0, + "content": ", and 40,000 episodes of training are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "regarded as one trail. 5 independent trials are performed and the equations with the highest validation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 443, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 458 + ], + "score": 1.0, + "content": "scores are selected as the final result. The uncovered equations are shown in Table E.2. They", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "score": 1.0, + "content": "are validated through interpolation on the testing set and compared with the smoothed derivatives,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "as shown in Figure E.4. The solution appears felicitous as the governing equations of the testing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 479, + 150, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 150, + 490 + ], + "score": 1.0, + "content": "responses.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 401, + 507, + 490 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 504, + 329, + 517 + ], + "lines": [ + { + "bbox": [ + 104, + 503, + 330, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 503, + 330, + 520 + ], + "score": 1.0, + "content": "F DISCUSSION AND FUTURE DIRECTIONS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 506, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 543 + ], + "score": 1.0, + "content": "The effectiveness of the proposed SPL machine is empowered by the following elements: (1) The use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 539, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 554 + ], + "score": 1.0, + "content": "of MCTS enables the flexible representation of search space with customized computational grammars,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "composed of a finite set of mathematical operators and symbols, to guide the search tree expansion.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 576 + ], + "score": 1.0, + "content": "(2) The exploration-exploitation trade-off nature of MCTS is remarkably useful for searching the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "optimal mathematical expression tree. (3) The key adjustments, including the greedy search, the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 584, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 598 + ], + "score": 1.0, + "content": "adaptive-scaled rewarding, the reward regularizer, and the expression tree module transplantation,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "make it possible to efficiently uncover the best path to formulate complex equations. (4) The SPL", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "machine straightforwardly accepts our prior or domain knowledge, or any sort of constraints of the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 617, + 460, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 460, + 630 + ], + "score": 1.0, + "content": "tasks in the grammar design while leveraging great flexibility in expression formulation.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 528, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 634, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "While SPL shows huge potential in both symbolic regression and governing equation discovery tasks,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "there are still some imperfections to be improved. In this appendix section, a few bottlenecks and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 656, + 261, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 261, + 668 + ], + "score": 1.0, + "content": "their potential solutions are presented:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 106, + 633, + 506, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 129, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "1. Computational cost. Computational cost for this framework is one of the major issues,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 141, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "especially when constant coefficient estimation is required. Evaluating the solution in the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 142, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 142, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "simulation phase happens very frequently for the MCTS algorithm where the policy selection", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 141, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "relies heavily on a large number of historical rewards. However, the constant coefficient", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 721, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 141, + 721, + 506, + 734 + ], + "score": 1.0, + "content": "value estimation, which requires repeated calls for an optimization process and can be slow,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 141, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "is needed for evaluation purposes. In particular, the constant coefficient value is estimated", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "via concurrently solving an optimization problem, e.g., by Powell’s conjugate direction", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 141, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "method (Powell, 1964). For example, when the tree structure changes or is updated, the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 142, + 116, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 142, + 116, + 506, + 127 + ], + "score": 1.0, + "content": "optimization of the constant coefficients should be re-performed simultaneously. The SPL", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 141, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "machine is not the only symbolic regressor suffering from the computational cost in constant", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 142, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 142, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "coefficient value estimation. In fact, the state-of-the-art symbolic regression model, the", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 141, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "neural-guided GP (NGGP) (Mundhenk et al., 2021), becomes much slower in the Nguyen’s", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 141, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "benchmark variant tasks (see Table E.3). 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BenchmarkSPL [s]NGGP[s]
Nguyen-18.7762.734
Nguyen-27.2963.296
Nguyen-381.2873.945
Nguyen-4567.0615.764
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Nguyen-2c295.7691188.215
Nguyen-5c2178.8911365.777
Nguyen-8c77.892129.349
Nguyen-9c2001.4023066.41
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BenchmarkSPL [s]NGGP[s]
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1 Input: Grammar G = (V,∑,R,S), measurement data D = {Y; Yi};
2 Parameters: discount/regularization factor n, exploration rate c, tmax;# n controls equation parsimony;
3 Output: Optimal governing equation f*;
for each episode do
5Selection: Initialize so = O,t = O, NT =[S];
6while St expandable andt <tmax do
7Choose at+1 = arg max A UCT(st, a);
8Take action at+1, observe s', NT;
9St+1 ← s' note as visited,t ← t+1;
10end
11Expansion: Randomly take an unvisited path with action a, observe s', NT;
12St+1 ← s' note as visited,t ←t+1;
13if NT=Othen
14Project f,Backpropagate rt+1 and visited count and finish the episode;
15end
16Simulation:Fix the starting point st,NT;
17foreach simulation do
18while St non-terminal andt < tmax do
19Randomly take an action a,observe s', NT;
20St+1←s',t←t+1;
21end
22if NT=Othen
23Project f and calculate rt+1;
24end
25end
26Backpropagate simulation results;
27 end
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BenchmarkExpressionSPLNGGPGP
Nguyen-1x²+x²+x100%100%99%
Nguyen-2x²+x²+x²+x100%100%90%
Nguyen-3+x4+x²+x²+x 25100%100%34%
Nguyen-4x+x+x4+x²+x²+x99%100%54%
Nguyen-5sin(x²) cos(𝑥) -195%80%12%
Nguyen-6sin(x²)+sin(x+x²)100%100%11%
Nguyen-7ln(𝑥 +1) + ln(x²+ 1)100%100%17%
Nguyen-8√x100%100%100%
Nguyen-9sin(x) + sin(y²)100%100%76%
Nguyen-102 sin(x) cos(y)100%100%86%
Nguyen-11xy100%100%13%
Nguyen-12x4-x²+¹y²-y28%4%0%
Nguyen-1c3.39x +2.12x² +1.78x100%100%0%
Nguyen-2c0.48x4 +3.39x+2.12x² +1.78x94%100%0%
Nguyen-5csin(x²) cos(x) -0.7595%98%1%
Nguyen-8c√1.23x100%100%56%
Nguyen-9csin(1.5x) + sin(0.5y²)96%90%0%
Average94.5%92.4%38.2%
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Physics ModelDerived model expression
Model-1H(t)= co+cit + c2t²+c3t3
Model-2H(t)= Co +Cit + C2eCt
Model-3H(t)= co + c1 log(cosh(c2t))
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Benchmark Candidate Operations
Nguyen-1 +,-,×,÷,cos(-),sin(-),exp(·),sin(+,-,×,÷,cos(-),sin(-),exp(-),log(),√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√+,-,×,÷,cos(-),sin(-),exp(-),log()，√+,-,×,÷,cos(-),sin(-),exp(-),log(-),√,const
Nguyen-2 +,-,×,÷,cos(·),sin(.),exp()
Nguyen-3×,÷,cos(·,sin(·
Nguyen-4X，÷,cos(,sin(·
Nguyen-5×，÷,cos(·,sin(·
Nguyen-6X,÷,cos(·,sin(·
Nguyen-7×÷,cos(·,sin(·,exp(·
Nguyen-8×÷,cos(·,sin(·,exp(·
Nguyen-9×÷cos(·,sin(·,exp(·
Nguyen-10×÷Cos,sin(·,exp(.
Nguyen-11×÷Cos,sin(·,exp(.
Nguyen-12×÷Cos,sin(·,exp(.
Nguyen-1c×÷Cos,sin(·,exp(.
Nguyen-2c×÷cos(.,sin(·,exp(.
Nguyen-5c十，一×÷Cos,sin(·,exp(·
Nguyen-8c十，一×÷cos,sin(·,exp(.
Nguyen-9c×÷cos(·,sin(·,exp(·
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BenchmarkMaximum Module TransplantationEpisodes Between Module TransplantationMaximum Tree SizeMaximum Augmented Grammars
Nguyen-12010,000505
Nguyen-22010,000505
Nguyen-320100,000505
Nguyen-420100,000505
Nguyen-520100,000505
Nguyen-62010,000505
Nguyen-7205,000505
Nguyen-8205,000505
Nguyen-92010,000505
Nguyen-102010,000505
Nguyen-112010,000505
Nguyen-1220100,000505
Nguyen-1c202,000505
Nguyen-2c2010,000505
Nguyen-5c2010.000505
Nguyen-8c202,000505
Nguyen-9c201,000505
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BenchmarkConstraints Utilization
Nguyen-1{1,3}
Nguyen-2{1,3}
Nguyen-3{1,3}
Nguyen-4{1,3}
Nguyen-5{2,3}
Nguyen-6{2,3}
Nguyen-7{2,3}
Nguyen-80
Nguyen-9{2,3,4}
Nguyen-10{2,3,4}
Nguyen-11{2,3}
Nguyen-12{1,3,4}
Nguyen-1c{1,3}
Nguyen-2c{1,3}
Nguyen-5c{2,3}
Nguyen-8c0
Nguyen-9c{2,3,4}
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BenchmarkExpressionSPLModel AModel BModel CModel D
Nguyen-1+ +x 2 2100%100%100%14%12%
Nguyen-2x4 + +x100%100%100%0%0%
Nguyen-3+x4 +x² +x 25100%100%100%0%0%
Nguyen-4+x+x4+x²+x²+x 2699%96%92%0%0%
Nguyen-5sin(𝑥²)cos(x)-195%95%92%92%88%
Nguyen-6sin(x²)+sin(x+x²)100%100%96%100%100%
Nguyen-7ln(x +1) +ln(x² + 1)100%100%100%100%100%
Nguyen-8√x100%100%100%100%100%
Nguyen-9sin(x)+sin(y²)100%100%100%100%100%
Nguyen-102 sin(x) cos(y)100%100%100%0%0%
Nguyen-11x100%96%96%92%87%
Nguyen-12x4 2 + 1 -y28%0%0%0%0%
Average93.5%90.58%89.67%49.83%48.92%
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TypeModelExpression
baseballSPL Model-1 Model-2H(t)= 47.8042 + 0.6253t- 4.5383t² H(t)= 47.682+ 1.456t- 5.629t² + 0.376t3 H(t)= 45.089- 8.156t + 5.448 exp(0t)
blue basketballModel-3 SPL Model-1 Model-2H(t)= 48.051- 183.467 log(cosh(0.217t)) H(t)= 46.4726-5.105t² + t³ -0.251t4 H(t)= 46.513-0.493t-3.912t² +0.03t H(t)= 43.522- 7.963t + 5.306 exp(0t)
green basketballModel-3 SPL Model-1H(t)= 46.402-84.791log(cosh(0.319t)) H(t)= 45.9087- 4.1465t² +log(cosh(1)) H(t)= 46.438-0.34t- 3.882t² - 0.055t3 H(t)= 43.512- 8.043t + 5.346 exp(0t)
volleyballModel-2 Model-3 SPL Model-1 Model-2H(t)= 46.391-124.424 log(cosh(0.263t)) H(t)= 48.0744-3.7772t² H(t)= 48.046+ 0.362t-4.352t² +0.218t
bowling ballModel-3 SPL Model-1H(t)= 45.32- 7.317t + 5.037exp(0t) H(t)= 48.124-107.816 log(cosh(0.27t)) H(t)= 46.1329-3.8173t² -0.2846t+4.14× 10-5 exp(20.7385t²)exp(-12.4538t3) H(t)= 46.139-0.091t-3.504t² -0.431t3
golf ballModel-2 Model-3 SPL Model-1H(t)= 43.336-8.525t + 5.676 exp(0t) H(t)= 46.342- 247.571log(cosh(0.189t)) H(t)= 49.5087-4.9633t² +log(cosh(t)) H(t)= 49.413+0.532t-5.061t²+0.102t3
tennis ballModel-2 Model-3 SPL Model-1H(t)= 46.356 -8.918t+ 5.964exp(0t) H(t)= 49.585-178.47log(cosh(0.23t)) H(t)= 47.8577-4.0574t² +log(cosh(0.121t))
whiffle ballModel-2 Model-3 SPL Model-1H(t)= 47.738+0.658t-4.901t²+0.325t H(t)= 45.016-7.717t+ 5.212 exp(0t) H(t)= 47.874- 114.19 log(cosh(0.269t)) H(t)= 4.1563t² -t + 47.0133exp(-0.1511t²) H(t)= 46.969+0.574t-4.505t²+0.522t
1 whiffle ballModel-2 Model-3 SPL Model-1H(t)= 44.259- 6.373t+ 4.689 exp(0t) H(t)= 47.062- 34.083log(cosh(0.462t)) H(t)= -18.6063 + 65.8583 exp(-0.0577t²) H(t)= 47.215+0.296t-4.379t²+0.421t3
2 yellow whiffleModel-2 Model-3 SPLH(t)= 44.443-6.744t + 4.813 exp(0t) H(t)= 47.255-38.29 log(cosh(0.447t)) H(t)= 148.9911/(log(cosh(t)) +3.065)-14.5828t²/(log(cosh(t)) +3.065)
ballModel-1 Model-2 Model-3+48.6092log(cosh(t))/(log(cosh(t)) + 3.065) H(t)= 48.613-0.047t-4.936t² +0.826t3 H(t)= 45.443- 6.789t+ 4.973exp(0t) H(t)= 48.594-12.49 log(cosh(0.86t))
orange whiffle ballSPL Model-1 Model-2 Model-3H(t)= -1.6626t + 47.8622 exp(-0.06815t²) H(t)= 47.836-1.397t-3.822t² +0.422t³ H(t)= 44.389-7.358t+ 5.152 exp(0t)
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ModelDiscovered Governing Equations
Eureqax= -0.56-9.02x +9.01y y= -0.047+18.79x+1.86y-0.046xy-0.74xz = -3.04-2.23z +0.88xy
pySINDyx = -0.46-9.18x +9.17y y= 22.32x+0.15y-0.85xz z = 6.04-2.83z +0.15x² + 0.81xy
NGGPx = 0.0047 -10.02x + 10.01y - 0.007x² + 0.007xy - 0.37x/z - 0.00074x²y +0.00063x +0.00018x²z +0.00011xy² -6.59e-5xyz -0.00011y² y = 26.36x-1.5y-0.83xz -7.20x/z + 13.08y/z + 4.52e-5x³ -0.0038xz² -44.25x/z²+0.00028x/z -0.00017xz + 5.98e 6x32 = -0.64- 0.036y － 2.64z + 1.038xy+ 0.00021xz + 0.0011yz － 0.00021x/z
SPL-8.04e-8y/x + 0.00022y/z - 8.04e-8z² - 0.00021xyz +0.00021xy/z -0.001y²z-0.00021y²/z+1.17y/z²-3.89e-7yz²/x+6.66e-9z²/x x = -9.966x + 9.964y y = 27.764x - 0.942y- 0.994xz
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BenchmarkSPL [s]NGGP[s]
Nguyen-18.7762.734
Nguyen-27.2963.296
Nguyen-381.2873.945
Nguyen-4567.0615.764
Nguyen-5431.22877.627
Nguyen-664.651104.588
Nguyen-714.9953.024
Nguyen-85.592.896
Nguyen-95.74313.229
Nguyen-1053.24586.497
Nguyen-1110.16344.399
Nguyen-12187.9334.757
Nguyen-1c452.734362.075
Nguyen-2c295.7691188.215
Nguyen-5c2178.8911365.777
Nguyen-8c77.892129.349
Nguyen-9c2001.4023066.41
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Networks + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 This paper investigates structure-preserving embedding for multi-layer networks +2 with community structure. We propose a novel generative tensor-based latent space +3 model (TLSM) that allows heterogeneity among vertices. It embeds vertices into +4 a low-dimensional latent space so that vertices within the same community are +5 close to each other in the ambient space, and captures layer heterogeneity through +6 a layer-effect factor matrix. With a general and flexible tensor decomposition +7 on the expected network adjacency tensor, TLSM is dedicated to preserving the +8 original vertex relations and layer-specific effects in the network embedding. An +9 efficient alternative updating scheme is developed to estimate the model parameters +10 and conduct community detection simultaneously. Theoretically, we establish the +11 asymptotic consistencies of TLSM in terms of both multi-layer network estimation +12 and community detection. The theoretical results are supported by extensive +13 numerical experiments on both synthetic and real-life multi-layer networks. + +# 14 1 Introduction + +15 Network has arisen as one of the most common structures to represent the relations among entities. +16 In many complex systems, entities can be multi-relational in that they may interact with each other +17 under various circumstances. A multi-layer network, which consists of a common vertex set across all +18 network layers representing the entities and an edge set at each layer to characterize a particular type +19 of relation among entities, is faithful to represent these relations. Examples of multi-layer networks +20 include social networks of multiple interaction channels [42, 15], biological networks of different +21 collaboration schemes [49, 31, 29] and world trading networks [1, 37] of various goods. +22 In this paper, we propose a structure-preserving embedding framework for multi-layer networks +23 via a tensor-based latent space model. Specifically, TLSM utilizes the factorization of network +24 adjacency tensor as a building block, embeds the vertices into a low dimensional latent space, and +25 captures the heterogeneity among different layers through a layer-effect factor matrix. Consequently, +26 the community structure of the multi-layer network can be detected from a network embedding +27 perspective, such that vertices within the same community are closer to one another in the ambient +28 space than those in different communities. In addition, one key feature of TLSM is that it introduces +29 a sparsity factor into the vanilla logit transformation of the network adjacency tensor, which allows +30 TLSM to model sparse multi-layer networks in a more explicit fashion and accommodate relatively +31 sparser multi-layer networks as the ones considered in literature [22]. More importantly, this sparsity +32 factor can be estimated from the network adjacency tensor directly. +33 The main contribution of this paper is three-fold. First, the proposed TLSM is flexible and general +34 in that it includes many popular network models as special cases. It also relaxes the layer-wise +35 positive semi-definite condition that has been frequently employed in literature [6, 35]. Second, a +36 joint modeling framework is constructed for TLSM, consisting of the multi-layer network likelihood +37 and a clustering type penalty, to estimate the multi-layer network and conduct community detection +38 simultaneously. Its advantages are supported by extensive numerical experiments on both synthetic +39 and real-life multi-layer networks. Third, the asymptotic consistencies of TLSM are established in +40 terms of both multi-layer network estimation and community detection. Notably, the established +41 theoretical results imply that the proposed methods can accommodate the sparsest multi-layer +42 networks considered in literature. +43 The rest of the paper is organized as follows. The remaining of Section 1 discusses related works and +44 introduces necessary notations. Section 2 presents the proposed TLSM and its estimation scheme with +45 an efficient algorithm. In Section 3, we establish the asymptotic consistencies of TLSM. Extensive +46 numerical performance of TLSM on synthetic and real-life multi-layer networks as well as ablation +47 studies on two novel components of the proposed method are carried out in Section 4. Section 5 +48 concludes the paper. The supplementary materials contains technique proofs and necessary lemmas, +49 additional simulation studies, detailed parameter tuning process, among others. + +# 50 1.1 Related work + +51 While there is a growing number of literature focusing on community detection in single-layer +52 network [48, 28, 13], community detection in multi-layer network is still in its infancy. One classical +53 approach is to detect community structure in each layer separately [4, 5], which fails to leverage +54 the homogeneity across different layers. Another approach is to aggregate multi-layer networks +55 into a single-layer one [41, 12, 35], which heavily relies on the assumption of homogeneous linking +56 pattern across multiple layers. Recently, [26] proposed to aggregate the biased-adjusted version of +57 the squared adjacency matrix in each layer to alleviate the information loss in aggregation. yet it +58 requires the average node degree to grow at a sub-optimal order. +59 In terms of multi-layer network generative models, [34] extended the seminal stochastic block +60 model (SBM; 19) to the multi-layer stochastic block model (MLSBM; 34), where the probability for +61 any two vertices to form an edge in a given layer depends only on their community memberships. +62 Clearly, MLSBM heavily relies on the assumption of homogeneous vertices within communities. +63 The framework of MLSBM has also been incorporated in degree-corrected network estimation [36], +64 spectral clustering [6, 35, 26], least square estimation [27] and likelihood-based approaches [45]. In +65 addition, network response regression model [46] and tensor factorization methods [8, 22] have also +66 been proposed to detect community structures in multi-layer networks. +67 To allow heterogeneous vertices, the latent space model [18] and random dot product graph model +68 [3] have been extended to multi-layer networks[47, 32, 2]. In addition, graph neural network and +69 graph convolutional networks has been extended to multi-layer network for learning the multi-layer +70 network embedding [14, 23, 17, 39]. + +# 71 1.2 Notations + +72 Throughout the paper, we use boldface calligraphic Euler scripts $( A )$ to denote tensors, boldface +73 capital letters $( A )$ or Greece letters $( \alpha , \beta )$ to denote matrices, boldface lowercase letters $( a )$ to +74 denote vectors, and regular letters $( a )$ to denote scalars. For an order three tensor $\pmb { \mathcal { A } } \in \mathbb { R } ^ { I _ { 1 } \times I _ { 2 } \times I _ { 3 } }$ , +75 $\mathcal { A } _ { i , . , . } \in \mathbb { R } ^ { I _ { 2 } \times I _ { 3 } } , \mathcal { A } _ { . , j , \cdot } \in \mathbb { R } ^ { I _ { 1 } \times I _ { 3 } }$ , and $\pmb { \mathscr { A } } _ { . , . , m } \in \mathbb { R } ^ { I _ { 1 } \times I _ { 2 } }$ are the $i$ -th horizontal slide, $j$ -th lateral slide +76 and $m$ -th frontal slide of $\mathcal { A }$ , respectively. Similarly, for a matrix $\pmb { A }$ , $A _ { i , }$ . denotes its $i$ -th row and $A _ { . , j }$ +77 denotes its $j$ -th column. For a vector $\textbf { \em a }$ , $\mathrm { d i a g } ( a )$ stands for the diagonal matrix whose diagonal is $\textbf { \em a }$ . +78 We use $| | \cdot | | , | | \cdot | | _ { \infty }$ , and $| | \cdot | | _ { F }$ to denote the $l _ { 2 }$ -norm, $l _ { \infty }$ -norm of a vector, and the Frobenius norm +79 of matrix or tensor, respectively. For any integer $n$ , denote $[ n ] = \{ 1 , 2 , . . . , n \}$ . + +80 81 1 product betsuch that its or -th $\pmb { \mathcal { A } } \in \mathbb { R } ^ { I _ { 1 } \times I _ { 2 } \times I _ { 3 } }$ an as $U \in \mathbb { R } ^ { J _ { 1 } \times I _ { 1 } }$ $\pmb { A } \times _ { 1 } \pmb { U } \in$ $\mathbb { R } ^ { J _ { 1 } \times I _ { 2 } \times I _ { 3 } }$ $( j _ { 1 } , i _ { 2 } , i _ { 3 } )$ $\begin{array} { r } { ( \pmb { \mathscr { A } } \times _ { 1 } \pmb { U } ) _ { j _ { 1 } , i _ { 2 } , i _ { 3 } } = \sum _ { i _ { 1 } = 1 } ^ { I _ { 1 } } \pmb { \mathscr { A } } _ { i _ { 1 } , i _ { 2 } , i _ { 3 } } U _ { j _ { 1 } , i _ { 1 } } } \end{array}$ The mode-2 or mode-3 product between $\pmb { A }$ and any matrix of appropriate dimension are defined 83 similarly. The CANDECOMP/PARAFAC (CP) decomposition of $\pmb { A }$ has the form + +$$ +\pmb { \mathcal { A } } = \sum _ { r = 1 } ^ { R } \pmb { a } ^ { ( r ) } \circ \pmb { b } ^ { ( r ) } \circ \pmb { c } ^ { ( r ) } , +$$ + +where 84 $\pmb { a } ^ { ( r ) } \in \mathbb { R } ^ { I _ { 1 } }$ , $\boldsymbol { b } ^ { ( r ) } \in \mathbb { R } ^ { I _ { 2 } }$ , and $\boldsymbol { c } ^ { ( r ) } \in \mathbb { R } ^ { I _ { 3 } }$ for $r \in [ R ]$ , and $\circ$ stands for the vector outer product. The CP-rank [24] of the tensor 85 $\pmb { a } ^ { ( r ) } \circ \pmb { b } ^ { ( r ) } \circ \pmb { c } ^ { ( r ) }$ is defined to be 1, for $r \in [ R ]$ . The minimal number + +86 of rank-1 tensors in the CP decomposition of $\pmb { A }$ is called the CP-rank of $\pmb { A }$ . Let $\pmb { \mathcal { T } } \in \{ 0 , 1 \} ^ { R \times R \times R }$ +87 be the identity tensor such that $\pmb { \mathcal { T } } _ { i _ { 1 } , i _ { 2 } , i _ { 3 } } = 1$ if $i _ { 1 } = i _ { 2 } = i _ { 3 }$ and 0 otherwise, and let $\pmb { A } \in \mathbb { R } ^ { I _ { 1 } \times R }$ , +88 $\boldsymbol { B } \in \mathbb { R } ^ { I _ { 2 } \times R }$ , and $C \in \mathbb { R } ^ { I _ { 3 } \times R }$ such that $\mathbf { \boldsymbol { A } } _ { \cdot , r } = \mathbf { \boldsymbol { a } } ^ { ( r ) }$ , $\mathbf { \delta } _ { B _ { \cdot , r } } = \mathbf { \delta } _ { \mathbf { \delta } } \mathbf { \delta } _ { B _ { \cdot , r } } ^ { ( r ) }$ , and $\boldsymbol { C } _ { \cdot , r } = \boldsymbol { c } ^ { ( r ) }$ . Equation (1) +89 then can be equivalently written as $\pmb { \mathcal { A } } = \pmb { \mathcal { T } } \times _ { 1 } \pmb { A } \times _ { 2 } \pmb { B } \times _ { 3 } \pmb { C }$ . + +# 90 2 Structure-preserving embedding + +91 In this paper, we consider multi-layer networks that can be represented as an undirected and un +92 weighted $M$ -layer graph $\mathcal { G } = ( V , \mathcal { E } )$ , where $V = [ n ]$ consists of the common $n$ vertices across +93 different layers, and $\mathcal { E } = \{ E ^ { ( m ) } \} _ { m = 1 } ^ { M }$ with $E ^ { ( m ) } \subset V \times V$ representing the $m$ -th relation network +94 among vertices. A order three adjacency tensor $\pmb { \mathcal { A } } = ( a _ { i , j , m } ) \in \{ 0 , 1 \} ^ { n \times n \times M }$ is then defined to +95 represent $\mathcal { G }$ with entries $a _ { i , j , m } = 1$ if $( i , j ) \in E ^ { ( m ) }$ and 0 otherwise. + +# 2.1 Tensor-based latent space model + +97 To fully characterize the multi-layer network structure, we propose the following generative tensor +8 based latent space model (TLSM). For any $i \leq j \in [ n ]$ , and $m \in [ M ]$ , + +$$ +\begin{array} { r l } & { a _ { i , j , m } = a _ { j , i , m } \overset { i n d . } { \sim } \mathrm { B e r n o u l l i } ( p _ { i , j , m } ) , \mathrm { ~ w i t h ~ } } \\ & { \theta _ { i , j , m } = \log \Big ( \frac { p _ { i , j , m } } { s _ { n } - p _ { i , j , m } } \Big ) , \mathrm { ~ a n d ~ } } \\ & { \Theta = \mathbb { Z } \times _ { 1 } \alpha \times _ { 2 } \alpha \times _ { 3 } \beta , \alpha \in \Omega _ { \alpha } , \beta \in \Omega _ { \beta } , } \end{array} +$$ + +99 where $\boldsymbol { \mathscr { x } }$ is the order three $R$ -dimensional identity tensor. Basically, (2) follows the standard routine +100 in the multi-layer network literature [34, 35, 27, 22] to model that $a _ { i , j , m } = a _ { j , i , m }$ are independently +101 generated from a Bernoulli distribution, for $i \leq j \in [ n ]$ and $m \in [ M ]$ . Denote $\pmb { \mathcal { P } } = ( p _ { i , j , m } ) \in$ +102 $\mathbb { R } ^ { n \times n \times M }$ as the network underlying probability tensor, and then $\Theta = ( \theta _ { i , j , m } ) \in \mathbb { R } ^ { n \times n \times M }$ is +103 the entry-wise transformation of $\mathcal { P }$ by (3). We call the transformation (3) as the modified logit +104 transformation in that the constant 1 in the standard logit transformation is replaced by a sparsity +105 factor $s _ { n }$ , which may vanish with $n$ and $M$ . We further assume all entries of $\mathcal { P }$ are of the order $s _ { n }$ ; that +106 is, there exists a constant $\textstyle { \frac { 1 } { 2 } } \leq \xi < 1$ such that $( 1 - \xi ) s _ { n } \leq p _ { i , j , m } \leq \xi s _ { n }$ , for $i , j \in [ n ]$ and $m \in [ M ]$ +107 Thus, the in $s _ { n }$ essval $\begin{array} { r } { [ - \log \frac { \xi } { 1 - \xi } , \log \frac { \xi } { 1 - \xi } ] } \end{array}$ overall network sparsity and the entries of . More importantly, (4) models the CP d $\Theta$ are ensured toomposition of $\Theta$ cate inby the +109 factor matrices $\pmb { \alpha } \in \mathbb { R } ^ { n \times R }$ and $\mathbf { \boldsymbol { \beta } } \in \mathbb { R } ^ { M \times R }$ with CP-rank $R$ , which can greatly reduce the number of +110 free parameters from $n ( n + 1 ) M / 2$ to $( n + M ) R$ . Throughout the paper, the CP-rank $R$ is allowed +111 to diverge with $n$ . In the CP decomposition of $\Theta$ , $_ \alpha$ is the vertex latent position matrix with each row +112 $\alpha _ i , $ . serving as the embedding of vertex $i$ , and $\beta$ captures heterogeneity across different layers. Herein, +113 we define the constraint sets for $_ { \pmb { \alpha } }$ and $\beta$ as $\begin{array} { r } { \Omega _ { \alpha } = \{ \alpha \in \mathbb { R } ^ { n \times R } : | | \alpha _ { i , \cdot } | | \leq \sqrt { \log \frac { \xi } { 1 - \xi } } } \end{array}$ , for $i \in [ n ] \}$ +114 and $\Omega _ { \beta } = \{ \beta \in \mathbb { R } ^ { M \times R } : | | \beta _ { \cdot , r } | | = 1 , r \in [ R ] \}$ . Note that the constraint on $\beta$ is necessary for +115 model identification, and detailed discussion will be presented shortly. The constraint set $\Omega _ { \alpha } \times \Omega _ { \beta }$ +116 is sufficient to maintain the bounded condition of $\Theta$ since a general Hölder inequality yields that +117 $\begin{array} { r } { | \theta _ { i , j , m } | = | \pmb { \mathcal { Z } } \times _ { 1 } \pmb { \alpha } _ { i , . } ^ { T } \times _ { 2 } \pmb { \alpha } _ { j , . } ^ { T } \times _ { 3 } \beta _ { m , . } ^ { T } | \le | | \pmb { \alpha } _ { i , . } | | | | \pmb { \alpha } _ { j , . } | | | | \beta _ { m , . } | | _ { \infty } \le \log \frac { \xi } { 1 - \xi } } \end{array}$ . To conclude this +118 paragraph, we remake that the parameter $\xi$ is introduced for theoretical purpose and it is not treated as +119 a tuning parameter. One can choose $\xi$ sufficiently close to 1 in empirical studies so that the restriction +120 on $_ { \pmb { \alpha } }$ will be alleviated. +121 We make several essential observations of the proposed TLSM. First and foremost, TLSM is flexible +122 and general. It includes the celebrated MLSBM [34, 43, 35, 27, 26, 36, 22] as special case. Specif +123 ically, suppose the vertices comes form $K$ disjoint communities, the standard MLSBM assumes +124 that the underlying network probability tensor ${ \pmb { \mathcal { P } } } = { \pmb { \mathcal { B } } } \times _ { 1 } { \pmb { Z } } \times _ { 2 } { \pmb { Z } }$ , where $\pmb { \mathscr { B } } \in \mathbb { R } ^ { K \times K \times M }$ is a +125 semi-symmetric core probability tensor with $\pmb { \mathscr { B } } _ { k _ { 1 } , k _ { 2 } , m } = \pmb { \mathscr { B } } _ { k _ { 2 } , k _ { 1 } , m }$ for $k _ { 1 } , k _ { 2 } \in [ K ]$ and $m \in [ M ]$ , +126 and $Z \in \{ 0 , 1 \} ^ { n \times K }$ is the community membership matrix with $Z _ { i , k } = 1$ if vertex $i$ comes from the +127 $k$ -th community and 0 otherwise. That is, the probability of any vertex pair to form an edge in a +128 particular layer depends only on their community memberships. Equivalently, under the modified +129 logit transformation (3), we have $\Theta = \widetilde { \pmb { \mathscr { B } } } \times _ { 1 } { Z } \times _ { 2 } { Z }$ , where $\widetilde { B }$ is the entry-wise transformation +130 of $_ { \pmb { B } }$ under (3). Taking $R$ to be the CP-rank of $\widetilde { B }$ , the CP-decomposition of $\widetilde { B }$ then has the form +131 $\widetilde { \pmb { \mathscr { B } } } = \pmb { \mathscr { T } } \times _ { 1 } \pmb { C } \times _ { 2 } \pmb { C } \times _ { 3 } \ \pmb { \beta }$ for some matrix $C \in \mathbb { R } ^ { K \times R }$ and $\beta \in \mathbb { R } ^ { M \times R }$ due to semi-symmetry. +132 This leads to the CP decomposition of $\Theta$ has the form (4) with $\mathbf { \alpha } _ { \alpha } = Z C$ . It is clear that MLSBM +133 requires vertices within the same community are homogeneous and exchangeable, while TLSM +134 allows vertices to have different embeddings even when they are in the same community. +135 Second, TLSM is identifiable when both $_ { \pmb { \alpha } }$ and $\beta$ have full column ranks. When both $_ { \pmb { \alpha } }$ and $\beta$ +136 have full column ranks, the Kruskal’s $\mathbf { k }$ -ranks [25] of $_ { \pmb { \alpha } }$ and $\beta$ satisfy $k _ { \alpha } = k _ { \beta } = R$ , then $\Theta$ has +137 CP-rank $R$ . Hence, $k _ { \alpha } + k _ { \alpha } + k _ { \beta } \geq 2 R + 2$ as long as $R \geq 2$ . By Theorem 1 of [40], the fixed +138 column $l _ { 2 }$ -norm constraint of $\beta$ implies that the tensor factorization in (4) is unique up to column +139 permutations of $_ { \pmb { \alpha } }$ and $\beta$ and column sign flip of $_ \alpha$ . It is important to remark that the community +140 structure encoded in $_ { \pmb { \alpha } }$ remains unchanged under any column permutation or sign flip. +141 Third, introducing a sparsity factor $s _ { n }$ via a modified logit transformation into the TLSM is non +142 trivial. We take a single-layer network as an example to illustrate the limitation of the standard +143 logit transformation in handling sparse network. Suppose a vanilla logit link is used to connect +144 the network underlying probability matrix $_ { r }$ and its transformation $\Theta$ , and the latent space model +145 usually assumes that $\breve { \Theta } = \alpha \alpha ^ { T }$ . A sparse network requires the entries of $\Theta$ diverge to negative +146 infinite due to the small magnitude of edge probability, which leads to unstable estimation of $_ { \pmb { \alpha } }$ in +147 numerical experiments. Moreover, this may conflict with the assumption that vertices within the same +148 community tend to be close in the embedding space and their inner product is likely to be positive. +149 These difficulties can be naturally circumvented when an appropriate $s _ { n }$ is chosen in (3). + +# 150 2.2 Regularized likelihood + +Given a network adjacency tensor $\mathcal { A }$ and number of communities $K$ , our goal is to estimate the multi-layer network embedding $( \alpha , \beta )$ and conduct community detection on the vertices. Throughout this paper, we assume the number of potential communities $K$ is given and may diverge with $n$ . Under the TLSM framework, with slight abuse of notation, we denote the average negative log-likelihood function of the multi-layer network $\mathcal { G }$ is $\mathcal { L } ( \alpha , \beta ; \mathcal { A } ) = \mathcal { L } ( \Theta ; \mathcal { A } )$ with + +$$ +\mathcal { L } ( \Theta ; \pmb { A } ) = \frac { 1 } { \varphi ( n , M ) } \sum _ { m = 1 } ^ { M } \sum _ { i \leq j } L ( \theta _ { i , j , m } ; a _ { i , j , m } ) , +$$ + +where 151 $\varphi ( n , M ) = { \textstyle { \frac { 1 } { 2 } } } n ( n { + } 1 ) M$ is the number of potential edges, and $\begin{array} { r } { L ( \theta ; a ) = \log \left( 1 + \frac { s _ { n } } { 1 - s _ { n } + e ^ { - \theta } } \right) - } \end{array}$ 152 $\begin{array} { r } { a \log \left( \frac { s _ { n } } { 1 - s _ { n } + e ^ { - \theta } } \right) } \end{array}$ is a negative log-density of a Bernoulli random variable $a$ . We now introduce a 153 novel regularization term to detect the potential communities in $\mathcal { G }$ , + +$$ +J ( \alpha ) = \operatorname* { m i n } _ { Z \in \Gamma , C \in \mathbb { R } ^ { K \times R } } \frac { 1 } { n } \| \alpha - Z C \| _ { F } ^ { 2 } , +$$ + +154 where $C$ encodes the vertex embedding centers and ${ \Gamma } ~ \subset ~ \{ 0 , 1 \} ^ { n \times K }$ is the set of all possible +155 community membership matrices; that is, for any $Z \in \Gamma$ , each row of $z$ consists of only one 1 +156 indicating the community membership and all others entries being 0. This leads to the proposed +157 regularized cost function, + +$$ +\begin{array} { r } { \mathcal L _ { \lambda } ( \boldsymbol { \alpha } , \beta ; \boldsymbol { \mathcal { A } } ) = \mathcal L ( \boldsymbol { \alpha } , \beta ; \boldsymbol { \mathcal { A } } ) + \lambda _ { n } J ( \boldsymbol { \alpha } ) , } \end{array} +$$ + +158 where $\lambda _ { n }$ is a positive tuning parameter that strikes the balance between network estimation and +159 community detection in the cost function. It is clear that the embeddings of vertices with similar +160 linking pattern will be pushed towards the same center, and thus close to each other in the ambient +161 space, leading to the desired community structure in $\mathcal { G }$ . + +# 2.3 Projected gradient descent algorithm + +163 We develop a scalable projected gradient descent (PGD) algorithm to optimize the penalized cost +164 function (6), which is highly non-convex and can be solved only locally. PGD, which alternatively +165 conducts gradient step and projection step, is one of the most popular and computationally fast +166 algorithm in tackling non-convex optimization problem [7, 33, 47, 9]. + +To compute the gradients of 167 $_ \alpha$ and $\beta$ , we introduce the following notations. Define $\pmb { \mathcal { T } } \in \mathbb { R } ^ { n \times n \times M }$ with entries 168 $\begin{array} { r } { \pmb { \mathcal { T } } _ { i , j , m } = \frac { \exp ( - \theta _ { i , j , m } ) } { 1 - s _ { n } + \exp ( - \theta _ { i , j , m } ) } ( p _ { i , j , m } - a _ { i , j , m } ) } \end{array}$ , and $\boldsymbol { X } _ { \mathcal { T } ( 2 , 3 ) } ^ { \alpha , \beta } \in \mathbb { R } ^ { n \times R }$ whose $i$ -th row + +169 170 al elements of the slic. Similarly, we define $( \mathcal { T } \times _ { 2 } \alpha ^ { T } \times _ { 3 } \beta ^ { T } ) _ { i , . , . }$ $X _ { \mathcal { T } ( 2 , 3 ) } ^ { \alpha , \beta } ( i , r ) ~ =$ $( \pmb { \mathcal { T } } \times _ { 2 } \pmb { \alpha } ^ { T } \times _ { 3 } \beta ^ { T } ) _ { i , r , r }$ $X _ { \mathcal { T } ( 1 , 2 ) } ^ { \alpha , \alpha } \in \mathbb { R } ^ { R \times M }$ $\boldsymbol { X } _ { \mathcal { T } ( 3 ) } ^ { \beta } \in \mathbb { R } ^ { n \times R }$ $X _ { T ( 1 , 2 ) } \in$ 171 $\mathbb { R } ^ { n \times M }$ , such that $X _ { \mathcal { T } ( 1 , 2 ) } ^ { \alpha , \alpha } ( r , m ) = ( \mathcal { T } \times _ { 1 } \alpha ^ { T } \times _ { 2 } \alpha ^ { T } ) _ { r , r , m }$ , $X _ { \mathcal { T } ( 3 ) } ^ { \beta } ( i , r ) = ( \mathcal { T } \times _ { 3 } \beta ^ { T } ) _ { i , i , r }$ , and 172 $X _ { \mathcal { T } ( 1 , 2 ) } ( i , m ) = \mathcal { T } _ { i , i , m }$ . Consequently, when the vertex membership matrix $z$ and the community 173 center matrix $C$ are fixed, we can derive the gradients of $\mathcal { L } _ { \lambda } ( \alpha , \beta ; \mathcal { A } )$ with respect to $_ { \pmb { \alpha } }$ and $\beta$ , as $\frac { 1 } { \varphi ( n , M ) } \big ( X _ { \mathcal { T } ( 2 , 3 ) } ^ { \alpha , \beta } + X _ { \mathcal { T } ( 3 ) } ^ { \beta } \ast \alpha \big ) + 2 \lambda _ { n } ( \alpha - Z C )$ and $\frac { 1 } { 2 \varphi ( n , M ) } \big ( ( X _ { \mathcal { T } ( 1 , 2 ) } ^ { \alpha , \alpha } ) ^ { T } + X _ { \mathcal { T } ( 1 , 2 ) } ^ { T } ( \alpha * \alpha ) \big ) ,$ 174 respectively. Herein, \* denotes the Hadamard product (entry-wise product) between two matrices. + +Let 175 $( { \tilde { \alpha } } , { \tilde { \beta } } )$ denote the solution given by one-step gradient descent, we then project $( { \tilde { \alpha } } , { \tilde { \beta } } )$ onto 176 $\Omega _ { \alpha } \times \Omega _ { \beta }$ in the following steps. + +Step 1. Multiply the $r$ -th column of $\tilde { \alpha } _ { . , r }$ by $| | \tilde { \beta } _ { . , r } | | ^ { 1 / 2 }$ for $r \in [ R ]$ . Denote the resultant matrix as $\tilde { \alpha } ^ { \prime }$ + +Step 2. Regularize each row of $_ { \pmb { \alpha } }$ as $\begin{array} { r } { \pmb { \alpha } _ { i , . } = \tilde { \pmb { \alpha } } _ { i , . } ^ { \prime } \operatorname* { m i n } \{ \sqrt { \log \frac { \xi } { 1 - \xi } } , | | \tilde { \pmb { \alpha } } _ { i , . } ^ { \prime } | | \} / | | \tilde { \pmb { \alpha } } _ { i , . } ^ { \prime } | | , \mathbf { f } } \end{array}$ or $i \in [ n ]$ . + +Step 3. Normalize the columns of 179 $\beta$ as $\beta _ { . , r } = \tilde { \beta } _ { . , r } / | | \tilde { \beta } _ { . , r } | |$ , for $r \in [ R ]$ . + +Next, when $( \alpha , \beta )$ are given, we apply a $( 1 + \delta )$ -approximation K-means algorithm on $\tilde { \alpha }$ to update the vertex community membership matrix $z$ and community center matrix $C$ . + +182 The above steps will be alternatively conducted until convergence or reaching the maximum number +183 of iterations. We further summarized the developed alternative updated scheme in Algorithm 1 in +184 Appendix A of the supplementary materials +185 Several remarks on the algorithm are in order. First, Algorithm 1 can only be guaranteed to converge +186 to a stationary point but not any local minimizer. We hence employ a transformed higher order +187 orthogonal iteration (HOOI) algorithm for warm initialization in all the numerical experiments in +188 Section 4 and 5. Specifically, given a user-specific value $\tau$ , we define $\widetilde { \Theta }$ to mimic the magnitude +189 of $\Theta$ such that $\widetilde { \Theta } _ { i , j , m } = - \tau$ if $a _ { i , j , m } = 0$ and $\widetilde { \Theta } _ { i , j , m } = \tau$ otherwise. A standard HOOI algorithm +190 [11] is applied to $\Theta$ to obtain $\pmb { \alpha } ^ { ( 0 ) }$ and $\beta ^ { ( 0 ) }$ . We set $\tau = 1 0 0$ in all the numerical experiments. +191 Second, the sparsity factor $s _ { n }$ is an intrinsic quantity of the multi-layer network data, and it should be +192 estimated from the network directly. Note that the minimal and maximal probabilities for any vertex +193 pair to form an edge in any layer are $p _ { \operatorname* { m i n } } = ( 1 - \xi ) s _ { n }$ and $p _ { \operatorname* { m a x } } = \xi s _ { n }$ , respectively. Interestingly, +194 $p _ { \operatorname* { m i n } } + p _ { \operatorname* { m a x } } = s _ { n }$ , which does not depend on $\xi$ any more. Therefore, we propose to estimate $s _ { n }$ as + +$$ +\hat { s } _ { n } = \operatorname* { m i n } _ { i \in [ n ] } \frac { 1 } { n M } \sum _ { m = 1 } ^ { M } \sum _ { j = 1 } ^ { n } a _ { i , j , m } + \operatorname* { m a x } _ { i \in [ n ] } \frac { 1 } { n M } \sum _ { m = 1 } ^ { M } \sum _ { j = 1 } ^ { n } a _ { i , j , m } , +$$ + +195 which is the sum of the minimal and maximal frequencies of a vertex to form edges with all other +196 vertices in all layers. Third, to optimally choose $\lambda _ { n }$ , we extend the network cross-validation by +197 edge sampling scheme in [30] to multi-layer networks. The detailed tuning procedure is relegated to +198 Appendix B in the supplementary materials. + +# 3 Asymptotic theory + +# 3.1 Consistency in estimating $\Theta ^ { * }$ + +201 Let $\begin{array} { r } { \lambda = \left\{ \Theta = \mathbb { Z } \times _ { 1 } { \pmb \alpha } \times _ { 2 } { \pmb \alpha } \times _ { 3 } \beta : { \pmb \alpha } \in \Omega _ { \pmb { \alpha } } , \beta \in \Omega _ { \beta } \right\} } \end{array}$ } be the parameter space of the problem and +202 203 $\Theta ^ { * } = \mathcal { T } \times _ { 1 } \pmb { \alpha } ^ { * } \times _ { 2 } \pmb { \alpha } ^ { * } \times _ { 3 } \beta ^ { * }$ $\begin{array} { r } { K L ( \boldsymbol { \Theta } ^ { * } | | \boldsymbol { \Theta } ) = \varphi ^ { - 1 } ( n , M ) \sum _ { m = 1 } ^ { M } \sum _ { i \leq j } E \bigl ( L ( \theta _ { i , j , m } ; a _ { i , j , m } ) - L ( \theta _ { i , j , m } ^ { * } ; a _ { i , j , m } ) \bigr ) } \end{array}$ ty tensor. Denote be the averaged +204 Kullback–Leibler divergence of the network generation distributions parametrized by and , for +205 any $\mathbf { \Theta } \Theta \in \Omega$ . The following large deviation inequality is derived to quantify the behavior of $\mathcal { L } _ { \lambda } ( \Theta ; \mathbf { \mathcal { A } } )$ +206 for any $\Theta$ in the neighborhood of $\Theta ^ { * }$ defined by $\dot { K } L ( \Theta ^ { * } | | \Theta )$ . + +Proposition 1. Suppose 207 $\lambda _ { n } J ( \alpha ^ { * } ) \leq \epsilon _ { n }$ , and $( n + M ) R \varphi ^ { - 1 } ( n , M ) \epsilon _ { n } ^ { - 1 } \log ( \epsilon _ { n } ^ { - 1 / 2 } ) \leq c _ { 1 }$ for some constant 208 $c _ { 1 }$ . Then with probability at lease $\begin{array} { r } { 1 - 2 \exp \Big ( - \frac { \varphi ( n , M ) \epsilon _ { n } } { 1 5 6 \frac { \xi } { 1 - \xi } + 2 8 \log 2 } \Big ) } \end{array}$ , we have + +$$ +\mathcal { L } _ { \lambda } ( \Theta ^ { * } ; \mathcal { A } ) \leq \operatorname* { i n f } _ { \substack { \{ \Theta \in \Omega \vert K L ( \Theta ^ { * } \vert \vert \Theta ) \geq 4 \epsilon _ { n } \} } } \mathcal { L } _ { \lambda } ( \Theta ; \mathcal { A } ) - \epsilon _ { n } . +$$ + +209 Proposition 1 basically states that any estimators with sufficiently small objective value should +210 be close enough to $\Theta ^ { * }$ in terms of $K \dot { L } ( \Theta ^ { * } | | \Theta )$ . We next study the asymptotic behavior of these +211 estimators more precisely. Let $( \hat { \alpha } , \hat { \beta } ) \in \Omega _ { \alpha } \times \Omega _ { \beta }$ be any estimator of $( \alpha ^ { * } , \beta ^ { * } )$ such that + +$$ +\begin{array} { r } { \mathcal L _ { \lambda } ( \hat { \alpha } , \hat { \beta } ; \mathcal A ) \le \mathcal L _ { \lambda } ( \alpha ^ { * } , \beta ^ { * } ; \mathcal A ) + \epsilon _ { n } , } \end{array} +$$ + +and denote 212 $\widehat { \Theta } = \mathcal { T } \times _ { 1 } \hat { \alpha } \times _ { 2 } \hat { \alpha } \times _ { 3 } \hat { \beta }$ . we have the following theorem. + +Theorem 1. Under the condition of Proposition $^ { l }$ , $i f \left( { \hat { \alpha } } , { \hat { \beta } } \right)$ satisfies (8), then with probability at least $\begin{array} { r } { 1 - 2 \exp \Big ( - \frac { \varphi ( n , M ) \epsilon _ { n } } { 1 5 6 \frac { \xi } { 1 - \xi } + 2 8 \log 2 } \Big ) } \end{array}$ , we have + +$$ +\frac { 1 } { n \sqrt { M } } \| \widehat { \Theta } - \Theta ^ { * } \| _ { F } \leq \frac { 4 \sqrt { 2 } \sqrt { \epsilon _ { n } } } { ( 1 - \xi ) \sqrt { \xi s _ { n } } } . +$$ + +213 The condition that $\lambda _ { n } J ( \Theta ^ { * } ) ~ \le ~ \epsilon _ { n }$ in Proposition 1 is mild. It implies that the true em +214 beddings of vertices within the same community are close to one another. We remark that +215 $\lambda _ { n } J ( \Theta ^ { * } )$ exactly equals to zero under the MLSBM discussed in Section 2.2. The condition that +216 $( n + M ) R \varphi ^ { - 1 } ( n , M ) \epsilon _ { n } ^ { - 1 } \log ( \epsilon _ { n } ^ { - 1 / 2 } )$ vanishes with $n$ is also mild. When $R = O ( 1 )$ , we can take any +217 ϵn such that ϵn ≫ log nn min{n,M} . Consequently, to ensure $\widehat { \Theta }$ converges to $\Theta ^ { * }$ , Theorem 1 implies the +218 smallest sparsity factor one can take is $\begin{array} { r } { s _ { n } \gg \epsilon _ { n } \gg \frac { \log n } { n \operatorname* { m i n } \{ n , M \} } } \end{array}$ log nn min{n,M} , which means that the average degree +219 of a vertex in any particular layer can be as small as $n s _ { n }$ . We remark that a common assumption +220 $M = O ( n )$ that appears in literature, such as [27] and [22], is not necessary in our theory. If we +221 further assume $\bar { M } \stackrel { } { = } O ( n )$ , we find that the average degree of a vertex in any layer under the +222 proposed TLSM set up can be smaller than that in [27] by a factor $( M \log n ) ^ { - 1 / 2 }$ and in [22] by a +223 factor $( \log n ) ^ { - 3 }$ , showing that our theoretical result accommodates sparser multi-layer networks. + +# 3.2 Consistency in community detection + +We now turn to establish the consistency of community detection in multi-layer network $\mathcal { G }$ . Let $\psi ^ { * } : [ n ] \ \longrightarrow \ [ K ]$ be the true community assignment function such that $\begin{array} { r l } { \psi ^ { * } } & { { } = } \end{array}$ $\begin{array} { r l } & { \arg \operatorname* { m i n } _ { \psi } \operatorname* { m i n } _ { C _ { 1 } , \ldots , C _ { K } } \sum _ { i = 1 } ^ { n } \| \pmb { \alpha } _ { i } ^ { * } - C _ { \psi _ { i } } \| ^ { 2 } } \end{array}$ , and then the community detection error of any estimated community assignment function $\hat { \psi }$ can be evaluated by the minimum scaled Hamming distance between $\hat { \psi }$ and $\psi ^ { * }$ under permutations, which is defined as + +$$ +\operatorname { e r r } ( \psi ^ { * } , \hat { \psi } ) = \operatorname* { m i n } _ { \pi \in S _ { K } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathbf { 1 } \{ \psi _ { i } ^ { * } \neq \pi ( \hat { \psi } _ { i } ) \} , +$$ + +230 where $\mathbf { 1 } \{ \cdot \}$ is the indicator function and $S _ { K }$ is the symmetric group of degree $K$ . Such a scaled +231 or unscaled Hamming distance has become a popular metric in quantifying the performance of +232 community detection [21, 22]. +233 Denote $N _ { k } ^ { * } = \{ i : \psi _ { i } ^ { * } = k \}$ be the $k$ -th true underlying community whose cardinality is $n _ { k }$ . Let +234 $C ^ { * } \in \mathbb { R } ^ { K \times R }$ be the true underlying community centers of the network embedding with $C _ { k . } ^ { * } =$ +235 $\begin{array} { r } { \frac { 1 } { n _ { k } } \sum _ { \psi _ { i } ^ { * } = k } \alpha _ { i . } ^ { * } } \end{array}$ , and let $\pmb { \mathcal { B } } ^ { \ast } = \pmb { \mathcal { T } } \times _ { 1 } \pmb { C } ^ { \ast } \times _ { 2 } \pmb { C } ^ { \ast } \times _ { 3 } \pmb { \beta } ^ { \ast }$ . The following assumptions are made to ensure +236 that communities within the multi-layer networks are asymptotically identifiable. + +Assumption A. Assume the difference between any two distinct horizontal slides of 37 ${ \pmb { \beta } } ^ { * }$ satisfies that + +$$ +\operatorname* { m i n } _ { k , k ^ { \prime } \in [ K ] , k \neq k ^ { \prime } } \frac { 1 } { \sqrt { K M } } \| \pmb { \mathscr { B } } _ { k , . , . } ^ { * } - \pmb { \mathscr { B } } _ { k ^ { \prime } , . , . } ^ { * } \| _ { F } \geq \gamma _ { n } , +$$ + +238 where $\gamma _ { n } > 0$ may vanish with $n$ + +Assumption B. Assume the tuning parameter $\lambda _ { n }$ satisfies that + +$$ +\begin{array} { r } { \lambda _ { n } \epsilon _ { n } s _ { n } ^ { - 2 } ( \log s _ { n } ^ { - 1 } ) ^ { - 1 } \geq c _ { 2 } , } \end{array} +$$ + +for an absolute constant $c _ { 2 }$ that does not depend on any model parameter. + +Assumption C. Denote $n _ { \mathrm { m i n } } = \mathrm { m i n } _ { k \in [ K ] } n _ { k }$ as the minimal community size. Assume + +$$ +\frac { \gamma _ { n } n _ { \mathrm { m i n } } \sqrt { \cal K } } { n } \geq c _ { \xi } \sqrt { \frac { \epsilon _ { n } } { s _ { n } } } , +$$ + +where cξ =240 $\begin{array} { r } { c _ { \xi } = \frac { 4 \sqrt 2 } { ( 1 - \xi ) \sqrt \xi } + c _ { 3 } \sqrt { \frac { ( 1 + \delta ) \operatorname* { m i n } \{ M , R \} } { M } } } \end{array}$ and $c _ { 3 }$ is a constant that depends on $\xi$ only. + +241 Assumption A is the minimal community separation requirement, and similar assumption has been +242 employed in [27] with a constant $\gamma _ { n }$ . Together with the condition $\lambda _ { n } J ( \alpha ^ { * } ) \leq \epsilon _ { n }$ in Proposition 1, +243 Assumption B gives a feasible interval for $\lambda _ { n }$ . Assumption $\textrm { C }$ allows for unbalanced communities +244 with vanishing $n _ { \mathrm { m i n } } / n$ if the network is not too sparse. Note that $c _ { \xi }$ can be further bounded by +245 $\begin{array} { r } { \frac { 4 \sqrt { 2 } } { ( 1 - \xi ) \sqrt { \xi } } + c _ { 3 } \sqrt { 1 + \delta } } \end{array}$ , and the first term of $c _ { \xi }$ will dominate the second term if $R = o ( M )$ . + +Theorem 2. Suppose all the assumptions in Theorem $^ { l }$ as well as Assumptions $A , B$ and $C$ are satisfied, it holds true that + +$$ +e r r ( \psi ^ { * } , \hat { \psi } ) \leq \frac { c _ { \xi } ^ { 2 } n \epsilon _ { n } } { n _ { \mathrm { m i n } } K \gamma _ { n } ^ { 2 } s _ { n } } , +$$ + +with probability at least 246 $\begin{array} { r } { 1 - \frac { 1 } { n ^ { 2 } } - 2 \exp \Big ( - \frac { \varphi ( n , M ) \epsilon _ { n } } { 1 5 6 \frac { \xi } { 1 - \xi } + 2 8 \log 2 } \Big ) . } \end{array}$ + +Theorem 2 assures that the community structure in a multi-layer network can be consistently recovered by the proposed TLSM. As a theoretical example, we consider a sparse case with $\begin{array} { r } { s _ { n } = \dot { \frac { ( \log n ) ^ { 1 + \tau _ { 1 } } } { n \operatorname* { m i n } \{ n , M \} } } } \end{array}$ , where $0 < \tau _ { 1 } < 1$ , $n _ { \mathrm { m a x } } = O ( n _ { \mathrm { m i n } } )$ , $\begin{array} { r } { \frac { 1 } { \sqrt { n } } | | \alpha ^ { * } - Z ^ { * } C ^ { * } | | _ { F } \leq ( \log n ) ^ { - 3 / 2 } } \end{array}$ , and both $\gamma _ { n }$ , $R$ and $K$ are of constant orders. With $\begin{array} { r } { \lambda _ { n } = \frac { ( \log n ) ^ { 2 + 2 \tau _ { 1 } } } { n \operatorname* { m i n } \{ n , M \} } } \end{array}$ , Theorems 1 and 2 imply that $\begin{array} { r } { \epsilon _ { n } = \frac { ( \log n ) ^ { 1 + \tau _ { 2 } } } { n \operatorname* { m i n } \{ n , M \} } } \end{array}$ with $0 < \tau _ { 2 } < \tau _ { 1 }$ and $e r r ( \psi ^ { * } , \hat { \psi } ) = o _ { p } ( 1 )$ . + +# 52 4 Numerical experiments + +In this section, we evaluate the numerical performance of the proposed TLSM in a variety of synthetic as well as real-life multi-layer networks, compare it against four competitors in literature, including the mean adjacency spectral embeddings (MASE; 16), least square estimation (LSE; 27), Tucker decomposition with HOSVD initialization (HOSVD-Tucker; 22), and spectral kernel (SPECK; 35), and conduct some ablation studies. The implementations of LSE and SPECK are available at the authors’ personal websites, HOSVD-Tucker is implemented in the routine “tucker" of the Python package “tensorly", and TLSM and MASE are implemented in Python by ourselves. + +# 4.1 Synthetic networks + +The multi-layer network $\mathcal { A } = ( a _ { i , j , m } ) \in \{ 0 , 1 \} ^ { n \times n \times M }$ is generated as follows. First, we randomly select $K = 4$ elements uniformly from $\{ 2 . 5 * ( b _ { 1 } , b _ { 2 } , \ldots , b _ { R } ) : b _ { r } \in \{ - 1 , 1 \} , r \in [ R ] \}$ as community centers, which are denoted as $c _ { k }$ , $k \in [ K ]$ . Second, the latent space embedding of vertex $i$ is generated as $\pmb { \alpha } _ { i } = \pmb { c } _ { \psi _ { i } } + \pmb { e } _ { i }$ with $\pmb { e } _ { i } \sim N ( \mathbf { 0 } _ { R } , 1 . 5 * I _ { R } )$ , and $\psi _ { i } \in [ K ]$ are independently drawn from the multinomial distribution $\mathbf { M u l t i } ( 1 ; \frac { 1 } { K } \mathbf { 1 } _ { K } )$ . Third, we generate $\beta = [ \beta _ { 1 } , \ldots , \beta _ { M } ] ^ { T }$ with $\beta _ { m , r }$ being independent standard normal random varibeles, for $m \in [ M ]$ and $r \in [ R ]$ . We then rescale the column norms of $\beta$ to be 1 for model identifiability. Finally, we generate $\mathcal { A }$ according to the proposed TLSM with $s _ { n } = 0 . 1$ . For the sake of fair comparisons, the embedding dimension $R$ is set as $K$ in all scenarios. We aim to illustrate the community detection performance of all methods as the number of vertices and number of layers increase. To this end, we consider $( n , M ) \in \{ 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 \} \times \{ 5 , 1 0 , 1 5 , 2 0 \}$ . The averaged hamming errors and their standard errors over 50 independent experiments of all methods are reported in Table 1. + +273 It is evident that TLSM consistently outperforms its competitors, and the performances of LSE +274 and HOSVD-Tucker are better than those of MASE and SPECK. This is expected since TLSM, +275 LSE and HOSVD-Tucker work on the multi-layer network adjacency tensor directly, while MASE +276 and SPECK are matrix aggregation methods that suffer form information loss. Furthermore, as the +277 number of vertices and number of layers increase, the community detection errors of all methods +278 decrease rapidly. Notably, TLSM and LSE converge faster than the other methods, and attain stable +279 performance even for relatively small $n$ and $M$ . Additional simulation studies for various network +280 sparsity and unbalanced community sizes are relegated to Appendix C in the supplementary materials. + +# 4.2 Real-life networks + +282 We also apply the proposed TLSM method to analyze three real-life multi-layer networks, including +283 a social network in the department of Computer Science at Aarhus University (AUCS) [38], a yeast +284 Saccharomyces cerevisiae gene co-expression (YSCGC) network [44], and a worldwide agriculture +285 trading network (WAT) [10]. Specifically, we conduct community detection on the first two networks +286 whose vertex community memberships are available, and carry out a link prediction task on the third +287 network whose vertex community memberships are unavailable. + +Table 1: The averaged hamming errors of various methods with their standard errors in Scenario I. The best performer in each case is bold-faced. + +
nMTLSMLSEMASEHOSVD-TuckerSPECK
20050.1180(0.0147)0.1405(0.0118)0.5086(0.0136)0.1623(0.0126)0.4254(0.0138)
100.0585(0.0046)0.0751(0.0050)0.4949(0.0131)0.1148(0.0106)0.2996(0.0141)
150.0551(0.0067)0.0593(0.0045)0.4910(0.0176)0.1040(0.0115)0.2505(0.0142)
200.0510(0.0037)0.0588(0.0043)0.4977(0.0161)0.1023(0.0110)0.1942(0.0156)
40050.0653(0.0066)0.1019(0.0087)0.3845(0.0193)0.1220(0.0106)0.3766(0.0195)
100.0608(0.0063)0.0636(0.0037)0.3859(0.0160)0.1012(0.0092)0.2244(0.0191)
150.0511(0.0031)0.0595(0.0036)0.3844(0.0221)0.0787(0.0051)0.1490(0.0123)
200.0536(0.0047)0.0551(0.0036)0.3985(0.0185)0.0795(0.0063)0.1409(0.0131)
60050.0607(0.0029)0.0909(0.0040)0.3665(0.0186)0.1221(0.0108)0.3038(0.0193)
100.0567(0.0029)0.0688(0.0031)0.3726(0.0179)0.1003(0.0081)0.1651(0.0127)
150.0558(0.0027)0.0630(0.0030)0.3803(0.0167)0.0918(0.0076)0.1231(0.0076)
200.0548(0.0028)0.0586(0.0029)0.3814(0.0185)0.0883(0.0078)0.1150(0.0088)
80050.0556(0.0056)0.0768(0.0055)0.3012(0.0194)0.1003(0.0103)0.2733(0.0171)
100.0560(0.0063)0.0583(0.0034)0.3004(0.0177)0.0788(0.0065)0.1424(0.0127)
150.0498(0.0030)0.0539(0.0033)0.3179(0.0195)0.0812(0.0068)0.1146(0.0098)
200.0485(0.0031)0.0516(0.0032)0.3184(0.0218)0.0803(0.0075)0.0979(0.0078)
+ +The AUCS dataset is publicly available at http://multilayer.it.uu.se/datasets.html, and it is a $6 1 \times 6 1 \times 5$ multi-layer network that records pairwise relationships of 5 types among 61 persons in AUCS, including current working relationships, repeated leisure activities, regularly eating lunch together, co-authorship of a publication, and friendship on Facebook. Since 54 persons in the dataset come from 7 research groups and the other 7 persons do not belong to any group, the dataset consists of 8 communities corresponding to 7 research groups and an outlier community. Applying TLSM and its competitors to the dataset, the number of misclassified vertices by TLSM, LSE, MASE, HOSVD-Tucker and SPECK, are 8, 21, 19, 23, 18, respectively. Clearly, TLSM significantly outperforms its competitors by at least reducing $1 6 . 3 9 \%$ of community detection error. + +The YSCGC dataset is publicly available at https://www.ncbi.nlm.nih.gov/pmc/articles/ $\mathtt { P M C 1 5 6 5 9 0 } /$ , and contains 205 genes of 4 functional categories, including protein metabolism and modification, carbohydrate metabolism and catabolism, nucleobase, nucleoside, nucleotide and nucleic acide metabolism, as well as transportation. We regard these four functional category labels as the community memberships of the genes. Further, the gene expression responses are measured by 20 systematic perturbations with varying genetic and environmental conditions in 4 replicated hybridizations. We thus constructed a gene co-expression network $\mathcal { A } = ( a _ { i , j , m } ) \in$ $\mathbb { R } ^ { 2 0 \bar { 5 } \times 2 0 5 \times 4 }$ based on the similarities of their expressions, where each layer represents one replicated hybridization. Specifically, the similarity between genes $i$ and $j$ in the $m$ -th replication is measured by $w _ { i , j , m } = \mathrm { e x p } \big ( - \| \pmb { x } _ { i } ^ { ( m ) } - \pmb { x } _ { j } ^ { ( m ) } \| \big )$ , where $\pmb { x } _ { i } ^ { ( m ) } \in \mathbb { R } ^ { 2 0 }$ contains the expression levels of 20 perturbations in the $m$ -th replicated hybridization for $i \in [ 2 0 5 ]$ and $m \in [ 4 ]$ . The binary value $a _ { i , j , m }$ is obtained by thresholding $w _ { i , j , m }$ with the thresholding value being the $60 \%$ quantile of all elements in $\{ w _ { i , j , m } : i \le j \in [ 2 0 5 ] , m \in [ 4 ] \}$ . Applying TLSM and its competitors to this dataset, the number of misclassified vertices by TLSM, LSE, MASE, HOSVD-Tucker and SPECK, are 6, 9, 12, 48, 13, respectively. TLSM again outperforms its competitors in this YSCGC dataset. + +312 The WAT dataset is publicly available at http://www.fao.org, and includes 364 agriculture +313 product trading relationships among 214 countries in 2010. To process the data, we extract 130 major +314 countries whose average degrees are greater than 9 from the 32 densest connected agriculture product +315 trading relations, leading to a $1 3 0 \times 1 3 0 \times 3 2$ multi-layer network. Investigating the eigen-structure +316 of the mode-1 matricization of the network adjacency tensor, we identify an elbow point [20] at the +317 7th largest eigen-value, suggesting there are 6 potential communities among the countries, and thus +318 we set $K = 6$ . The corresponding eigen-value plot is attached in Appendex D of the supplementary +319 materials. We then randomly selected $8 0 \%$ of the entries of the adjacency tensor as the training set, +320 and conduct link prediction on the remaining $2 0 \%$ of the entries. Specifically, we employ TLSM +321 and the adaptations of its competitors to estimate the network expected tensor $\mathcal { P }$ and generate +322 estimations for the missing entries by independent Bernoulli random variables accordingly. The +323 averaged link prediction accuracy of TLSM, LSE, MASE, HOSVD-Tucker and SPECK over 50 +324 independent replications are $7 9 . 6 0 \%$ , $7 6 . 6 6 \%$ , $7 5 . 9 6 \%$ , $7 7 . 7 8 \%$ and $7 9 . 0 8 \%$ , respectively, where the +325 link prediction accuracy is defined as the percentile of the correctly predicted entries. Clearly, all 5 +326 methods are comparative in terms of link prediction, while TLSM still deliver highest averaged link +327 prediction accuracy. + +# 328 4.3 Ablation studies + +In this subsection, we carry out some ablation studies on two novel components of the proposed method, namely the sparsity factor $s _ { n }$ and the community-inducing regularizer $J ( \alpha )$ . To study the effectiveness of $s _ { n }$ , we generate a $3 0 0 \times 3 0 0 \times 5$ multi-layer network with 3 communities and the true network sparsity $s _ { n } = 0 . 3$ . The blue curve in the left panel of Figure 1 shows the average Hamming error of 50 independent replications given by the proposed method when employing $\hat { s } _ { n } \in \{ 0 . 0 5 i : i \in [ 2 0 ] \}$ in the optimization algorithm, and the red line indicates the averaged Hamming error of the proposed method with $\hat { s } _ { n }$ estimated via the proposed data-adapted estimation scheme. It is clear that the Hamming error at $s _ { n } = 1$ is much larger than that when $s _ { n }$ is close to 0.3, showing the advantages of the modified logit transformation by $s _ { n }$ over the standard logit transformation when the network indeed reveals sparse pattern. Moreover, we observe that the red line is even lower than the minimum Hamming error in the blue curve. This further confirms the effectiveness of the proposed data-adapted estimation scheme for estimating $s _ { n }$ . + +![](images/c331ba0dbd825a27deb1e7af3e215fd861180ab5cae8dc4dc8e45ed78969d719.jpg) +Figure 1: Ablation studies on $s _ { n }$ (left) and community-inducing regularizer (right). + +341 To study the effectiveness of the community-inducing regularizer in the proposed objective function, +342 we generate an $n \times n \times 5$ multi-layer network with 2 communities, for $\overline { { n } } \in \{ 5 0 , 1 0 \mathrm { { 0 } } , 2 0 0 , 4 0 0 \}$ . In +343 the right panel of Figure 1, the black pillars indicate the network estimation error $\frac { 1 } { n \sqrt { 5 } } \| \widehat { \Theta } - \Theta ^ { * } \| _ { F }$ +344 given by the proposed method with $\lambda _ { n } = 0$ which corresponds to the absence of $J ( \alpha )$ , while the +345 red ones indicate the counterparts given by the proposed method with $\lambda _ { n }$ is selected by network +346 cross-validation. There is a clear improvement when the community-inducing regularizer is enforced +347 in all scenarios, particularly for small $n$ . This showcases the helpfulness of the community-inducing +348 regularizer in detecting network community structure. + +# 349 5 Conclusions + +50 In this paper, we propose a novel tensor-based latent space model for community detection in +51 multi-layer networks. 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[No] There should be no negative societal impacts. +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3. (b) Did you include complete proofs of all theoretical results? [Yes] All technical proofs are provided in Appendix E of the supplementary materials. + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The URLs for data are included in Section 4.2, and codes with instructions are included in the supplementary materials. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 2.3 and Appendix B in the supplementary materials. +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We show the standard erros in Table 1 and $9 5 \%$ confident intervals of additional simulation studies in Appendix C in the supplementary materials. +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] We used publicly available datasets and cite the creators. +(b) Did you mention the license of the assets? [Yes] All datasets we used are publicly available. +(c) Did you include any new assets either in the supplemental material or as a URL? [No] +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] All data we used do not contains personally identifiable information or offensive content. + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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