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+# INVERTIBLE GENERATIVE MODELS FOR INVERSE PROBLEMS: MITIGATING REPRESENTATION ERROR AND DATASET BIAS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Trained generative models have shown remarkable performance as priors for inverse problems in imaging. For example, Generative Adversarial Network priors permit recovery of test images from $5 \mathrm { - } 1 0 \mathrm { x }$ fewer measurements than sparsity priors. Unfortunately, these models may be unable to represent any particular image because of architectural choices, mode collapse, and bias in the training dataset. In this paper, we demonstrate that invertible neural networks, which have zero representation error by design, can be effective natural signal priors at inverse problems such as denoising, compressive sensing, and inpainting. Our formulation is an empirical risk minimization that does not directly optimize the likelihood of images, as one would expect. Instead we optimize the likelihood of the latent representation of images as a proxy, as this is empirically easier. For compressive sensing, our formulation can yield higher accuracy than sparsity priors across almost all undersampling ratios. For the same accuracy on test images, they can use 10-20x fewer measurements. We demonstrate that invertible priors can yield better reconstructions than sparsity priors for images that have rare features of variation within the biased training set, including out-of-distribution natural images.
+
+# 1 INTRODUCTION
+
+![](images/be78d18a239aa32bc7621313d5c99c936a6f56f84817bd8cb6cad51402c3ea77.jpg)  
+Figure 1: We train an invertible generative model with CelebA images (including those at left). When used as a prior for compressed sensing, it can yield higher quality image reconstructions than Lasso and a trained DCGAN, even on out-of-distribution images. Note that the DCGAN reflects biases of the training set by removing the man’s glasses and beard, whereas our invertible prior does not.
+
+Generative deep neural networks have shown remarkable performance as natural signal priors in imaging inverse problems, such as denoising, inpainting, compressed sensing, blind deconvolution, and phase retrieval. These generative models can be trained from datasets consisting of images of particular natural signal classes, such as faces, fingerprints, MRIs, and more (Karras et al., 2017; Minaee and Abdolrashidi, 2018; Shin et al., 2018; Chen et al., 2018). Some such models, including variational autoencoders (VAEs) and generative adversarial networks (GANs), learn an explicit low-dimensional manifold that approximates a natural signal class (Goodfellow et al., 2014; Kingma and Welling, 2013; Rezende et al., 2014). We will refer to such models as GAN priors. With an explicit parameterization of the natural signal manifold by a low dimensional latent representation, these generative models allow for direct optimization over a natural signal class. Consequently, they can obtain significant performance improvements over non-learning based methods. For example, GAN priors have been shown to outperform sparsity priors at compressed sensing with $5 \mathrm { - } 1 0 \mathrm { x }$ fewer measurements. Additionally, GAN priors have led to theory for signal recovery in the linear compressive sensing and nonlinear phase retrieval problems (Bora et al., 2017; Hand and Voroninski, 2017; Hand et al., 2018), and they have also shown promising results for the nonlinear blind image deblurring problem (Asim et al., 2018).
+
+A significant drawback of GAN priors for solving inverse problems is that they can have representation error or bias due to architecture and training. This can happen for many reasons, including because the generator only approximates the natural signal manifold, because the natural signal manifold is of higher dimensionality than modeled, because of mode collapse, or because of bias in the training dataset itself. As many aspects of generator architecture and training lack clear principles, representation error of GANs may continue to be a challenge even after substantial hand crafting and engineering. Additionally, learning-based methods are particularly vulnerable to the biases of their training data, and training data, no matter how carefully collected, will always contain degrees of bias. As an example, the CelebA dataset (Liu et al., 2015) is biased toward people who are young, who do not have facial hair or glasses, and who have a light skin tone. As we will see, a GAN prior trained on this dataset learns these biases and exhibits image recovery failures because of them.
+
+In contrast, invertible neural networks can be trained as generators with zero representation error. These networks are invertible (one-to-one and onto) by architectural design (Dinh et al., 2016; Gomez et al., 2017; Jacobsen et al., 2018; Kingma and Dhariwal, 2018). Consequently, they are capable of recovering any image, including those significantly out-of-distribution relative to a biased training set; see Figure 1. We call the domain of an invertible generator the latent space, and we call the range of the generator the signal space. These must have equal dimensionality. Flow-based invertible generative models are composed of a sequence of learned invertible transformations. Their strengths include: their architecture allows exact and efficient latent-variable inference, direct loglikelihood evaluation, and efficient image synthesis; they have the potential for significant memory savings in gradient computations; and they can be trained by directly optimizing the likelihood of training images. This paper emphasizes an additional strength: because they lack representation error, invertible models can mitigate dataset bias and improve performance on inverse problems with out-of-distribution data.
+
+In this paper, we study generative invertible neural network priors for imaging inverse problems. We will specifically use the Glow architecture, though our framework could be used with other architectures. A Glow-based model is composed of a sequence of invertible affine coupling layers, 1x1 convolutional layers, and normalization layers. Glow models have been successfully trained to generate high resolution photorealistic images of human faces (Kingma and Dhariwal, 2018).
+
+We present a method for using pretrained generative invertible neural networks as priors for imaging inverse problems. The invertible generator, once trained, can be used for a wide variety of inverse problems, with no specific knowledge of those problems used during the training process. Our method is an empirical risk formulation based on the following proxy: we penalize the likelihood of an image’s latent representation instead of the image’s likelihood itself. While this may be couterintuitive, it admits optimization problems that are easier to solve empirically. In the case of compressive sensing, our formulation succeeds even without direct penalization of this proxy likelihood, with regularization occuring through initialization of a gradient descent in latent space.
+
+We train a generative invertible model using the CelebA dataset. With this fixed model as a signal prior, we study its performance at denoising, compressive sensing, and inpainting. For denoising, it can outperform BM3D (Dabov et al., 2007). For compressive sensing on test images, it can obtain higher quality reconstructions than Lasso across almost all subsampling ratios, and at similar reconstruction errors can succeed with $1 0 { - } 2 0 \mathrm { x }$ fewer measurements than Lasso. It provides an improvement of about $2 \mathbf { x }$ fewer linear measurements when compared to Bora et al. (2017). Despite being trained on the CelebA dataset, our generative invertible prior can give higher quality reconstructions than Lasso on out-of-distribution images of faces, and, to a lesser extent, unrelated natural images. Our invertible prior outperforms a pretrained DCGAN (Radford et al., 2015) at face inpainting and exhibits qualitatively reasonable results on out-of-distribution human faces. We provide additional experiments in the appendix, including for training on other datasets.
+
+# 2 METHOD AND MOTIVATION
+
+We assume that we have access to a pretrained generative invertible neural network $G : \mathbb { R } ^ { n }  \mathbb { R } ^ { n }$ . We write $x = G ( z )$ and $z = G ^ { - 1 } ( \dot { x } )$ , where $x \in \mathbb { R } ^ { n }$ is an image that corresponds to the latent representation $z \in \mathbb { R } ^ { n }$ . We will consider a $G$ that has the Glow architecture introduced in Kingma and Dhariwal (2018). It can be trained by direct optimization of the likelihood of a collection of training images of a natural signal class, under a standard Gaussian distribution over the latent space. We consider recovering an image $x$ from possibly-noisy linear measurements given by $A \in \mathbb { R } ^ { \bar { m } \times n }$ ,
+
+$$
+y = A x + \eta ,
+$$
+
+where $\eta \in \mathbb { R } ^ { m }$ models noise. Given a pretrained invertible generator $G$ , we have access to likelihood estimates for all images $x \in \mathbb { R } ^ { n }$ . Hence, it is natural to attempt to solve the above inverse problem by a maximum likelihood formulation given by
+
+$$
+\operatorname* { m i n } _ { x \in \mathbb { R } ^ { n } } \| A x - y \| ^ { 2 } - \gamma \log p _ { G } ( x ) ,
+$$
+
+where $p _ { G }$ is the likelihood function over $x$ induced by $G$ , and $\gamma$ is a hyperparameter. We have found this formulation to be empirically challenging to optimize; hence we study the following proxy:
+
+$$
+\operatorname* { m i n } _ { z \in \mathbb { R } ^ { n } } \| A G ( z ) - y \| ^ { 2 } + \gamma \| z \| .
+$$
+
+Unless otherwise stated, we initialize (2) at $z _ { 0 } = 0$ .
+
+The motivation for formulation (2) is as follows. As a proxy for the likelihood of an image $x \in \mathbb { R } ^ { n }$ , we will use the likelihood of its latent representation $z \stackrel { - } { = } G ^ { - 1 } ( x )$ . Because the invertible network $G$ was trained to map a standard normal in $\mathbb { R } ^ { n }$ to a distribution over images, the log-likelihood of a point $z$ is proportional to $\| z \| ^ { 2 }$ . Instead of penalizing $\| z \| ^ { 2 }$ , we alternatively penalize the unsquared $\| z \|$ . In Appendix B, we show comparable performance for both the squared and unsquared formulations.
+
+In principle, our formulation has an inherent flaw: some high-likelihood latent representations $z$ correspond to low-likelihood images $x$ . Mathematically, this comes from the Jacobian term that relates the likelihood in $z$ to the likelihood in $x$ upon application of the map $G$ . For multimodel distributions, such images must exist, which we will illustrate in the discussion. This proxy formulation relies on the fact that the set of such images has low probability and that they are inconsistent with enough provided measurements. Surprisingly, despite this potential weakness, we will observe image reconstructions that are superior to BM3D and GAN-based methods at denoising, and superior to GAN-based and Lasso-based methods at compressive sensing.
+
+In the case of compressive sensing and inpainting, we take $\gamma = 0$ in formulation (2). The motivation for such a formulation initialized at $z _ { 0 } = 0$ is as follows. There is a manifold of images that are consistent with the provided measurements. We want to find the image $x$ of highest likelihood on this manifold. Our proxy turns the likelihood maximization task over an affine space in $x$ into the geometric task of finding the point on a manifold in $z$ -space that is closest to the origin with respect to the Euclidean norm. In order to approximate that point, we run a gradient descent in $z$ down the data misfit term starting at $z _ { 0 } = 0$ .
+
+In the case of GAN priors for $G : \mathbb { R } ^ { k }  \mathbb { R } ^ { n }$ , we will use the formulation from Bora et al. (2017), which is the formulation above in the case where the optimization is performed over $\mathbb { R } ^ { k }$ , $\gamma = 0$ , and initialization is selected randomly.
+
+All the experiments that follow will be for an invertible model we trained on the CelebA dataset of celebrity faces, as in Kingma and Dhariwal (2018). Similar results for models trained on birds and flowers (Wah et al., 2011; Nilsback and Zisserman, 2008) can be found in the appendix. Due to computational considerations, we run experiments on $6 4 \times 6 4$ color images with the pixel values scaled between [0, 1]. The train and test sets contain a total of 27,000 and 3,000 images, respectively. We trained a Glow architecture (Kingma and Dhariwal, 2018); see Appendix A for details. Once trained, the Glow prior is fixed for use in each of the inverse problems below. We also trained a DCGAN for the same dataset. We solve (2) using LBFGS, which was found to outperform Adam (Kingma and Ba, 2014). DCGAN results are reported for an average of 3 runs because we observed some variance due to random initialization.
+
+# 3 APPLICATIONS
+
+# 3.1 DENOISING
+
+We consider the denoising problem with $A = I$ and $\eta \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I )$ , for images $x$ in the CelebA test dataset. We evaluate the performance of a Glow prior, a DCGAN prior, and BM3D for two different noise levels. Figure 2 shows the recovered PSNR values as a function of $\gamma$ for denoising by the Glow and DCGAN priors, along with the PSNR by BM3D. The figure shows that the performance of the regularized Glow prior increases with $\gamma$ , and then decreases. If $\gamma$ is too low, then the network fits to the noise in the image. If $\gamma$ is too high, then data fit is not enforced strongly enough. The left panel reveals that an appropriately regularized Glow prior can outperform BM3D by almost 2 dB. The experiments also reveal that appropriately regularized Glow priors outperform the DCGAN prior, which suffers from representation error and is not aided by the regularization. The right panel confirms that with smaller noise levels, less regularization is needed for optimal performance. A visual comparison of the recoveries at the noise level $\sigma = 0 . 1$ using Glow, DCGAN priors, and BM3D can be seen in Figure 3. Note that the recoveries with Glow are sharper than BM3D. See Appendix B for more quantitative and qualitative results.
+
+![](images/2075a5ed2083f8349c7cdfc0c3f413e46d9251728f3659d14eea35f11431121c.jpg)  
+Figure 2: Recovered PSNR values as a function of $\gamma$ for denoising by the Glow and DCGAN priors. All the results are averaged over 12 test set images. For reference, we show the average PSNRs of the original noisy images, after applyig BM3D, and under the Glow prior in the noiseless case $( \sigma = 0$ ).
+
+![](images/3318d6b6c6d5602cd03471c3a0e78fe9c4563377c5109e9d2ea3c40230eb8a0a.jpg)  
+Figure 3: Denoising results using the Glow prior, the DCGAN prior, and BM3D at noise level $\sigma = 0 . 1$ . Note that the Glow prior gives a sharper image than BM3D.
+
+# 3.2 COMPRESSED SENSING
+
+In compressed sensing, one is given undersampled linear measurements of an image, and the goal is to recover the image from those measurements. In our notation, $A \in \mathbb { R } ^ { m \times n }$ with $m < n$ . As the image $x$ is undersampled, there is an affine space of images consistent with the measurements, and an algorithm must select which is most ‘natural.’ A common proxy for naturalness in the literature has been sparsity with respect to the DCT or wavelet bases. With a GAN prior, an image is considered natural if it lies in or near the range of the GAN. For an invertible prior under our proxy for likelihood, we consider an image to be natural if it has a latent representation of small norm.
+
+We study compressed sensing in the case that $A$ is an $m \times n$ matrix of i.i.d. $\mathcal { N } ( 0 , 1 / m )$ entries, and $x$ is an image from the CelebA test set. Here, $n = 6 4 \times 6 4 \times 3 = 1 2 2 8 8$ . We consider the case where $\eta$ is standard iid Gaussian random noise normalized such that $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . We compare Glow, DCGAN, and Lasso1 with respect to the DCT and wavelet bases.
+
+Our main result is that the Glow prior with $\gamma = 0$ and initialization $z _ { 0 } = 0$ outperforms both DCGAN and Lasso in reconstruction quality over all undersampling ratios, as shown in the left panel of Figure 4. Surprisingly, in the case of extreme undersampling, Glow substantially outperforms these methods even though it does not maintain a direct low-dimensional parameterization of the signal manifold. The Glow prior (1) can result in 15 dB higher PSNRs than DCGAN, and (2) can give comparable recovery errors with $2 { - } 3 \mathbf { x }$ fewer measurements at high undersampling ratios. This difference is explained by the representation error of DCGAN, which has been shown to be the dominant source of error in DCGAN by Bora et al. (2017). Additional plots and visual comparisons, available in Appendix C, show notable improvements in quality of in- and out-of-distribution images using an invertible prior relative to DCGAN and Lasso.
+
+![](images/c165f08eb9db10dea04f05a19030b559a8d7a310ed2002b1c112b095bd53d9b6.jpg)  
+Figure 4: The left panel shows recovered PSNRs averaged over 12 test set images under the Glow, and DCGAN prior with $\gamma = 0$ ; and the Lasso with respect to the DCT and a Wavelet Transform. We initialize with $z _ { 0 } = 0$ . See Appendix C for a zoom-in of the case of small $m$ . The right panel shows the resulting PSNR when $m = 5 0 0 0$ with a Glow prior after different initialization strategies, as described in the text. The highest PSNR was recovered with initialization $z _ { 0 } = 0$ and $\gamma = 0$ .
+
+We conducted several additional experiments to understand the regularizing effects of $\gamma$ and the initialization $z _ { \mathrm { 0 } }$ . The right panel of Figure 4 shows the PSNRs under multiple initialization strategies: $z _ { 0 } = 0$ , $z _ { 0 } \sim \dot { \mathcal { N } } ( 0 , 0 . 1 ^ { \bar { 2 } } I )$ , $z _ { 0 } \sim \mathcal { N } ( \bar { 0 , } 0 . 7 ^ { 2 } I )$ , $z _ { 0 } = G ^ { - 1 } ( x _ { 0 } )$ with $x _ { 0 }$ given by the solution to Lasso with respect to the wavelet basis, and $z _ { 0 } = G ^ { - 1 } ( x _ { 0 } )$ where $x _ { 0 }$ is $x$ perturbed by a random point in the null space of $A$ . The best performance was observed with initialization $z _ { 0 } = 0$ . The hyperparameter $\gamma$ can be taken to be zero, which is surprising because then there is no direct penalization of likelihood for this noisy compressive sensing problem. In the case of $\gamma = 0$ , we observe that larger initializations result in recovered images of lower PSNR. See Appendix C for additional experiments that show this effect. We observe that initialization strategy can have a strong qualitative effect on the recovery formulation. For example, if the optimization is initialized by the solution to the Lasso, then directly penalizing the likelihood of $z$ can improve reconstruction PSNR, though those reconstruction are still worse than with initialization $z _ { 0 } = 0$ and $\gamma = 0$ . Suboptimal initialization procedures apparently benefit from direct penalization of likelihood, whereas the $z _ { 0 } = 0$ initialization apparently does not.
+
+Finally, we observe that the Glow prior is much more robust to out-of-distribution examples than the GAN Prior. Figure 5 shows recovered images using (2) for compressive sensing for images not belonging to the CelebA dataset. DCGAN’s performance reveals biases of the underlying dataset and limitations of low-dimensional modeling. For example, projecting onto the CelebA-trained DCGAN can cause incorrect skin tone, gender, and age. It’s performance on out-of-distribution images is poor.
+
+In contrast, the Glow prior mitigates this bias, even demonstrating image recovery for natural images that are not representative of the CelebA training set, including people who are older, have darker skin tones, wear glasses, have a beard, or have unusual makeup. The Glow prior’s performance also extends to significantly out-of-distribution images, such as animated characters and natural images unrelated to faces. See Appendix C.2 for additional experiments.
+
+![](images/8dcb148392a47110339a14cdc438533f289af3646091310acee72aac32c1034a.jpg)  
+Figure 5: Compressed sensing (CS) with a number $m = 2 , 5 0 0 ( \approx 2 0 \% )$ of measurements of outof-distribution images. Visual comparisons: CS under the Glow prior, DCGAN prior, Lasso-WVT, and Lasso-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance. We use $\gamma = 0$ for both DCGAN and Glow priors and $\gamma = 0 . 0 1$ for Lasso-WVT, and Lasso-DCT, respectively.
+
+# 3.3 INPAINTING
+
+In inpainting, one is given a masked image of the form $y = M \odot x$ , where $M$ is a masking matrix with binary entries and $x \in \mathbb { R } ^ { n }$ is an n-pixel image. The goal is to find $x$ . We could rewrite (2) with $\gamma = 0$ as
+
+$$
+\operatorname* { m i n } _ { z \in \mathbb { R } ^ { n } } \| y - M \odot G ( z ) \| ^ { 2 }
+$$
+
+There is an affine space of images consistent with the measurements, and an algorithm must select which is most natural. As before, using the minimizer $\hat { z }$ , the estimated image is given by $G ( \hat { z } )$ . Our experiments reveal the same story as for compressed sensing. If initialized at $z _ { 0 } = 0$ , then the empirical risk formulation with $\gamma = 0$ exhibits high PSNRs on test images. Algorithmic regularization is again occurring due to initialization. In contrast, DCGAN is limited by its representation error. See Figure 6, and Appendix D for more results, including visually reasonable face inpainting, even for out-of-distribution human faces.
+
+# 4 DISCUSSION
+
+![](images/ce5bcdfd53ea689ffcabd97003a69735e8bb28253af120803aafc6f4689d6744.jpg)  
+Figure 6: Inpainting: Recoveries under DCGAN and Glow, both with $\gamma = 0$ .
+
+We have demonstrated that pretrained generative invertible models can be used as natural signal priors in imaging inverse problems. Their strength is that every desired image is in the range of an invertible model, and the challenge that they overcome is that every undesired image is also in the range of the model and no explicit low-dimensional representation is kept. We study a regularization for empirical loss minimization that promotes recovery of images that have a high value of a proxy for image likelihood under the generative model. We demonstrate that this formulation can quantitatively and qualitatively outperform BM3D at denoising. Additionally, it has lower recovery errors than Lasso across all levels of undersampling, and it can get comparable errors from 10-20x fewer measurements, which is a $2 \mathbf { x }$ reduction from Bora et al. (2017). The superior recovery performance of the invertible prior at very extreme undersampling ratios is particularly surprising given that invertible nets do not maintain explicit low dimensional representations, as GANs do. Additionally, our trained invertible model yields significantly better reconstructions than Lasso even on out-of-distribution images, including images with rare features of variation, and on unrelated natural images.
+
+The idea of analyzing inverse problems with invertible neural networks has appeared in Ardizzone et al. (2018). The authors study estimation of the complete posterior parameter distribution under a forward process, conditioned on observed measurements. Specifically, the authors approximate a particular forward process by training an invertible neural network. The inverse map is then directly available. In order to cope with information loss, the authors augment the measurements with additional variables. This work differs from ours because it involves training a separate net for every particular inverse problem. In contrast, our work studies how to use a pretrained invertible generator for a variety of inverse problems not known at training time. Training invertible networks is challenging and computationally expensive; hence, it is desirable to separate the training of off-the-shelf invertible models from potential applications in a variety of scientific domains.
+
+# Why optimize a proxy for image likelihood instead of optimizing image likelihood directly?
+
+As noted in Section 2, the immediate formulation one would write down for inverse problems under an invertible prior is to optimize a data misfit term together with an image log-likelihood term. Unfortunately, we found it difficult to get this optimization to converge in practice. The likelihood term can exhibit rapid variation due to the Jacobian of the transformation $z \mapsto x = G ( z )$ ; additionally the likelihood term may in principle even contain local minima or other geometric properties that make gradient descent difficult. Figure 7 compares the loss landscapes in $x$ and $z$ , illustrating that the learned likelihood function in $x$ may lead to difficulty in choosing appropriate step sizes for gradient descent algorithms.
+
+![](images/58f55a127e7ab3bfa9eec056d0f5860a5bee1181578e8033346229aefd18d7a5.jpg)  
+Figure 7: Landscapes of (a) the loss surface in $x$ -space, (b) just the image likelihood in $x$ -space, and (c) the loss surface in $z$ -space, as functions of two random directions in either $x$ or $z$ , as appropriate.
+
+In contrast, there are nice geometric properties that appear in latent space from an invertible model. As an illustration, consider the compressive sensing problem with noiseless measurements. Here, the formulation corresponds to a gradient descent down the data misfit term $\| A G ( z ) - y \| ^ { 2 }$ starting at $z _ { 0 } = 0$ . This data misfit term has a favorable geometry for optimization in that all local minima are global minima. This is because the level sets in $z$ of $\| A G ( \bar { z } ) - y \| ^ { 2 }$ are given by $G ^ { - 1 }$ applied to the level sets in $x$ of $\| A x - y \| ^ { 2 }$ , which have a simple structure because of the linearity of the measurements in $x$ . There may be additional benefits due to optimizing in $z$ because the invertible net learns representations that permit interpolation between images and semantically meaningful arithmetic, as reported in Kingma and Dhariwal (2018).
+
+# Why is the likelihood of an image’s latent representation a reasonable proxy for the image’s likelihood?
+
+The training process for an invertible generative model attempts to learn a target distribution in images space by directly maximizing the likelihood of provided samples from that distribution, given a standard Gaussian prior in latent space. High probability regions in latent space map to regions in image space of equal probability. Hence, broadly speaking, regions of small values of $\lVert z \rVert$ are expected to map to regions of large likelihoods in image space. There will be exceptions to this property. For example, natural image distributions have a multimodal character. The preimage of high probability modes in image space will correspond to high likelihood regions in latent space. Because the generator $G$ is invertible and continuous, interpolation in latent space of these modes will provide images of high likelihood in $z$ but low likelihood in the target distribution. To illustrate this point, we trained a Real-NVP (Dinh et al., 2016) invertible neural network on the two dimensional set of points depicted in Figure 8 (left panel). The middle and right panels show that high likelihood regions in latent space generally correspond to higher likelihood regions in image space, but that there are some regions of high likelihood in latent space that map to points of low likelihood in image space and in the target distribution. We see that the spurious regions are of low total probability and would be unlikely to be the desired outcomes of an inverse problem arising from the target distribution.
+
+![](images/ebdcd435587b520bdd424f2c78ba74e26e823e4d86718bb7b5ae8b5b865cbba0.jpg)  
+Figure 8: An invertible net was trained on the data points in $x$ -space (left), resulting in the given plots of latent $z$ -likelihood versus $x$ (middle), and $x$ -likelihood versus latent representation $z$ (right).
+
+How can solving compressive inverse problems be successful without direct penalization of the proxy image likelihood?
+
+If there are fewer linear measurements than the dimensionality of the desired signal, an affine space of images is consistent with the measurements. In our formulation, regularization does not occur by direct penalization of our proxy for image likelihood; instead, it occurs implicitly by performing the optimization in $z$ -space with an initialization of $z _ { 0 } = 0$ . The set of latent representations $z$ that are consistent with the compressive measurements define a $m$ -dimensional nonlinear manifold. As per the likelihood proxy mentioned above, the spirit of our formulation is to find the point on this manifold that is closest to the origin with respect to the Euclidean norm. Our specific way of estimating this point is to perform a gradient descent down a data misfit term in $z$ -space, starting at the origin. While a gradient flow typically will not find the closest point on the manifold, it empirically finds a reasonable approximation of that point. In practice, one could further do a local search to refine the output of this gradient flow, but we elect not to do so for the sake of simplicity.
+
+Why does the invertible prior do so well, especially on out-of-distribution images?
+
+One reason that the invertible prior performs so well is because it has no representation error. The lack of representation error of invertible nets presents a significant opportunity for imaging with a learned prior. Any image is potentially recoverable, even if the image is significantly outside of the training distribution. In contrast, methods based on projecting onto an explicit low-dimensional representation of a natural signal manifold will have representation error, perhaps due to modeling assumptions, mode collapse, or bias in a training set. Such methods will see performance prematurely saturate as the number of measurements increases. In contrast, an invertible prior would not see performance saturate. In the extreme case of having a full set of exact measurements, an invertible prior could in principle recover any image exactly.
+
+It is natural to wonder which images can be effectively recovered using an invertible prior trained on a particular signal class. As expected, we see the best reconstruction errors on in-distribution images and performance degrades as images get further out-of-distribution. Nonetheless, we observe that reconstruction errors of unrelated natural images are still of higher quality than with the Lasso. It appears that the invertible generator learns some general attributes of natural images. This leads to several questions: when a generative invertible net is trained, how far out-of-distribution can an image be while maintaining a high likelihood? How do invertible nets learn useful statistics of natural images? Is that due primarily to training, or is there architectural bias toward natural images, as with the Deep Image Prior and Deep Decoder (Ulyanov et al., 2018; Heckel and Hand, 2018)?
+
+The results of this paper provide further evidence that reducing representational error of generators can significantly enhance the performance of generative models for inverse problems in imaging. This idea was also recently explored in Athar et al. (2018), where the authors trained a GAN-like prior with a high-dimensional latent space. The high dimensionality of this space lowers representational error, though it is not zero. In their work, the high-dimensional latent space had a structure that was difficult to directly optimize, so the authors successfully modeled latent representations as the output of an untrained convolutional neural network whose parameters are estimated at test time. Their paper and ours raises several questions: Which generator architectures provide a good balance between low representation error, ease of training, and ease of inversion? Should a generative model be capable of producing all images in order to perform well on out-of-distribution images of interest? Are there cheaper architectures that perform comparably? These questions are quite important, as solving equation 2 in our $6 4 \times 6 4$ pixel color images experiments took 15 GPU-minutes. New developments are needed on architectures and frameworks in between low-dimensional generative priors and fully invertible generative priors. Such methods could leverage the strengths of invertible models while being much cheaper to train and use.
+
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+
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+
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+
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+
+# A EXPERIMENTAL SETUP
+
+Simulations were completed mainly on CelebA-HQ dataset, used in Kingma and Dhariwal (2018); it has 30,000 color images that were resized to $6 4 \times 6 4$ for computational reasons, and were split into 27,000 training and 3000 test images. We also provide some additional experiments on the Flowers Nilsback and Zisserman (2008), and Birds Wah et al. (2011) datasets. Flowers dataset contains 8189 color images resized to $6 4 \times 6 4$ out of which 500 images are spared for testing. Birds dataset contains a total of 11,788 images, which were center aligned and resized to $6 4 \times 6 4$ out of which 5794 images are set aside for testing.
+
+We specifically model our invertible networks after the recently proposed Glow Kingma and Dhariwal (2018) architecture, which consists of a multiple flow steps. Each flow step comprises of an activation normalization layer, a $1 \times 1$ convolutional layer, and an affine coupling layer, each of which is invertible. Let $K$ be the number of steps of flow before a splitting layer, and $L$ be the number of times the splitting is performed. To train over CelebA, we choose the network to have $K = 4 8$ , $L = 4$ and affine coupling, and train it with a learning rate 0.0001, and a batch size 6 at resolution $6 4 \times 6 4 \times 3$ The model was trained over 5 bit images with 10,000 warmup iterations as in Kingma and Dhariwal (2018), but when solving inverse problems using Glow original 8−bit images were used. We refer the reader to Kingma and Dhariwal (2018) for specific details on the operations performed in each of the network layer.
+
+We use LBFGS to solve the inverse problem. For best performance, we set the number of iterations and learning rate for denoising, compressed sensing, and inpainting to be 20, 1; 30, 0.1; and 20, 1; respectively. we use Pytorch to implement Glow network training and solve the inverse problem. Glow training was conducted on a single Titan Xp GPU using a maximum allowable (under given computational constraints) batch size of 6. In case of CS, recovering a single image on Titan $\mathrm { X p }$ using LBFGS solver with 30 steps takes 889.125 seconds (14.82 minutes). However, we can solve 6 inverse problems in parallel on the given hardware platform.
+
+Unless specified otherwise, inverse problem under Glow prior is always initialized with $z _ { 0 } = 0$ Whereas under DCGAN prior, we initialize with $z _ { 0 } \sim \mathcal { N } ( 0 , 0 . 1 ^ { 2 } I )$ and report average over three random restarts. In all the quantitative experiments over, the reported quality metrics such as PSNR, and reconstruction errors are averaged over 12 randomly drawn test set images.
+
+![](images/955fd240fafc26b3316d95f35f39a6b70ccd00fbba9276fceb5c2c496e0d981c.jpg)  
+Figure 9: Samples from training set of CelebA downsampled to $6 4 \times 6 4 \times 3$ .
+
+B DENOISING: ADDITIONAL EXPERIMENTS
+
+We present additional quantitative experiments on image denoising here. Complete set of experiments on average PSNR over 12 CelebA (within distribution2) test set images versus penalization parameter $\gamma$ under noise levels $\sigma = 0 . 0 1$ , 0.05, 0.1, and 0.2 are presented in Figure 10 below. The central message is that Glow prior outperforms DCGAN prior uniformly across all $\gamma$ due to the representation limit of DCGAN. In addition, striking the right balance between the misfit term and the penalization term by appropriately choosing $\gamma$ improves the performance of Glow, and it also approaches stateof-the-art BM3D algorithm at low noise levels, and clearly visible in higher noise, for example, at a noise level of $\sigma = 0 . 2$ , the Glow prior improves upon BM3D by 2dB. Visually the results of Glow prior are clearly even superior to BM3D recoveries that are generally blurry and over smoothed as can be spotted in the qualitative results below. To avoid fitting the noisy image using the Glow model, we force the recoveries to be natural by choosing large enough $\gamma$ .
+
+![](images/e8143864a374eac9935d79ca9eb577ebb147d619a55e58e9add32c282eb35cff.jpg)  
+Figure 10: Image Denoising — Recovered PSNR values as a function of $\gamma$ under Glow prior, and DCGAN prior on (within-distribution) test set CelebA images. For reference, we show the average PSNRs of the original noisy images, and under the Glow prior in the noiseless case $( \sigma = 0$ ) in both panels. The average PSNR after applying BM3D, and the average PSNR under the Glow prior at noise levels $\sigma = 0 . 0 1 , 0 . 0 5 , 0 . 1 0 , 0 . 2 0$ are reported.
+
+Recall that we are solving a regularized empirical risk minimization program
+
+$$
+\operatorname * { a r g m i n } _ { z \in \operatorname { D o m a i n } ( G ) } \| y - A G ( z ) \| ^ { 2 } + \gamma \| z \| .
+$$
+
+In general, one can instead solve arg min $\| y - A G ( z ) \| ^ { 2 } + H ( \| z \| )$ , where $H ( \cdot )$ is a monotonically $z \in \operatorname { D o m a i n } ( G )$   
+increasing function. Figure 11 shows the comparison of most common choices of linear (already used in the rest of the paper), and quadratic $H$ in the context of densoing. We find that the highest achievable PSNR remains the same in both the cases, however, the penalization parameter $\gamma$ has to be adjusted accordingly.
+
+We train Glow and DCGAN on CelebA. Additional qualitative image denosing results under higher noise level $\sigma = 0 . 1$ and 0.2 comparing Glow prior against DCGAN prior, and BM3D are presented below in Figure 12, and 13.
+
+We also trained Glow model on Flowers dataset. Below we present its qualitative denoising performance against BM3D on the test set Flowers images. We also show the effect of varying $\gamma$ — smaller $\gamma$ leads to overfitting and vice versa.
+
+![](images/2ad9829eac2dc3e182e51ebb89d1a5cba84c6831402b3f494eb1a4f27527c34b.jpg)  
+Figure 11: Image Denoising — Recovered PSNR values as a function of $\gamma$ under Glow prior with $\| z \|$ and $\| z \| ^ { 2 }$ penalization on (within-distribution) test set CelebA images. Comparison is provided with BM3D denoising at noise level $\sigma = 0 . 1$
+
+![](images/41f01163c85a103655d3531c153eee9c57707ddd5118652c0c7b02968c3e4d42.jpg)  
+Figure 12: Image Denoising — Visual comparisons under the Glow prior, the DCGAN prior, and BM3D at a noise level $\sigma = 0 . 1$ on CelebA (within-distribution) test set images. Under DCGAN prior, we only show the case of $\gamma = 0$ as this consistently gave the best performance for DCGAN. Under Glow prior, the best performance over is achieved with $\gamma = 1$ , overfitting of the image occurs with $\gamma = 0$ and underfitting occurs at $\gamma = 5$ . Note that the Glow prior with $\gamma = 1$ also gives a sharper image than BM3D.
+
+![](images/a03d29d9d948d47871144213268a3409bf2ca459095e94c73ab4ef772138c735.jpg)  
+Figure 13: Image Denoising — Visual comparisons under the Glow prior, the DCGAN prior, and BM3D at noise level $\sigma = 0 . 2$ on CelebA (within-distribution) test set images. Under DCGAN prior, we only show the case of $\gamma = 0$ as this consistently gives the best performance. Under Glow prior, the best performance is achieved with $\gamma = 2 . 5$ , overfitting of the image occurs with $\gamma = 0$ and underfitting occurs with $\gamma = 5$ . Note that the Glow prior with $\gamma = 2 . 5$ also gives a sharper image than BM3D.
+
+![](images/7aef579d0bac6f9a8de50967709e4b7bdb26535e382989b74b27fbe4da9c8a07.jpg)  
+Figure 14: Image Denoising — Visual comparisons under the Glow prior, and BM3D at noise level $\sigma = 0 . 1$ on (within-distribution) test set Flowers images. Under Glow prior, the best performance is obtained with $\gamma = 1$ . Note that the Glow prior with $\gamma = 1$ also gives a sharper image than BM3D.
+
+# C COMPRESSED SENISNG: ADDITIONAL EXPERIMENTS
+
+Some additional quantitative image recovery results on test set of CelebA dataset are presented in Figure 15; it depicts the comparison of Glow prior, DCGAN prior, LASSO-DCT, and LASSO-WVT at compresimage and . We plot the reconstruction is the number of pixels in the $\begin{array} { r } { \vdots = \frac { 1 } { n } \| x - \hat { x } \| _ { 2 } ^ { 2 } } \end{array}$ , where A image $\hat { x }$ is the recovered Glow uniformly $n = 1 2 2 8 8$ $6 4 \times 6 4 \stackrel { \because } { \times } 3$   
+outperforms DCGAN, and LASSO across entire range of the number of measuremnts. LASSODCT and LASSO-WVT eventually catch up to Glow but only when observed measurements are a significant fraction of the total number of pixels. On the other hand, DCGAN is initially better than LASSO but prematurely saturates due to limited representation capacity.
+
+![](images/43b0c87fc2af4dddfbbd9442cf9ea822c2fdd1e3d9d6217ec49a7e702fd5856f.jpg)  
+Figure 15: Compressed sensing — Reconstruction error vs. number of measurements under Glow prior, DCGAN prior, LASSO-DCT and LASSO-WVT on CelebA (within-distribution) test set images. Noise $\eta$ is scaled such that $\mathbb { E } \Vert \eta \Vert ^ { 2 } = 0 . 0 1$ and the penalization parameter $\gamma = 0$ for Glow, and DCGAN; and $\gamma = 0 . 0 1$ for LASSO-DCT, and LASSO-WVT.
+
+![](images/8dab39d68ea6ee99aeb0ea3e34e3bb90967dceaefcff6cb9055c1f7468defaed.jpg)  
+Figure 16: Compressed sensing — Zoomed-in version of the left panel of Figure 4 in the main paper in the low measurement regime for CelebA. PSNR vs. number of measurements under Glow prior, DCGAN prior, LASSO-DCT and LASSO-WVT on the CelebA (within distribution) test set images. Noise $\eta$ is scaled such that $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ and the penalization parameter $\gamma = 0$ for Glow and DCGAN; and $\gamma = 0 . 0 1$ for LASSO-DCT, and LASSO-WVT.
+
+![](images/696ee38b29c21cf8ccde3b8c599cefa16684e9ca2645049db3c34a3e138a72bf.jpg)  
+Figure 17: Compressed sensing under Glow prior. Performance comparison between LBFGS and Adam solver for the inverse problem. For Adam solver, 2000 gradient steps were taken with learning rate chosen to be 0.01. The rest of the parameters were fixed to be the same as with LBFGS.
+
+![](images/3c9fa2f02465bfe86e387d71324e440706e69879299a2110769adbed7728c0be.jpg)  
+Figure 18: Residual error vs. number of iterations. Left panel compares DCGAN and Glow priors. Both converge roughly at the same rate to their respective saturation levels. The right panel compares LBFGS and Adam solvers for compressed sensing under Glow prior. LBFGS tends to converge far more quickly than Adam. We choose $\gamma = 0$ in both the experiments.
+
+Surprisingly, we observe that no explicit penalization of likelihood is necessary for compressive sensing with an invertible generative prior under formulation equation 2. That is, we can take $\gamma = 0$ when the optimization is initialized at $z _ { 0 } = 0$ . This indicates that algorithmic regularization is occurring and that initialization plays a role.We performed some additional experiments to study the role of initialization. The left panel in Figure 19 shows that as the norm of the latent initialization increases, the norm of the recovered latent representation increases and the PSNR of the recovered image decreases. Moreover, the right panel in Figure 19 shows the norm of the estimated latent representation at each iteration of the optimization. In all our experiments, it monotonically grows versus iteration number. These experiments provide further evidence that smaller latent initializations lead to outputs that are more natural and have smaller latent representations.
+
+![](images/e0ecd5540b80dceba2a7ff2cd98c38ddde5d13ef9fd8f0fe79648672ff8d5945.jpg)  
+Figure 19: The left panel shows the average PSNR over 12 test set images and norm of the optimizer $\hat { z }$ as a function of the norm of the initialization for the LBFGS solver to equation 2 for Compressed sensing under Glow prior with $\gamma = 0$ . The initialization $z _ { \mathrm { 0 } }$ was chosen randomly and rescaled to the desired norm. The right panel shows the norm of the estimated latent representation as a function of iteration number for multiple initializations. The Adam solver behaves similarly.
+
+Recall that the natural face images correspond to smaller $z _ { \mathrm { 0 } }$ . In Figure 20, we plot the norm of the latent codes of the iterates of each algorithm vs. the number of iterations. The central message is that initializing with smaller norm $z _ { 0 }$ tends to yield natural (smaller latent representations) recoveries. This is one explanation as to why in compressed sensing, one is able to obtain the true solution out of the affine space of solutions without penalizing the unnaturalness of the recoveries.
+
+![](images/067f8b34c6c6bcb6e0bbf6e0a80b92a7d5c0dbc03689b5ba08a5019b544e7dc6.jpg)  
+Figure 20: Compressed sensing — Norm of the latent codes with iterations. Left panel shows how the norm of the latent codes evolves over iterations of the LBFGS solver under different size initializations. Right panel shows the same experiment for the Adam solver (although over much larger number of iterations as Adam requires comparatively more iterations to converge). Each point is averaged over 12 test set images under random rescaled initializations $z _ { 0 }$ . We set the penalization parameter $\gamma = 0$ in both experiments.
+
+We now present visual recovery results on test images from the CelebA dataset under varying number of measurements in compressed sesing. We compare recoveries under Glow prior, DCGAN prior, LASSO-DCT, and LASSO-WVT.
+
+![](images/98d62d97c16ae3f39114f55808b270f93aa18297f33fb8723ce3078ed0b5ea58.jpg)  
+Figure 21: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 2 0 0$ $( \approx 1 . 5 \% )$ of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/00b50e5d8df1f2af59e3981f5abc8a53d5c8526464a7818839ab5f32793d7e82.jpg)  
+Figure 22: Compressed sensing visual comparisons — Recoveries on the (within-distribution) test set images with a number $m = 3 0 0$ $( \approx 2 \% )$ ) of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/15ef5daded80724b6920bd9544b35dea745fce6db1c6b95d1fa7cfe13a82afb7.jpg)  
+Figure 23: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 4 0 0$ $( \approx 3 \% )$ of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/24fef2258238ffd66998b0b2597823dc2d0837f787acc9c8cb96a16c090105a4.jpg)  
+Figure 24: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 5 0 0$ $( \approx 4 \% )$ ) of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/100b00c309d34eb5fe2a0bcebf09f89044147bdea47b1e919b538b8e73cd41cf.jpg)  
+Figure 25: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 7 5 0$ $( \approx 6 \% )$ of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/d53004567f25bbe44175c52ef9e023f8ad37f245e4c9969cd9ec5ea2ac773eee.jpg)  
+Figure 26: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 1 0 0 0$ $( \approx 8 \% )$ ) of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/d2d25b379d78e879ec4ce7e0de288c6cd0a7eb31b985616ae8e5f7c8d3b9c436.jpg)  
+Figure 27: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 2 5 0 0$ $( \approx 2 0 \% )$ ) of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/ea9e6bd012b6c14fb7ca61846101384811380f46e021d8a08747b62562acbec7.jpg)  
+Figure 28: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 5 0 0 0$ $( \approx 4 1 \% )$ ) of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/8c492d6b71718e1c826fc7d5138944c4d0f9986336f674ea5105daf146006e94.jpg)  
+Figure 29: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 7 5 0 0$ $( \approx 6 1 \% )$ of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/806a9eb9ea27b7899a9001f2834b9a4b546f305a92650af0f8be59849579beb6.jpg)  
+Figure 30: Compressed sensing visual comparisons — Recoveries on (within-distribution) test set images with a number $m = 1 0$ , 000 $( \approx 8 1 \% )$ of measurements under the Glow prior, the DCGAN prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for both DCGAN, and Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+# C.1 COMPRESSED SENSING ON FLOWER AND BIRD DATASET
+
+We also performed compressed sensing experiments similar to those on CelebA dataset above on Birds dataset, and Flowers dataset. We trained a Glow invertible network for each dataset, and present below the quantitative and qualitative recoveries for each dataset.
+
+![](images/6284130a02df6938dc287230d507a5a058ccefb37de1e023446473b09a405ddf.jpg)  
+Figure 31: PSNR vs. number of measurements $m$ in compressed sensing under Glow prior, LASSODCT and LASSO-WVT on Birds dataset (left panel) and Flowers dataset (right panel). Noise $\eta$ is scaled such that $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ and the penalization parameter $\gamma = 0$ for Glow, and $\gamma = 0 . 0 1$ for LASSO-DCT, and LASSO-WVT.
+
+![](images/63326ff6af0b497af549b4d3afb343ea50674058e4ba90b0bdac53b2add1adde.jpg)  
+Figure 32: Compressed sensing — Visual comparisons on (within-distribution) test set images from Birds and Flowers dataset with a number $m = 2 0 0$ $( \approx 1 . 5 \%$ ) of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/03573da6f930f27172eb34510f993dccf4988b579ceb83983a46f91c9e5a0deb.jpg)  
+Figure 33: Compressed sensing — Visual comparisons on (within-distribution) test set images from Birds and Flowers dataset with a number $m = 3 0 0$ $( \approx 2 \% )$ ) of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/8174a45f17918a415ab7e616e679a939285da4c3a42e736c140af368baba9cd0.jpg)  
+Figure 34: Compressed sensing — Visual comparisons on (within-distribution) test set images from Birds and Flowers dataset with a number $m = 4 0 0$ $( \approx 3 \% )$ ) of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/7da4c84c1afc59e49fee9a7acc285aa7342e9b865b00d6887f188cc1f245b32c.jpg)  
+Figure 35: Compressed sensing — Visual comparisons on (within-distribution) test set images from Birds and Flowers dataset with a number $m = 5 0 0$ $( \approx 4 \%$ ) of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/03c5968c2d19e4c07cee33911c78f66f0b99d2ab1cf62e02a2f52446429e679e.jpg)  
+Figure 36: Compressed sensing — Visual comparisons on the test set images from Birds and Flowers dataset with a number $m = 7 5 0$ $( \approx 6 \% )$ ) of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/94a211008edf311f61018ec07d4a5fe4d5f333debd6948778f4405a77025ba38.jpg)  
+Figure 37: Compressed sensing — Visual comparisons on the test set images from Birds and Flowers dataset with a number $m = 1 , 0 0 0 ( \approx 8 \% )$ of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/4626111bbf52d6faff8b1ac44cc582758cba63f12b4fa0f245f71adaae690c15.jpg)  
+Figure 38: Compressed sensing — Visual comparisons on the test set images from Birds and Flowers dataset with a number $m = 2 , 5 0 0 ( \approx 2 0 \% )$ of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/0c450f5c25ab017d6d1a7081bf90672ceea0ce0a47bf84dd659e8b279094ef74.jpg)  
+Figure 39: Compressed sensing — Visual comparisons on the test set images from Birds and Flowers dataset with a number $m = 5 , 0 0 0 ( \approx 4 1 \% )$ of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/ac3df8a922da1a10bd106e094fb1acd9300d9423751d6fd1dbac03e1a4d2e7d9.jpg)  
+Figure 40: Visual comparisons of compressed sensing of the test set images from Birds and Flowers dataset with a number $m = 7 , 5 0 0 ( \approx \mathrm { { \bar { 6 } 1 \% } ) }$ of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+![](images/ea58019eb61b48f74f32ca10e2fff8f7a4a50bf60a2ace960c94a0797e47bf7b.jpg)  
+Figure 41: Visual comparisons of compressed sensing of the test set images from Birds and Flowers dataset with a number $m = 1 0 , 0 0 0 ( \approx 8 1 \% )$ of measurements under the Glow prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance among the tested values. We use $\gamma = 0$ for Glow prior and $\gamma = 0 . 0 1$ for LASSO-WVT, and LASSO-DCT, respectively.
+
+# C.2 COMPRESSED SENSING ON OUT OF DISTRIBUTION IMAGES
+
+Lack of representation error in invertible nets leads us to an important and interesting question: does the trained network fit related natural images that are underrepresented or even unrepresented in the training dataset? Specifically, can a Glow network trained on CelebA faces be a good prior on other faces; for example, those with dark-skin tone, faces with glasses or facial hair, or even animated faces? In general, our experiments show that Glow prior has an excellent performance on such out-of-distribution images that are semantically similar to celebrity faces but not representative of the CelebA dataset. In particular, we have been able to recover faces of darker skin tone, older people with beards, eastern women, men with hats, and animated characters such as Shrek, from compressed measurements under the Glow prior. Recoveries under the Glow prior convincingly beat the DCGAN prior, which shows a definite bias due to training. Not only that, the Glow prior also outperforms unbiased methods such as LASSO-DCT, and LASSO-WVT.
+
+Can we expect the Glow prior to continue to be an effective proxy for arbitrarily out-of-distribution images? To answer this question, we tested arbitrary natural images such as car, house door, and butterfly wings that are semantically unrelated to CelebA images. In general, we found that Glow is an effective prior at compressed sensing of out-of-distribution natural images, which are assigned a high likelihood score (small normed latent representations). On these images, Glow also outperforms LASSO.
+
+Recoveries of natural images that are assigned very low-likelihood scores by the Glow model generally run into instability issues. During training, invertible nets learn to assign high likelihood scores to the training images. All the network parameters such as scaling in the coupling layers of Glow network are learned to behave stably with such high likelihood representations. However, on very low-likelihood representations, unseen during the training process, the networks becomes unstable and outputs of network begin to diverge to very large values; this may be due to several reasons, such as normalization (scaling) layers not being tuned to the unseen representations. An LBFGS search for the solution of an inverse problem to recover a low-likelihood image leads the iterates into neighborhoods of low-likelihood representations that may lead the network to instability.
+
+We find that Glow network has the tendency to assign higher likelihood scores to arbitrarily outof-distribution natural images. This means that invertible networks have at least partially learned something more general about natural images from CelebA dataset — may be some high level features that face images share with other natural images such as smooth regions followed by discontinuities, etc. This allows Glow prior to extend its effectiveness as a prior to other natural images beyond just the training set.
+
+Figure 42, 43 , 44, 45, and 46 compare the performance of LASSO-DCT, LASSO-WVT, DCGAN prior, and Glow prior on the compressed sensing of out-of-distribution images under varying number of measurements.
+
+![](images/4963cac0a068434dfcc93134fbd5fa894d8d24dd05fa77b474b1d00c1f04b809.jpg)  
+Figure 42: Compressed sensing $m = 1 0 0 0 \approx 8 \%$ of $n$ ) visual comparisons on out-of-distribution images. We compare the recoveries under Glow (trained on CelebA) prior, DCGAN (trained on CelebA) prior, LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance. We use $\gamma = 0$ for both DCGAN, and Glow prior and and optimize $\gamma$ for each recovery using LASSO-WVT, and LASSO-DCT.
+
+![](images/dc5b122075b21f68a67bee24e992fccaf217228d884b5a1d700b59d36cedad64.jpg)  
+Figure 43: Compressed sensing $m = 2 5 0 0 \approx 2 0 \%$ of $n$ ) visual comparisons on out-of-distribution images. We compare the recoveries under Glow (trained on CelebA) prior, DCGAN (trained on CelebA) prior, LASSO-WVT, and LASSO-DCT at a noise level $\begin{array} { r } { \sqrt { \mathbb { E } \| \eta \| ^ { 2 } } = 0 . 1 . } \end{array}$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance. We use $\gamma = 0$ for both DCGAN, and Glow prior and and optimize $\gamma$ for each recovery using LASSO-WVT, and LASSO-DCT.
+
+![](images/ef07d66ffd1d391ee0d18ed8d89c4a9ccdee651ce5e0793e4fceb217ff6a2f6b.jpg)  
+Figure 44: Compressed sensing $m = 5 0 0 0 \approx 4 1 \%$ of $n$ ) visual comparisons on out-of-distribution images. We compare the recoveries under Glow prior (trained on CelebA), DCGAN prior (trained on CelebA), LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance. We use $\gamma = 0$ for both DCGAN, and Glow prior and and optimize $\gamma$ for each recovery using LASSO-WVT, and LASSO-DCT.
+
+![](images/6178feda9556245d70ce21098802acb87902b163b491a12ba423c78af8bfb726.jpg)  
+Figure 45: Compressed sensing $m = 7 5 0 0 \approx 6 1 \%$ of $n$ ) visual comparisons on out-of-distribution images. We compare the recoveries under Glow prior (trained on CelebA), DCGAN prior (trained on CelebA), LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance. We use $\gamma = 0$ for both DCGAN, and Glow prior and and optimize $\gamma$ for each recovery using LASSO-WVT, and LASSO-DCT.
+
+![](images/40f2fba7218ce1aaefa63a49fada8fe56732d325faec62007a7a8771733ba6ab.jpg)  
+Figure 46: Compressed sensing $( m = 1 0 , 0 0 0 , \approx 8 1 \%$ of $n$ ) visual comparisons on out-of-distribution images. We compare the recoveries under Glow prior (trained on CelebA), DCGAN prior (trained on CelebA), LASSO-WVT, and LASSO-DCT at a noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ . In each case, we choose values of the penalization parameter $\gamma$ to yield the best performance. We use $\gamma = 0$ for both DCGAN, and Glow prior and and optimize $\gamma$ for each recovery using LASSO-WVT, and LASSO-DCT.
+
+# D IMAGE INPAINITING
+
+Our experiments with inpainting reveal a similar story as with compressed sensing. Compared to DCGAN, the recovered PSNRs using Glow prior are much higher under appropriate $\gamma$ as depicted in the right panel in Figure 47. If improperly initialized, then performance for $\gamma = 0$ could be poor. Even if improperly initialized, sufficiently large $\gamma$ leads to higher PSNRs.
+
+As with compressive sensing, if the initialization is from a small latent variable, then the empirical risk formulation with $\gamma = 0$ exhibits high PSNRs. Algorithmic regularization is again occurring due to the small latent variable initialization.
+
+![](images/557518d4a67391827b39f438c085c5de03098e7e8b8a8ebee9d6ca2c1ea03e30.jpg)  
+Figure 47: Inpainiting: PSNR (averaged over 12 test images of CelebA) vs. penalization parameter $\gamma$ under Glow prior and DCGAN prior (left panel) and using different initializations under Glow prior (right panel).
+
+We present here qualitative results on image inpainting under the DCGAN prior, and the Glow prior on the CelebA test set. Compared to DCGAN, the reconstructions from Glow are of noticeably higher visual quality.
+
+![](images/6d0876d1bac9a70fd4452b3123aa2d5cba36f47329ac0c824fda2eb4280aaabc.jpg)  
+Figure 48: Image inpainiting results on CelebA test set. Masked images are recovered under DCGAN prior and Glow prior. Recoveries under DCGAN prior are skewed and blurred whereas Glow prior leads to sharper and coherent inpainted images. For both Glow and DCGAN, we set $\gamma = 0$ .
+
+# D.1 IMAGE INPAINTING ON OUT OF DISTRIBUTION IMAGES
+
+We now perform image inpainiting under Glow prior, and DCGAN prior each trained on CelebA. Figure 49 shows the visuals of out-of-distribution inpainiting. As before, DCGAN continues to suffer due to representation limits and data bias while Glow achieves reasonable reconstructions on out-of-distribution images semantically similar to CelebA faces. As one deviates to other natural images such as houses, doors, and butterfly wings, the inpainting performance deteriorates. At compressed sensing, Glow performed much better on such arbitrarily out-of-distribution images as good recoveries there only require the network only to assign a higher likelihood score to the true image compared to the all the candidate static images given by the null space of the measurement operator.
+
+![](images/bdda75d6c0aee9abe5cf6491294d1ef54b63a013642d788fbdbb431b76307786.jpg)  
+Figure 49: Image inpainiting results on out-of-distribution images. Masked images are recovered under DCGAN prior and Glow prior. Recoveries under DCGAN prior are skewed and blurred whereas Glow prior leads to sharper and coherent inpainted images. For both Glow and DCGAN, we set $\gamma = 0$ .
+
+# E DISCUSSION
+
+Figure 50 confirms the intuition brought up in the Discussion Section of the main paper that trained Glow network assigns lower likelihoods (larger latent representations) to noisy images. Histograms show that noisy images are generally occupy the less likelihood regimes or equivalently, the larger norm latent representations.
+
+![](images/a91e877bd33d699e23e42696d5f0d33c1a531ffe336d841587755824e765e4f9.jpg)  
+Figure 50: Histograms of the norm of the latent representation, $z$ , over 3000 test images under additive Gaussian noise with $\sigma = 0 . 1$ (left), $\sigma = 0 . 0 5$ (middle), and $\sigma = 0 . 0 1$ (right).
+
+Our experiments verify that natural images have smaller latent representations than unnatural images. Here we also show that adding noise to natural images increases the norm of their latent representations, and that higher noise levels result in larger increases. Additionally we provide evidence that random perturbations in image space induce larger changes in $z$ than comparable natural perturbations in image space. Figure 51 shows a plot of the norm of the change in image space, averaged over 100 test images, as a function of the size of a perturbation in latent space. Natural directions are given by the interpolation between the latent representation of two test images. For the denoising problem, this difference in sensitivity indicates that the optimization algorithm might obtain a larger decrease in $\| z \|$ by an image modification that reduces unnatural image components than by a correspondingly large modification in a natural direction.
+
+![](images/0bc07e8fafef3e4171b2fdd64c758834d036331e66c9c37aaf0676ff1704be78.jpg)  
+Figure 51: The magnitude of the change in image space as a function of the size of a perturbation in latent space. Solid lines are the mean behavior and shaded region depicts $9 5 \%$ confidence interval.
+
+# F LOSS LANDSCAPE: DCGAN VS. GLOW
+
+In Figure 52, we plot $\lVert y - A G ( z ^ { * } + \alpha \delta _ { v } + \beta \delta _ { w } ) \rVert ^ { 2 }$ versus $( \alpha , \beta )$ where $\delta _ { v }$ and $\delta _ { w }$ are scaled to have the same norm as $z ^ { * }$ , the latent representation of a fixed test image. For DCGAN, we plot the loss landscape versus two pairs of random directions. For Glow, we plot the loss landscape versus a pair of random directions and a pair of directions that linearly interpolate in latent space between $z ^ { * }$ and another test image.
+
+![](images/7ff6943bc1183243b03ebd84a66a23eb43ce128061e966c5717597b66ba2e600.jpg)  
+Figure 52: Loss landscapes for $\| A G ( z ) - y \| _ { 2 } ^ { 2 } + \gamma \| z \| _ { 2 }$ with $\gamma = 0$ around the latent representation of a fixed image and with respect to either random latent directions or latent directions that interpolate between images.
+
+# G IMAGE AND LATENT SPACE FORMULATIONS
+
+As mentioned in the main paper, a natural formulation of the inverse problem is
+
+$$
+\operatorname* { m i n } _ { x \in \mathbb { R } ^ { n } } \| A x - y \| ^ { 2 } - \gamma \log p _ { G } ( x ) ,
+$$
+
+where $p _ { G } ( x )$ is the target density. We instead formulate the inverse problem as
+
+$$
+\operatorname* { m i n } _ { z \in \mathbb { R } ^ { n } } \| A G ( z ) - y \| _ { 2 } ^ { 2 } + \gamma \| z \| _ { 2 } ;
+$$
+
+a measurement misfit combined with a Gaussian prior on the latent space.
+
+We will denote the target distribution by $p _ { G } ( x )$ and the latent Gaussian distribution by $p ( z )$ . To illustrate the differences between equation 3 and equation 4, we train a Real-NVP model Dinh et al. (2016) on a synthetic two-dimensional dataset, visualize both the $\log p _ { G } ( x )$ and $\log p ( z )$ in latent and image space, and solve a simple compressive sensing recovery problem. Our two dimensional data points are generated by sampling the first coordinate $x _ { 1 }$ from a bimodel Gaussian distribution and the second coordinate $x _ { 2 }$ from a uniform distribution as shown in Figure 53.
+
+![](images/083ec2d5a351c98002e3914ca04589f1ba776ec5edba052ee1946fa71a4a70a8.jpg)  
+Figure 53: A point cloud of the synthetically generated data $\boldsymbol { x } \in \mathbb { R } ^ { 2 }$
+
+For comparison, we plot the $x$ -likelihood versus $x$ (left), latent $z$ -likelihood versus $x$ (middle), and $x$ -likelihood versus $z$ (right) in Figure 54. These plots illustrate that generally high-likelihood $x$ points are also given higher latent $z$ -likelihood, however, some low $x$ -likelihood might be assigned a higher Gaussian $z$ -likelihood; these are, for example, the points living on the darker contour spearing through the Gaussian bowl in the right plot. Figure 55 shows some of the points in the $x$ -likelihood (left) that map to this contour in the $z$ -space (right).
+
+![](images/d2a19770c0150c098899d81b0f8389cae87bd04d28a4352c8b06de771c6a9878.jpg)  
+Figure 54: $x$ -likelihood versus $x$ (left), $z$ -likelihood versus $x$ (middle), and $x$ -likelihood versus $z$ (right).
+
+![](images/8a18b4621eda54536fd49a52d2603f08eb39e249f23ad7b1704269799e1e8bb7.jpg)  
+Figure 55: Some points (red-crosses) in $x$ -space mapped to $z$ -space. The (unwanted, as it corresponds to low-likelihood points) bridge connecting the models of the learned bimodal distribution is mapped to the contour in the $z$ -space.
+
+# G.1 COMPRESSIVE SENSING IN 2D
+
+To compare latent-space formulation equation 4 and data-space formulation equation 3, we construct a simple compressive sensing recovery problem for this two-dimensional data and illustrate the difference under both good and bad initializations. Specifically, we want to recover a vector $x =$ $[ x _ { 1 } \ x _ { 2 } ] ^ { \mathrm { T } }$ from a single linear measurement $y = \langle a , x \rangle = x _ { 2 }$ , where $a = [ 0 \ 1 ] ^ { \mathrm { T } }$ . Figure 56 shows the gradient descent path, and final solution, while solving equation 4 (left column), and equation 3 (right column) from a good and a bad initialization. $x$ -likelihood formulation seems more robust to a bad initialization in this case compared to $z$ -likelihood as $z$ -likelihood might not be a good proxy for $x$ -likelihood for some points. This bad case is carefully crafted to illustrate the difference between the two formulations, however, in practice, it seems unlikely that a low $x$ -likelihood points that somehow achieves higher $z$ -likelihood will also obey the measurement constraints.
+
+![](images/0573b317db73250631df0962f205498d4a41ee6358bb22e0081ddd73269dec6b.jpg)  
+Figure 56: We show gradient descent path from the initialization to the final estimate along with true solution. In the first row, we initialized from $z = 0$ (good initialization) and in the second row we used low likelihood data points as intializations (bad initialization).
+
+# G.2 COMPRESSIVE SENSING FOR CELEBA
+
+In case of CelebA images, we found that optimizing over direct likelihood of images proved very hard to tune. To better understand why equation 4 is easier compared to equation 3, we draw the landscape of the loss surfaces of equation 4 versus $z$ and equation 3 versus $x$ under different $\gamma$ in two random directions around an the ground truth in $z$ , or $x$ , as appropriate; see Figure 57. In the $x$ -formulation the loss surfaces (first row) have a sharp dip at the ground truths, which comes from $- \log p _ { G } ( x )$ term. We believe that sharp dip in the loss surface makes it difficult to tune the $\gamma$ parameter, the learning rate, and makes the optimization using equation 3 numerically more challenging as observed in our experiments. On the other hand, the loss surfaces for equation 4 (second row) appear smoother.
+
+We now show a quantitative comparison of the $x$ -likelihood formulation in equation 3, and $z$ - likelihood formulation in equation 4 on compressive sensing for CelebA test images versus $m$ for fixed values of $\gamma$ ; see Figure 58. We initialize with $z _ { 0 } = 0$ , and $x _ { 0 } = G ( z _ { 0 } )$ , as appropriate. We simply choose $\gamma = 0$ in equation 4. However, we need to choose $\gamma$ more carefully in equation 3, and different values of $\gamma$ are appropriate across different undersampling ratios. Even if one ignores the difficulty of choosing the hyperparameter $\gamma$ , the formulation in equation 4 generally performs much better than equation 3 as evident from the plots.
+
+![](images/4b7a613cf8e80b0ed28efdebb433fe36b97e4124c2624aae0138f6d505b3b195.jpg)  
+Figure 57: Landscapes of the loss surfaces in the $x$ -space (first row), the loss surfaces of in the $z$ -space (second row) for various values of $\gamma$ , and loss surface of $x$ -likelihood $- \log p ( x )$ (third row).
+
+![](images/d9e9f2b6ef360f48d7b19e21665c34eade9b38dd6a9b29db4fc60994dd86d152.jpg)  
+Figure 58: We report PSNR against number of measurements $m$ when optimizing in the latent space equation 4 with $\gamma = 0$ and the image space equation 3 with $\gamma$ set to 10, 50 and 100.
+
+To show the effect of noise on recovery in compressive sensing under different values of $\gamma$ and noise levels, we plot PSNR of the iterates when solving equation 4 against iterations in Figure 59. This plot shows, perhaps surprisingly, that even under noisy compressed measurements it is a good idea to solve the inverse compressed sensing problem equation 4 with $\gamma = 0$ .
+
+# G.3 DENOISING FOR CELEBA
+
+For completeness, we also compare denoising using our latent space formulation equation 4, our image space forumation equation 3 under different noise levels $\sigma = 0 . 0 5$ and $\sigma = 0 . 1 0$ ; see Figure 60 and Figure 61 respectively. For both noise levels, we observe equal performance (indicated by the highest PSNR) when optimizing in the latent or image space. We do not report results over $\sigma = 0 . 2 0$ as it was hard to tune hyper paremeters for higher noise levels in equation 3.
+
+![](images/fac54caf5e63650079bffd146bb2eafe3ebe9b9cd5323d4649c47d8f5a8e188e.jpg)  
+Figure 59: We plot PSNR against gradient iterations for compressive sensing at $m = 5 0 0 0$ on a single image under the presence and absence of noise with different values of $\gamma$ with noise level $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 0 . 1$ (left) and $\sqrt { \mathbb { E } \lVert \eta \rVert ^ { 2 } } = 1$ (right).
+
+![](images/1afd59fd6eb1c49016738eeba5c8ca0c3f5d7a4fb211ae792edb820f48e9888b.jpg)  
+Figure 60: Denoising comparision at $\sigma = 0 . 0 5$ when optimizing over latent space (left) versus image space (right).
+
+![](images/150fb21f7087ae9053eed779c52c3638b2b10bbe560542fa1d6e8140d3e98129.jpg)  
+Figure 61: Denoising comparision at $\sigma = 0 . 1 0$ when optimizing over latent space (left) versus image space (right).

parse/train/de11dbHzAMF/de11dbHzAMF.md

@@ -0,0 +1,490 @@
+# CONDITIONALLY ADAPTIVE MULTI-TASK LEARNING: IMPROVING TRANSFER LEARNING IN NLP USING FEWER PARAMETERS & LESS DATA
+
+Jonathan Pilault1∗, Amine El hattami1∗, Christopher Pal1,2,3 1Polytechnique Montreal & Mila, 2Element AI, 3Canada CIFAR AI Chair {jonathan.pilault,amine.elhattami,christopher.pal}@polymtl.ca
+
+# ABSTRACT
+
+Multi-Task Learning (MTL) networks have emerged as a promising method for transferring learned knowledge across different tasks. However, MTL must deal with challenges such as: overfitting to low resource tasks, catastrophic forgetting, and negative task transfer, or learning interference. Often, in Natural Language Processing (NLP), a separate model per task is needed to obtain the best performance. However, many fine-tuning approaches are both parameter inefficient, i.e., potentially involving one new model per task, and highly susceptible to losing knowledge acquired during pretraining. We propose a novel Transformer based Adapter consisting of a new conditional attention mechanism as well as a set of task-conditioned modules that facilitate weight sharing. Through this construction, we achieve more efficient parameter sharing and mitigate forgetting by keeping half of the weights of a pretrained model fixed. We also use a new multi-task data sampling strategy to mitigate the negative effects of data imbalance across tasks. Using this approach, we are able to surpass single task fine-tuning methods while being parameter and data efficient (using around $66 \%$ of the data for weight updates). Compared to other BERT Large methods on GLUE, our 8-task model surpasses other Adapter methods by $2 . 8 \%$ and our 24-task model outperforms by $0 . 7 \mathrm { - } 1 . 0 \%$ models that use MTL and single task fine-tuning. We show that a larger variant of our single multi-task model approach performs competitively across $2 6 \mathrm { N L P }$ tasks and yields state-of-the-art results on a number of test and development sets. Our code is publicly available at https://github.com/CAMTL/CA-MTL.
+
+# 1 INTRODUCTION
+
+The introduction of deep, contextualized Masked Language Models $( \mathbf { M L M } ) ^ { 1 }$ trained on massive amounts of unlabeled data has led to significant advances across many different Natural Language Processing (NLP) tasks (Peters et al., 2018; Liu et al., 2019a). Much of these recent advances can be attributed to the now well-known BERT approach (Devlin et al., 2018). Substantial improvements over previous state-of-the-art results on the GLUE benchmark (Wang et al., 2018) have been obtained by multiple groups using BERT models with task specific fine-tuning. The “BERT-variant $^ +$ fine-tuning” formula has continued to improve over time with newer work constantly pushing the state-of-the-art forward on the GLUE benchmark. The use of a single neural architecture for multiple NLP tasks has shown promise long before the current wave of BERT inspired methods (Collobert & Weston, 2008) and recent work has argued that autoregressive language models (ARLMs) trained on large-scale datasets – such as the GPT family of models (Radford et al., 2018), are in practice multi-task learners (Brown et al., 2020). However, even with MLMs and ARLMs trained for multi-tasking, single task fine-tuning is usually also employed to achieve state-of-the-art performance on specific tasks of interest. Typically this fine-tuning process may entail: creating a task-specific fine-tuned model (Devlin et al., 2018), training specialized model components for task-specific predictions (Houlsby et al., 2019) or fine-tuning a single multi-task architecture (Liu et al., 2019b).
+
+Single-task fine-tuning overall pretrained model parameters may have other issues. Recent analyses of such MLM have shed light on the linguistic knowledge that is captured in the hidden states and attention maps (Clark et al., 2019b; Tenney et al., 2019a; Merchant et al., 2020). Particularly, BERT has middle Transformer (Vaswani et al., 2017) layers that are typically the most transferable to a downstream task (Liu et al., 2019a). The model proxies the steps of the traditional NLP pipeline in a localizable way (Tenney et al., 2019a) — with basic syntactic information appearing earlier in the network, while high-level semantic information appearing in higher-level layers. Since pretraining is usually done on large-scale datasets, it may be useful, for a variety of downstream tasks, to conserve that knowledge. However, single task fine-tuning causes catastrophic forgetting of the knowledge learned during MLM (Howard & Ruder, 2018). To preserve knowledge, freezing part of a pretrained network and using Adapters for new tasks have shown promising results (Houlsby et al., 2019).
+
+![](images/99726a0f1d90af840ba8dc9ac3c50e988152aa3febfdb46640b3bc908655f6dc.jpg)  
+Figure 1: CA-MTL base architecture with our uncertainty-based sampling algorithm. Each task has its own decoder. The input embedding layer and the lower Transformer layers are frozen. The upper Transformer layer and Conditional Alignment module are modulated with the task embedding.
+
+Inspired by the human ability to transfer learned knowledge from one task to another new task, Multi-Task Learning (MTL) in a general sense (Caruana, 1997; Rajpurkar et al., 2016b; Ruder, 2017) has been applied in many fields outside of NLP. Caruana (1993) showed that a model trained in a multi-task manner can take advantage of the inductive transfer between tasks, achieving a better generalization performance. MTL has the advantage of computational/storage efficiency (Zhang & Yang, 2017), but training models in a multi-task setting is a balancing act; particularly with datasets that have different: (a) dataset sizes, (b) task difficulty levels, and (c) different types of loss functions. In practice, learning multiple tasks at once is challenging since negative transfer (Wang et al., 2019a), task interference (Wu et al., 2020; Yu et al., 2020) and catastrophic forgetting (Serrà et al., 2018) can lead to worse data efficiency, training stability and generalization compared to single task fine-tuning.
+
+Using Conditionally Adaptive Learning, we seek to improve pretraining knowledge retention and multi-task inductive knowledge transfer. Our contributions are the following:
+
+• A new task conditioned Transformer that adapts and modulates pretrained weights (Section 2.1). • A novel way to prioritize tasks with an uncertainty based multi-task data sampling method that helps balance the sampling of tasks to avoid catastrophic forgetting (Section 2.2).
+
+Our Conditionally Adaptive Multi-Task Learning (CA-MTL) approach is illustrated in Figure 1. To the best of our knowledge, our work is the first to explore the use of a latent representation of tasks to modularize and adapt pretrained architectures. Further, we believe our work is also the first to examine uncertainty sampling for large-scale multi-task learning in NLP. We show the efficacy of CA-MTL by: (a) testing on 26 different tasks and $\mathbf { ( b ) }$ presenting state-of-the-art results on a number of test sets as well as superior performance against both single-task and MTL baselines. Moreover, we further demonstrate that our method has advantages over (c) other adapter networks, and (d) other MTL sampling methods. Finally, we provide ablations and separate analysis of the MT-Uncertainty Sampling technique in section 4.1 and of each component of the adapter in 4.2.
+
+# 2 METHODOLOGY
+
+This section is organized according to the two main MTL problems that we will tackle: (1) How to modularize a pretrained network with latent task representations? (2) How to balance different tasks in MTL? We define each task as: $\mathbb { T } _ { i } \triangleq \{ p _ { i } ( \mathbf { y } _ { i } | \mathbf { x } _ { i } , \mathbf { z } _ { i } ) , \mathcal { L } _ { i } , \tilde { p } _ { i } ( \mathbf { x } _ { i } ) \}$ , where $\mathbf { z } _ { i }$ is task $i$ ’s learnable shallow embedding, $\mathcal { L } _ { i }$ is the task loss, and $\tilde { p } _ { i } ( \mathbf { x } _ { i } )$ is the empirical distribution of the training data pair $\left\{ \mathbf { x } _ { i } , \mathbf { y } _ { i } \right\}$ , for $i \in \{ 1 , \ldots , T \}$ and $T$ the number of supervised tasks. The MTL objective is:
+
+$$
+\operatorname* { m i n } _ { \phi ( \mathbf { z } ) , \theta _ { 1 } , \dots , \theta _ { T } } \sum _ { i = 1 } ^ { T } \mathcal { L } _ { i } \big ( f _ { \phi ( \mathbf { z } _ { i } ) , \theta _ { i } } \big ( \mathbf { x } _ { i } \big ) , \mathbf { y } _ { i } \big )
+$$
+
+where $f$ is the predictor function (includes encoder model and decoder heads), $\phi ( \mathbf { z } )$ are learnable generated weights conditioned on $\mathbf { z }$ , and $\theta _ { i }$ are task-specific parameters for the output decoder heads. $\mathbf { z }$ is constructed using an embedding lookup table.
+
+# 2.1 TASK CONDITIONED TRANSFORMER
+
+Our task conditioned Transformer architecture is based on one simple concept. We either add conditional layers or modulate existing pretrained weights using a task representation by extending Feature Wise Linear Modulation (Perez et al., 2018) functions in several ways depending on the Transformer layer. We define our framework below.
+
+Definition 1 (Conditional Weight Transformations). Given a neural network weight matrix $W$ , we compute transformations of the form $\phi ( \mathbf { W } | z _ { i } ) = \gamma _ { i } ( z _ { i } ) \mathbf { W } + \beta _ { i } ( z _ { i } )$ , where $\gamma _ { i }$ and $\beta _ { i }$ are learned functions that transform the weights based on a learned vector embedding $z _ { i }$ , for task $i$ .
+
+Definition 2 (Conditionally Adaptive Learning). In our setting, Conditionally Adaptive Learning is the process of learning a set of φs for the conditionally adaptive modules presented below along with a set of task embedding vectors $z _ { i }$ for $T$ tasks, using a multi-task loss (see equation 1).
+
+In the subsections that follow: We introduce a new Transformer Attention Module using blockdiagonal Conditional Attention that allows the original query-key based attention to account for task-specific biases (section 2.1.1). We propose a new Conditional Alignment method that aligns the data of diverse tasks and that performs better than its unconditioned and higher capacity predecessor (section 2.1.2). We adapt layer normalization statistics to specific tasks using a new Conditional Layer Normalization module (section 2.1.3). We add a Conditional Bottleneck that facilitates weight sharing and task-specific information flow from lower layers (section 2.1.4). In our experiments we provide an ablation study of these components (Table 1) examining performance in terms of GLUE scores.
+
+# 2.1.1 CONDITIONAL ATTENTION
+
+Given $d$ , the input dimensions, the query Q, the key $\mathbf { K }$ , and the value $\mathbf { V }$ as defined in Vaswani et al. (2017), we redefine the attention operation:
+
+$$
+\begin{array} { l } { { \displaystyle \mathrm { A t t e n t i o n } ( { \bf Q } , { \bf K } , { \bf V } , { \bf z } _ { i } ) ) = \mathrm { s o f t m a x } \left[ { \cal M } ( { \bf z } _ { i } ) + \frac { { \bf Q } { \bf K } ^ { T } } { \sqrt { d } } \right] { \bf V } } } \\ { { \displaystyle ~ { \cal M } ( { \bf z } _ { i } ) = \bigoplus _ { n = 1 } ^ { N } A _ { n } ^ { \prime } ( { \bf z } _ { i } ) , ~ A _ { n } ^ { \prime } ( { \bf z } _ { i } ) = A _ { n } \gamma _ { i } ( { \bf z } _ { i } ) + \beta _ { i } ( { \bf z } _ { i } ) } } \end{array}
+$$
+
+where is the direct sum operator (see section A.6), $N$ is the number of block matrices $A _ { n } \in \mathbb { R } ^ { ( L / N ) \times ( L / N ) }$ along the diagonal of the attention matrix, $L$ is the input sequence, $M ( \mathbf { z } _ { i } ) = \operatorname { d i a g } ( A _ { 1 } ^ { \prime } , \ldots , A _ { N } ^ { \prime } )$ is a block diagonal conditional matrix. Note that $A _ { n }$ is constructed using $L / N$ trainable and randomly initialized $L / N$ dimensional vectors. While the original attention matrix depends on the hidden states $h$ , ${ \cal M } ( { \bf z } _ { i } )$ is a learnable weight matrix that only depends on the task embedding $\mathbf { z } _ { i } \in \mathbb { R } ^ { d }$ . $\gamma _ { i } , \beta _ { i } : \mathbb { R } ^ { d } \mapsto \mathbb { R } ^ { L ^ { 2 } / N ^ { 2 } }$ are Feature Wise Linear Modulation (Perez et al., 2018) functions. We also experimented with full-block Conditional Attention $\in \mathbb { R } ^ { L \times L }$ . Not only did it have $N ^ { 2 }$ more parameters compared to the block-diagonal variant, but it also performed significantly worse on the GLUE development set (see FBA variant in Table 10). It is possible that GLUE tasks derive a certain benefit from localized attention that is a consequence of ${ M } ( { \bf { z } } _ { i } )$ . With ${ \cal M } ( { \bf z } _ { i } )$ , each element in a sequence can only attend to other elements in its subsequence of length $L / N$ . In our experiments we used $N = d / L$ . The full Conditional Attention mechanism used in our experiments is illustrated in Figure 2.
+
+![](images/3b3eac1dc937ff8fb579e293306567ba6d8e9a4dff6957bbd5820cbb602c2fe6.jpg)  
+Figure 2: Conditional Attention Module
+
+# 2.1.2 CONDITIONAL ALIGNMENT
+
+Wu et al. (2020) showed that in MTL having $T$ separate alignment modules $R _ { 1 } , \ldots , R _ { T }$ increases BERTLARGE avg. scores on five GLUE tasks (CoLA, MRPC, QNLI, RTE, SST-2) by $2 . 3 5 \%$ . Inspired by this work, we found that adding a task conditioned alignment layer between the input embedding layer and the first BERT Transformer layer improved multi-task model performance. However, instead of having $T$ separate alignment matrices $R _ { i }$ for each $T$ task, one alignment matrix $\hat { R }$ is generated as a function of the task embedding $z _ { i }$ . As in Wu et al. (2020), we tested this module on the same five GLUE tasks and with BERTLARGE. Enabling task conditioned weight sharing across covariance alignment modules allows us to outperforms $\mathbf { B E R T _ { L A R G E } }$ by $3 . 6 1 \%$ . This is $1 . 2 6 \%$ higher than having $T$ separate alignment matrices. Inserting $\hat { R }$ into BERT, yields the following encoder function $\hat { f }$ :
+
+$$
+\hat { f } = \sum _ { t = 1 } ^ { T } g _ { \theta _ { i } } ( E ( \mathbf { x } _ { i } ) \hat { R } ( \mathbf { z } _ { i } ) B ) , \qquad \hat { R } ( \mathbf { z } _ { i } ) = R \gamma _ { i } ( \mathbf { z } _ { i } ) + \beta _ { i } ( \mathbf { z } _ { i } )
+$$
+
+where $\mathbf { x } _ { i } \in \mathbb { R } ^ { d }$ is the layer input, $g _ { \boldsymbol { \theta } _ { i } }$ is the decoder head function for task $i$ with weights $\theta _ { i }$ , $E$ the frozen BERT embedding layer, $B$ the BERT Transformer layers and $R$ the linear weight matrix of a single task conditioned alignment matrix. $\gamma _ { i } , \beta _ { i } : \mathbb { R } ^ { d } \mapsto \mathbb { R } ^ { \dot { d } }$ are Feature Wise Linear Modulation functions.
+
+# 2.1.3 CONDITIONAL LAYER NORMALIZATION (CLN)
+
+We extend the Conditional Batch Normalization idea from de Vries et al. (2017) to Layer Normalization (Ba et al., 2016). For task $\mathcal { T } _ { i }$ , $i \in \{ 1 , \ldots , T \}$ :
+
+$$
+\mathbf { h } _ { i } = { \frac { 1 } { \sigma } } \odot ( \mathbf { a } _ { i } - { \boldsymbol { \mu } } ) * { \hat { \gamma } } _ { i } ( \mathbf { z } _ { i } ) + \beta _ { i } ( \mathbf { z } _ { i } ) , \qquad { \hat { \gamma } } _ { i } ( \mathbf { z } _ { i } ) = { \gamma } ^ { \prime } { \gamma } _ { i } ( \mathbf { z } _ { i } ) + { \beta } ^ { \prime }
+$$
+
+where $\mathbf { h } _ { i }$ is the CLN output vector, ${ \bf { a } } _ { i }$ are the preceding layer activations associated with task $i$ , $\mu$ and $\sigma$ are the mean and the variance of the summed inputs within each layer as defined in Ba et al. (2016). Conditional Layer Normalization is initialized with BERT’s Layer Normalization affine transformation weights and bias $\gamma ^ { \prime }$ and $\beta ^ { \prime }$ from the original formulation: $\begin{array} { r } { \mathbf { h } = \frac { 1 } { \sigma } \odot ( \mathbf { a } - \boldsymbol { \mu } ) * \boldsymbol { \gamma } ^ { \prime } + \boldsymbol { \beta } ^ { \prime } } \end{array}$ . During training, the weight and bias functions of $\gamma _ { i } ( * )$ and $\beta _ { i } ( * )$ are always trained, while the original Layer Normalization weight may be kept fixed. This module was added to account for task specific rescaling of individual training cases. Layer Normalization normalizes the inputs across features. The conditioning introduced in equation 2.1.3 allows us to modulate the normalization’s output based on a task’s latent representation.
+
+# 2.1.4 CONDITIONAL BOTTLENECK
+
+We created a task conditioned two layer feed-forward bottleneck layer (CFF up/down in Figure 3). The conditional bottleneck layer follows the same transformation as in equation 2. The module in Figure 3a is added to the top most Transformer layers of CA-MTL and uses a CLN. For CA-MTLLARGE this module is the main building block of the skip connection added alongside all Transformer layers seen in Figure 3b. The connection at layer $j$ takes in the matrix sum of the Transformer layer output at $j$ and the previous connection’s output at $j - 1$ . The Conditional bottleneck allows lower layer information to flow upwards depending on the task. Our intuition for introducing this component is related to recent studies (Tenney et al., 2019a) that showed that the “most important layers for a given task appear at specific positions”. As with the other modules described so far, each task adaptation is created from the weights of a single shared adapter that is modulated by the task embedding.
+
+![](images/29f9dad5e585bb0bc4752354077750eb90ed63d6d495c3045a18ce4f47033415.jpg)  
+Figure 3: a) Conditional Bottleneck for CA-MTLBASE. b) Conditional Bottleneck for CA-MTLLARGE.
+
+# 2.2 MULTI-TASK UNCERTAINTY SAMPLING
+
+MT-Uncertainty Sampling is a task selection strategy that is inspired by Active Learning techniques. Our algorithm 1 is outlined in the Appendix, Section A.2. Similar to Active Learning, our algorithm first evaluates the model uncertainty. MT-Uncertainty Sampling uses Shannon Entropy, an uncertainty measure, to choose training examples by first doing forward pass through the model with $b \times T$ input samples. For an output classification prediction with $C _ { i }$ possible classes and probabilities $( p _ { i , 1 } , \ldots , p _ { i , C _ { i } } )$ , the Shannon Entropy $H _ { i }$ , for task $\mathbb { T } _ { i }$ and $i \in \{ 1 , \ldots , T \}$ , our uncertainty measure $\mathbb { U } ( \mathbf { x } )$ are given by:
+
+$$
+H _ { i } = H _ { i } { \big ( } f _ { \phi ( \mathbf { z } _ { i } ) , \theta _ { i } } ( \mathbf { x } ) { \big ) } = - \sum _ { c = 1 } ^ { C _ { i } } p _ { c } \log p _ { c } , \qquad \mathbb { U } ( x _ { i } ) = { \frac { H _ { i } { \big ( } f _ { \phi ( \mathbf { z } _ { i } ) , \theta _ { i } } ( \mathbf { x } ) { \big ) } } { \hat { H } \times H _ { i } ^ { \prime } } }
+$$
+
+$$
+\hat { H } = \operatorname* { m a x } _ { i \in \{ 1 , \dots , T \} } \bar { H } _ { i } = \operatorname* { m a x } \Bigg [ \frac { 1 } { b } \sum _ { { \bf x } \in { \bf x } _ { i } } H _ { i } \Bigg ] , \qquad H _ { i } ^ { \prime } = - \sum _ { c = 1 } ^ { C _ { i } } \frac { 1 } { C _ { i } } \log \Bigg [ \frac { 1 } { C _ { i } } \Bigg ]
+$$
+
+where ${ \bar { H } } _ { i }$ is the average Shannon Entropy across $b$ samples of task t, $H _ { i } ^ { \prime }$ , the Shannon entropy of choosing classes with uniform distribution and $\hat { H }$ , the maximum of each task’s average entropy over $b$ samples. $H _ { i } ^ { \prime }$ is normalizing factor that accounts for differing number of prediction classes (without the normalizing factor $H _ { i } ^ { \prime }$ , tasks with a binary classification $C _ { i } = 1$ were rarely chosen). Further, to limit high entropy outliers and to favor tasks with highest uncertainty, we normalize with $\hat { H }$ . The measure in eq. 4 allows Algorithm 1 to choose $b$ samples from $b \times T$ candidates to train the model.
+
+# 3 RELATED WORK
+
+Multi-Tasking in NLP. To take advantage of the potential positive transfer of knowledge from one task to another, several works have proposed carefully choosing which tasks to train as an intermediate step in NLP before single task fine-tuning (Bingel & Søgaard, 2017; Kerinec et al., 2018; Wang et al., 2019a; Standley et al., 2019; Pruksachatkun et al., 2020; Phang et al., 2018). The intermediate tasks are not required to perform well and are not typically evaluated jointly. In this work, all tasks are trained jointly and all tasks used are evaluated from a single model. In Natural Language Understanding (NLU), it is still the case that to get the best task performance one often needs a separate model per task (Clark et al., $2 0 1 9 \mathrm { c }$ ; McCann et al., 2018). At scale, Multilingual NMT systems (Aharoni et al., 2019) have also found that MTL model performance degrades as the number of tasks increases. We notice a similar trend in NLU with our baseline MTL model. Recently, approaches in MTL have tackled the problem by designing task specific decoders on top of a shared model (Liu et al., 2019b) or distilling multiple single-task models into one (Clark et al., 2019c). Nonetheless, such MTL approaches still involves single task fine-tuning. In this paper, we show that it is possible to achieve high performance in NLU without single task fine-tuning.
+
+Adapters. Adapters are trainable modules that are attached in specific locations of a pretrained network. They provide another promising avenue to limit the number of parameters needed when confronted with a large number of tasks. This approach is useful with pretrained MLM models that have rich linguistic information (Tenney et al., 2019b; Clark et al., 2019b; Liu et al., 2019a; Tenney et al., 2019a). Recently, Houlsby et al. (2019) added an adapter to a pretrained BERT model by fine-tuning the layer norms and adding feed forward bottlenecks in every Transformer layer. However, such methods adapt each task individually during the fine-tuning process. Unlike prior work, our method harnesses the vectorized representations of tasks to modularize a single pretrained model across all tasks. Stickland et al. (2019) and Tay et al. (2020) also mix both MTL and adapters with BERT and T5 encoder-decoder (Raffel et al., 2019) respectively by creating local task modules that are controlled by a global task agnostic module. The main drawback is that a new set of non-shared parameters must be added when a new task is introduced. CA-MTL shares all parameters and is able to re-modulate existing weights with a new task embedding vector.
+
+Active Learning, Task Selection and Sampling. Our sampling technique is similar to the ones found in several active learning algorithms (Chen et al., 2006) that are based on Shannon entropy estimations. Reichart et al. (2008) and Ikhwantri et al. (2018) examined Multi-Task Active Learning (MTAL), a technique that chooses one informative sample for $T$ different learners (or models) for each $T$ tasks. Instead we choose $T$ tasks samples for one model. Moreover, the algorithm weights each sample by the corresponding task score, and the Shannon entropy is normalized to account for various losses (see equation 5). Also, our algorithm is used in a large scale MTL setup $\gg 2$ tasks). Recently, Glover & Hokamp (2019) explored task selection in MTL using learning policies based on counterfactual estimations (Charles et al., 2013). However, such method considers only fixed stochastic parameterized policies while our method adapts its selection criterion based on model uncertainty throughout the training process.
+
+# 4 EXPERIMENTS AND RESULTS
+
+We show that our adapter of section 2 achieve parameter efficient transfer for 26 NLP tasks. Our implementation of CA-MTL is based on HuggingFace (Wolf et al., 2019). Hyperparameters and our experimental set-up are outlined in A.5. To preserve the weights of the pretrained model, CA-MTL’s bottom half Transformer layers are frozen in all experiments (except in section 4.4). We also tested different layer freezing configurations and found that freezing half the layers worked best on average (see Section A.8).
+
+# 4.1 MULTI-TASK UNCERTAINTY SAMPLING
+
+Our MT-Uncertainty sampling strategy, from section 2.2, is compared to 3 other task selection schemes: a) Counterfactual b) Task size c) Random. We used a BERTBASE (no adapters) on $2 0 0 \mathrm { k }$ iterations and with the same hyperparameters as in Glover & Hokamp (2019). For more information on Counterfactual task selection, we invite the reader to consult the full explanation in Glover & Hokamp (2019). For $T$ tasks and the dataset $D _ { i }$ for tasks $i \in \{ 1 , \ldots , T \}$ , we rewrite the definitions of Random πrand and Task size π|task| sampling:
+
+$$
+\pi _ { r a n d } = 1 / T , \pi _ { | t a s k | } = | D _ { i } | \biggl [ \sum _ { i = 1 } ^ { T } | D _ { i } | \biggl ] ^ { - 1 }
+$$
+
+![](images/34eb77740d33c2b3598990f689875121cccead2186770f4cae1a14a288a3439e.jpg)  
+Figure 4: MT-Uncertainty vs. other task sampling strategies: median dev set scores on 8 GLUE tasks and using $\mathbf { B E R T _ { B A S E } }$ . Data for the Counterfactual and Task Size policy $\pi | t a s k |$ (eq. 6) is from Glover & Hokamp (2019).
+
+In Figure 4, we see from the results that MTUncertainty converges faster by reaching the $80 \%$ average GLUE score line before other task sampling methods. Further, MT-Uncertainty maximum score on $2 0 0 \mathrm { k }$ iterations is at 82.2, which is $1 . 7 \%$ higher than Counterfactual sampling. The datasets in the GLUE benchmark offers a wide range of dataset sizes. This is useful to test how MT-Uncertainty manages a jointly trained low resource task (CoLA) and high resource task (MNLI). Figure 5 explains how catastrophic forgetting is curtailed by sampling tasks before performance drops. With $\pi _ { r a n d }$ , all of CoLA’s tasks are sampled by iteration 500, at which point the larger MNLI
+
+![](images/c0edcd9dbfb63940afee1cee5e485a34f2983bf79768a803581334598893f550.jpg)  
+Figure 5: CoLA/MNLI Dev set scores and Entropy for $\pi _ { r a n d }$ (left) and MT-Uncertainty (right).
+
+dataset overtakes the learning process and CoLA’s dev set performance starts to diminish. On the other hand, with MT-Uncertainty sampling, CoLA is sampled whenever Shannon entropy is higher than MNLI’s. The model first assesses uncertain samples using Shannon Entropy then decides what data is necessary to train on. This process allows lower resource tasks to keep performance steady. We provide evidence in Figure 8 of A.2 that MT-Uncertainty is able to manage task difficulty — by choosing the most difficult tasks first.
+
+# 4.2 ABLATION AND MODULE ANALYSIS
+
+In Table 1, we present the results of an ablation study to determine which elements of CA-MTLBERT-BASE had the largest positive gain on average GLUE scores. Starting from a MTL BERTBASE baseline trained using random task sampling $( \pi _ { r a n d } )$ . Apart for the Conditional Adapter, each module as well as MTUncertainty lift overall performance and reduce variance across tasks. Please note that we also included accuracy/F1 scores for QQP, MRPC and Pearson/ Spearman correlation for STS-B to calculate score standard deviation Task $\sigma$ . Intuitively, when negative task transfer occurs between two tasks, either (1) task interference is bidirectional and scores are both impacted, or (2) interference is unidirectional and only one score is impacted. We calculate Task $\sigma$ to characterize changes in the dynamic range of performance across multiple tasks. We do this to asses the degree to which performance improvements are distributed across all tasks or only subsets of tasks. As we can see from Table 1, Conditional Attention, Conditional Alignment, Conditional Layer Normalization, MT-Uncertainty play roles in reducing Task $\sigma$ and increasing performance across tasks. This provides partial evidence of CA-MTL’s ability to mitigating negative task transfer.
+
+Table 1: Model ablation studya on the GLUE dev set. All models have the bottom half layers frozen.   
+
+<table><tr><td>Model changes</td><td>Avg GLUE</td><td>Task σ GLUE</td><td>% data used</td></tr><tr><td>BERTBASEMTL(πrand)</td><td>80.61</td><td>14.41</td><td>100</td></tr><tr><td>+ Conditional Attention</td><td>82.41</td><td>10.67</td><td>100</td></tr><tr><td>+ Conditional Adapter</td><td>82.90</td><td>11.27</td><td>100</td></tr><tr><td> + CA and CLN</td><td>83.12</td><td>10.91</td><td>100</td></tr><tr><td>+ MT-Uncertainty (CA-MTLBERT-BASE)</td><td>84.03</td><td>10.02</td><td>66.3</td></tr></table>
+
+$\overline { { ^ \mathrm { a } C \mathrm { A } = } }$ Conditional Alignment, CLN=Conditional Layer Normalization, Task $\sigma { = } 1$ scores standard deviation across tasks.
+
+We show that Conditional Alignment can learn to capture covariate distribution differences with task embeddings co-learned from other adapter components of CA-MTL. In Figure 6, we arrive at similar conclusions as Wu et al. (2020), who proved that negative task transfer is reduced when task covariances are aligned. The authors provided a “covariance similarity score” to gauge covariance alignment. For task $i$ and $j$ with $m _ { i }$ and $m _ { j }$ data samples respectively, and given $d$ dimensional inputs to the first Transformer layer $X _ { i } ~ \in ~ \mathbb { R } ^ { m _ { i } \times d }$ and $\boldsymbol { X } _ { j } \in \mathbb { R } ^ { m _ { j } \times d }$ , we rewrite the steps to calculate the covariance similarity score between task $i$ and $j$ : (a) Take the covariance matrix $X _ { i } ^ { \top } X _ { i }$ , (b) Find its best rank- $\cdot r _ { i }$ approximation $U _ { i , r _ { i } } D _ { i , r _ { i } } U _ { i , r _ { i } } ^ { \top }$ , where $r _ { i }$ is chosen to contain $9 9 \%$ of the singular values. (c) Apply steps (a), (b) to $X _ { j }$ , and compute the covariance similarity score $C o v S i m _ { i , j }$ :
+
+![](images/0b711cc530d06279a62112811821ec097b1eb1de96f6d8a2e794718e2b146d0e.jpg)  
+Figure 6: Task performance vs. avg. covariance similarity scores (eq. 7) for MTL and CA-MTL.
+
+$$
+C o v S i m _ { i , j } : = \frac { | | ( U _ { i , r _ { i } } D _ { i , r _ { i } } ^ { 1 / 2 } ) ^ { \top } U _ { j , r _ { j } } D _ { j , r _ { j } } ^ { 1 / 2 } | | _ { F } } { | | U _ { i , r _ { i } } D _ { i , r _ { i } } ^ { 1 / 2 } | | _ { F } \cdot | | U _ { j , r _ { j } } D _ { j , r _ { j } } ^ { 1 / 2 } | | _ { F } } . \ C o v S i m _ { i } = \frac { 1 } { T - 1 } \sum _ { j \neq i } C o v S i m _ { i , j }
+$$
+
+Since we are training models with $T$ tasks, we take the average covariance similarity score $C o v S i m _ { i }$ between task $i$ and all other tasks. We measure $C o v S i m _ { i }$ using equation 7 between 9 single-task models trained on individual GLUE tasks. For each task in Figure 6, we measure the similarity score on the MTL trained BERTBASE baseline, e.g., CoLA (MTL), or CA-MTLBERT-BASE model, e.g., MNLI (CA-MTL). Our score improvement measure is the $\%$ difference between a single task model and MTL or CA-MTL on the particular task. We find that covariance similarity increases for 9 tasks and that performance increases for 7 out 9 tasks. These measurements confirm that the Conditional Alignment is able to align task covariance, thereby helping alleviate task interference.
+
+# 4.3 JOINTLY TRAINING ON 8 TASKS: GLUE
+
+In Table 2, we evaluate the performance of CA-MTL against single task fine-tuned models, MTL as well as the other BERT-based adapters on GLUE. As in Houlsby et al. (2019), $\bf { M N L I } _ { \mathrm { m } }$ and $\mathrm { M N L I _ { m m } }$ are treated as separate tasks. Our results indicate that CA-MTL outperforms both the BASE adapter,
+
+Table 2: Adapters with layer freezing vs. ST/MT on GLUE test set. F1 scores are reported for QQP/MRPC, Spearman’s correlation for STS-B, accuracy on the matched/mismatch sets for MNLI, Matthew’s correlation for CoLA and accuracy for other tasks. \* Individual scores not available. ${ \mathrm { S T } } { = } { \mathrm { : } }$ Single Task, MTL $\vartriangle$ Multitask, g.e.= greater or equal to. Results from: 1Devlin et al. (2018) 2Stickland et al. (2019). 3Houlsby et al. (2019) .
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Type</td><td rowspan="2">Total params</td><td rowspan="2">Trained params/task</td><td rowspan="2">#tasks g.e.ST</td><td colspan="8">GLUE MRPC QNLI QQP RTE SST-2 STS-B</td></tr><tr><td>MNLI</td><td></td><td></td><td></td><td></td><td></td><td></td><td>Avg</td></tr><tr><td colspan="10">Base Models- Test Server Results</td><td colspan="7"></td></tr><tr><td>BERTBASE</td><td>ST</td><td>9.0×</td><td>100%</td><td>丨</td><td>52.1</td><td>84.6/83.4</td><td>88.9</td><td>90.5</td><td>71.2</td><td>66.4</td><td>93.5</td><td></td><td>85.8</td><td>79.6</td></tr><tr><td>BERTBASE2 2</td><td>MTL</td><td>1.0×</td><td>11.1%</td><td>2</td><td>51.2</td><td>84.0/83.4</td><td>86.7</td><td></td><td>89.3</td><td>70.8</td><td>76.6</td><td>93.4</td><td>83.6</td><td>79.9</td></tr><tr><td>PALs+Anneal Samp. 2</td><td>MTL</td><td>1.13×</td><td>12.5%</td><td>4</td><td>51.2</td><td>84.3/83.5</td><td></td><td>88.7</td><td>90.0</td><td>71.5</td><td>76.0</td><td>92.6</td><td>85.8</td><td>80.4</td></tr><tr><td>CA-MTLBERT-BASE（ (ours)</td><td>MTL</td><td>1.12×</td><td>5.6%</td><td>5</td><td>53.1</td><td>85.9/85.8</td><td></td><td>88.6</td><td>90.5</td><td>69.2</td><td>76.4</td><td>93.2</td><td>85.3</td><td>80.9</td></tr><tr><td colspan="10">LargeModels TestServer Results</td><td colspan="7"></td></tr><tr><td>BERTLARGE 1</td><td>ST</td><td>9.0×</td><td>100%</td><td></td><td></td><td>60.5</td><td>86.7/85.9</td><td>89.3</td><td>92.7</td><td>72.1</td><td>70.1</td><td>94.9</td><td>86.5</td><td>82.1</td></tr><tr><td> Adapters-2563</td><td>ST</td><td>1.3×</td><td>3.6%</td><td>1 3</td><td>59.5</td><td>84.9/85.1</td><td></td><td>89.5</td><td>90.7</td><td>71.8</td><td>71.5</td><td>94.0</td><td>86.9</td><td>80.0</td></tr><tr><td>CA-MTLBERT-LARGE (ours)</td><td>MTL</td><td>1.12×</td><td>5.6%</td><td>3</td><td>59.5</td><td>85.9/85.4</td><td></td><td>89.3</td><td>92.6</td><td>71.4</td><td>79.0</td><td>94.7</td><td>87.7</td><td>82.8</td></tr></table>
+
+PALS $+$ Anneal Sampling (Stickland et al., 2019), and the LARGE adapter, Adapters-256 (Houlsby et al., 2019). Against single task (ST) models, CA-MTL is $1 . 3 \%$ higher than $\mathbf { B E R T _ { B A S E } }$ , with 5 out 9 tasks equal or greater performance, and $0 . 7 \%$ higher than BERTLARGE, with 3 out 9 tasks equal or greater performance. ST models, however, need 9 models or close to $9 \times$ more parameters for all 9 tasks. We noted that CA-MTLBERT-LARGE’s average score is driven by strong RTE scores. While RTE benefits from MTL, this behavior may also be a side effect of layer freezing. In Table 10, we see that CA-MTL has gains over ST on more and more tasks as we gradually unfreeze layers.
+
+# 4.4 TRANSFER TO NEW TASKS
+
+In Table 3 we examine the ability of our method to quickly adapt to new tasks. We performed domain adaptation on SciTail (Khot et al., 2018) and SNLI (Bowman et al., 2015) datasets, using a CA-MTLBASE model trained on GLUE and a new linear decoder head. We
+
+Table 3: Domain adaptation results on dev. sets for BASE models. 1Liu et al. (2019b), 2Jiang et al. (2020)   
+
+<table><tr><td rowspan="2">% data used</td><td colspan="4">SciTail</td><td colspan="4">SNLI</td></tr><tr><td>0.1%</td><td>1%</td><td>10%</td><td>|100%</td><td>0.1%</td><td>1%|10%</td><td></td><td>100%</td></tr><tr><td>BERTBASE</td><td>51.2</td><td>82.2</td><td>90.5</td><td>94.3</td><td>52.5</td><td>78.1</td><td>86.7</td><td>91.0</td></tr><tr><td>MT-DNN1</td><td>81.9</td><td>88.3</td><td>91.1</td><td>95.7</td><td>81.9</td><td>88.3</td><td>91.1</td><td>95.7</td></tr><tr><td>MT-DNNSMART2 2</td><td>82.3</td><td>88.6</td><td>91.3</td><td>96.1</td><td>82.7</td><td>86.0</td><td>88.7</td><td>91.6</td></tr><tr><td>CA-MTLBERT</td><td>83.2</td><td>88.7</td><td>91.4</td><td>95.6</td><td>82.8</td><td>86.2</td><td>88.0</td><td>91.5</td></tr></table>
+
+tested several pretrained and randomly initialized task embeddings in a zero-shot setting. The complete set of experiments with all task embeddings can be found in the Appendix, Section A.4. We then selected the best task embedding for our results in Table 3. STS-B and MRPC MTL-trained task embeddings performed best on SciTail and SNLI respectively. CA-MTLBERT-BASE has faster adaptation than MT-DNNSMART (Jiang et al., 2020) as evidenced by higher performances in low-resource regimes ( $0 . 1 \%$ and $1 \%$ of the data). When trained on the complete dataset, CA-MTLBERT-BASE is on par with MT-DNNSMART. Unlike MT-DNNSMART however, we do not add context from a semantic similarity model – MT-DNNSMART is built off HNN (He et al., 2019). Nonetheless, with a larger model, CA-MTL surpasses MT-DNNSMART on the full SNLI and SciTail datasets in Table 6.
+
+# 4.5 JOINTLY TRAINING ON 24 TASKS: GLUE/SUPER-GLUE, MRQA AND WNUT2017
+
+Effects of Scaling Task Count. In Figure 7 we continue to test if CA-MTL mitigates task interference by measuring GLUE average scores when progressively adding 9 GLUE tasks, 8 Super-GLUE tasks (Wang et al., 2019b), 6 MRQA tasks (Fisch et al., 2019). Tasks are described in Appendix section A.9. The results show that adding 23 tasks drops the performance of our baseline MTL BERTBASE $( \pi _ { r a n d } )$ . MTL BERT increases by $4 . 3 \%$ when adding MRQA but, with 23 tasks, the model performance drops by $1 . 8 \%$ . The opposite is true when CA-MTL modules are integrated into the model. CA-MTL continues to show gains with a large number of tasks and surpasses the baseline MTL model by close to $4 \%$ when trained on 23 tasks.
+
+![](images/ec21abfc0af4768516a38bacdcfe935a0d7b0afe0997309b7dfe520b668e403b.jpg)  
+Figure 7: Effects of adding more datasets on avg GLUE scores. Experiments conducted on 3 epochs. When 23 tasks are trained jointly, performance of CA-MTLBERT-BASE continues to improve.
+
+24-task CA-MTL. We jointly trained large MTL baselines and CA-MTL models on GLUE/Super-GLUE/MRQA and Named Entity Recognition (NER) WNUT2017 (Derczynski et al., 2017). Since some dev. set scores are not provided and since RoBERTa results were reported with a median score over 5 random seeds, we ran our own single seed ST/MTL baselines (marked “ReImp”) for a fair comparison. The dev. set numbers reported in Liu et al. (2019c) are displayed with our baselines in Table 9. Results are presented in Table 4.
+
+Table 4: 24-task CA-MTL vs. ST and vs. 24-task MTL with frozen layers on GLUE, SuperGLUE, MRQA and NER development sets. ${ \mathrm { S T } } { = } { \mathrm { S } }$ ingle Task, MTL Multitask, g.e.= greater or equal to. Details in section A.5.   
+
+<table><tr><td>Model</td><td colspan="3">Task Grouping GLUE SuperGLUE MRQA</td><td>Avg</td><td>#tasks e.g.ST</td><td>Total Params</td></tr><tr><td colspan="7">BERT-LARGEmodels</td></tr><tr><td>STReImp</td><td>84.5</td><td>68.9</td><td>79.7 54.1</td><td>76.8</td><td></td><td>24×</td></tr><tr><td>MTLReImp</td><td>83.2</td><td>72.1</td><td>77.8 42.2</td><td>76.4</td><td>9/24</td><td>1×</td></tr><tr><td>CA-MTL</td><td>86.6</td><td>74.1</td><td>79.5 49.0</td><td>79.1</td><td>17/24</td><td>1.12×</td></tr><tr><td colspan="7">RoBERTa-LARGEmodels</td></tr><tr><td>STReImp</td><td>88.2</td><td>76.5</td><td>83.6 57.8</td><td>81.9</td><td></td><td>24×</td></tr><tr><td>MTLReImp</td><td>86.0</td><td>78.6</td><td>80.7</td><td>49.3 80.7</td><td>7/24</td><td>1×</td></tr><tr><td>CA-MTL</td><td>89.4</td><td>80.0</td><td>82.4 55.2</td><td>83.1</td><td>15/24</td><td>1.12×</td></tr></table>
+
+We notice in Table 4 that even for large models, CA-MTL provides large gains in performance on average over both ST and MTL models. For the BERT based models, CA-MTL provides $2 . 3 \%$ gain over ST and higher scores on 17 out 24 tasks. For RoBERTa based models, CA-MTL provides $1 . 2 \%$ gain over ST and higher scores on 15 out 24 tasks. We remind the reader that this is achieved with a single model. Even when trained with 16 other tasks, it is interesting to note that the MTL baseline perform better than the ST baseline on Super GLUE where most tasks have a small number of samples. Also, we used NER to test if we could still outperform the ST baseline on a token-level task, significantly different from other tasks. Unfortunately, while CA-MTL performs significantly better than the MTL baseline model, CA-MTL had not yet overfit on this particular task and could have closed the gap with the ST baselines with more training cycles.
+
+Comparisons with other methods. In Table 5, CA-MTLBERT is compared to other Large BERT based methods that either use $\mathbf { M T L } + \mathbf { S T }$ , such as MT-DNN (Liu et al., 2019b), intermediate tasks $+ \mathrm { \Omega } \mathrm { S T }$ , such as STILTS (Phang et al., 2018) or MTL model distillation $+ \mathrm { \bf ~ \nabla ~ } \mathrm { S T }$ , such as BAM! (Clark et al., 2019c). Our method scores higher than MT-DNN on 5 of 9 tasks and by $1 . 0 \%$ on avg. Against STILTS, CA-MTL realizes a $0 . 7 \%$ avg. score gain, surpassing scores on 6 of 9 tasks. We also show
+
+Table 5: Our 24-task CA-MTL vs. other large models on GLUE. F1 is reported for QQP/MRPC, Spearman’s corr. for STS-B, Matthew’s corr. for CoLA and accuracy for other tasks. \*Split not available. \*\*Uses intermediate task fine-tuning $+ \thinspace \mathrm { S T }$ .   
+
+<table><tr><td>Model</td><td colspan="7">GLUE tasks</td><td rowspan="2">Avg</td></tr><tr><td></td><td>CoLA MNLI</td><td>MRPC QNLI QQP RTE SST-2 STS-B</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="9">BERT-LARGE based models on Dev set.</td></tr><tr><td>MT-DNN</td><td>63.5 87.1/86.7</td><td>91.0</td><td>92.9</td><td>89.2</td><td>83.4</td><td>94.3</td><td>90.6</td><td>85.6</td></tr><tr><td>STILTS **</td><td>62.1 86.1*</td><td>92.3</td><td>90.5</td><td>88.5</td><td>83.4</td><td>93.2</td><td>90.8</td><td>85.9</td></tr><tr><td>BAM！</td><td>61.8 87.0*</td><td>1</td><td>92.5</td><td>1</td><td>82.8</td><td>93.6</td><td>89.7</td><td>1</td></tr><tr><td>24-task CA-MTL</td><td>63.8 86.3/86.0</td><td>92.9</td><td>93.4</td><td>88.1</td><td>84.5</td><td>94.5</td><td>90.3</td><td>86.6</td></tr><tr><td colspan="9">RoBERTa-LARGEbasedmodelson Testset. RoBERTA** with</td></tr><tr><td>Ensemble</td><td>67.8 91.0/90.8</td><td>91.6</td><td>95.4</td><td>74.0</td><td>87.9</td><td>97.5</td><td>92.5</td><td>87.3</td></tr><tr><td>24-task CA-MTL</td><td>62.2 89.0/88.4</td><td>92.0</td><td>94.7</td><td>72.3</td><td>86.2</td><td>96.3</td><td>89.8</td><td>85.7</td></tr></table>
+
+that CA-MTLRoBERTa is within only $1 . 6 \%$ of a RoBERTa ensemble of 5 to 7 models per task and that uses intermediate tasks. Using our 24-task CA-MTL large RoBERTa-based model, we report NER F1 scores on the WNUT2017 test set in Table 6a. We compare our result with RoBERTaLARGE and XLM-RLARGE (Nguyen et al., 2020) the current state-of-the-art (SOTA). Our model outperforms XLM-RLARGE by $1 . 6 \%$ , reaching a new state-of-the-art. Using domain adaptation as described in Section 4.4, we report results on the SciTail test set in Table 6b and SNLI test set in Table 6b. For SciTail, our model matches the current $\mathrm { S O T A } ^ { 2 }$ ALUM (Liu et al., 2020), a RoBERTa large based model that additionally uses the SMART (Jiang et al., 2020) fine-tuning method. For SNLI, our model outperforms SemBert, the current $\mathsf { S O T A } ^ { 3 }$ .
+
+Table 6: CA-MTL test performance vs. SOTA.   
+
+<table><tr><td>(b) SciTail</td><td>% Acc</td></tr><tr><td>MT-DNN ALUMRoBERTa</td><td>94.1 96.3</td></tr><tr><td>ALUMRoBERTa-SMART</td><td>96.8</td></tr><tr><td>CA-MTLRoBERTa (ours)</td><td>96.8</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>(a)WNUT2017</td><td rowspan=1 colspan=1>F1</td></tr><tr><td rowspan=1 colspan=1>RoBERTaLARGEXLM-RLARGE</td><td rowspan=1 colspan=1>56.957.1</td></tr><tr><td rowspan=1 colspan=1>CA-MTLRoBERTa(ours)</td><td rowspan=1 colspan=1>58.0</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>(C) SNLI</td><td rowspan=1 colspan=1>% Acc</td></tr><tr><td rowspan=1 colspan=1>MT-DNNMT-DNNSMARTSemBERT</td><td rowspan=1 colspan=1>91.691.791.9</td></tr><tr><td rowspan=1 colspan=1>CA-MTLRoBERTa(ours)</td><td rowspan=1 colspan=1>92.1</td></tr></table>
+
+# 5 CONCLUSION
+
+We believe that our experiments here have helped demonstrate the potential of task conditioned adaptive learning within a single model that performs multiple tasks. In a large-scale 24-task NLP experiment, CA-MTL outperforms fully tuned single task models by $2 . 3 \%$ for BERT Large and by $1 . 2 \%$ for RoBERTa Large using 1.12 times the number of parameters, while single task fine-tuning approach requires 24 separately tuned single task models or 24 times the number of parameters. When a BERT vanilla MTL model sees its performance drop as the number of tasks increases, CA-MTL scores continue to climb. Performance gains are not driven by a single task as it is often the case in MTL. Each CA-MTL module that adapts a Transformer model is able to reduce performance variances between tasks, increasing average scores and aligning task covariances. This evidence shows that CA-MTL is able to mitigate task interference and promote more efficient parameter sharing. We showed that MT-Uncertainty is able to avoid degrading performances of low resource tasks. Tasks are sampled whenever the model sees entropy increase, helping avoid catastrophic forgetting. Overall, CA-MTL offers a promising avenue to dynamically adapt and modularize knowledge embedded in large monolithic pretrained models. Extending such ideas will be an objective for future work.
+
+# ACKNOWLEDGMENTS
+
+This research was supported by the Canada CIFAR AI Chairs Program, NSERC and PROMPT. Experiments in this article were conducted with Compute Canada and MILA computational infrastructure and we thank them for their support. We would like to thank Colin Raffel, Sandeep Subramanian, and Nicolas Gontier for their useful feedback and the anonymous reviewers for helpful comments, discussions and suggestions.
+
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+# A APPENDIX
+
+# A.1 SUMMARY OF ACRONYMS
+
+Acronyms of datasets and descriptions can be found below in section A.9.
+
+Table 7: List of acronyms used in this paper.   
+
+<table><tr><td>Acronym</td><td>Description</td></tr><tr><td>ARLM CA-MTL</td><td>AutoregressiveLanguageModels</td></tr><tr><td></td><td>Conditional Adaptive Multi-Task Learning:our architecture</td></tr><tr><td>CFF</td><td>Conditional Feed-Forward:a feed-forward layer modulated by a conditioning vector</td></tr><tr><td>CLN</td><td>Conditional Layer Normalization in section 2.1.3</td></tr><tr><td>EDM</td><td>Evolutionary Data Measures (Collins et al.,2O18):a task difficulty estimate</td></tr><tr><td>GLUE</td><td>General Language Understanding Evaluation Wang et al. (2O18):a benchmark with multiple datasets</td></tr><tr><td>QA</td><td>Question Answering</td></tr><tr><td>MT</td><td>Multi-Task</td></tr><tr><td>MTAL</td><td>Multi-Task Active Learning: finding the most informative instance for multiple learners (or models)</td></tr><tr><td>MLM</td><td>Masked Language Model: BERT Devlin et al.(2O18) is an example of an MLM</td></tr><tr><td>MTL</td><td>Multi-Task Learning:&quot;learning tasks in paralel while using a shared representation&quot;(Caruana,1997)</td></tr><tr><td>MRQA</td><td>Machine Reading for Question Answering Fisch et al.(2O19):a benchmark with multiple datasets</td></tr><tr><td>NER</td><td>Named Entity Recognition</td></tr><tr><td>NLP</td><td>Natural Language Processing</td></tr><tr><td>SOTA</td><td>State of the art</td></tr><tr><td>ST</td><td>Single Task fine-tuning:all weights are typically updated</td></tr><tr><td>ST-A</td><td>ST withAdapter modules:one adapter per task is trained and pretrained weights are optionally updated</td></tr></table>
+
+A.2 UNCERTAINTY SAMPLING: ALGORITHM AND ADDITIONAL RESULTS
+
+# Algorithm 1: Multi-task Uncertainty Sampling
+
+Input: Training data $D _ { t }$ for task $t \in [ 1 , \ldots , T ]$ ; batch size $b$ ; $C _ { t }$ possible output classes   
+for task $t$ ; $f : = f _ { \phi ( \mathbf { z } _ { i } ) , \theta _ { i } }$ our model with weights $\phi , \theta _ { i }$ ;   
+Output: $B ^ { \prime }$ - multi-task batch of size $b$   
+1 $B  \emptyset$   
+2 for $t \gets 1$ to $T$ do   
+3 Generate $\mathbf { x } _ { t } : = \{ x _ { t , 1 } , \ldots , x _ { t , b } \} \overset { \mathrm { i . i . d . } } { \sim } D _ { t }$   
+4 for $i \gets 1$ to $b$ do   
+5 $\begin{array} { r } { \mathcal { H } _ { t , i } \gets - \sum _ { c = 1 } ^ { C _ { i } } p _ { c } ( f ( x _ { t , i } ) ) \log p _ { c } ( f ( x _ { t , i } ) ) } \end{array}$ . Entropy of each sample   
+6   
+7 end   
+8 Compute $\begin{array} { r } { \bar { \mathcal { H } } _ { t } \gets \frac { 1 } { b } \sum _ { \mathbf { x } \in \mathbf { x } _ { i } } \mathcal { H } _ { t , i } } \end{array}$ . Average entropy for task t   
+9   
+10 Compute $\begin{array} { r } { H _ { t } ^ { \prime } \gets - \sum _ { c = 1 } ^ { C _ { t } } \frac { 1 } { C _ { t } } \log \bigg [ \frac { 1 } { C _ { t } } \bigg ] } \end{array}$ . Max entropy (uniform distribution)   
+11   
+12 $B  B \cup { \bf x } _ { t }$ and $D _ { t } \gets D _ { t } \setminus { \bf x } _ { t }$   
+13 if $D _ { t } = \varnothing$ then   
+14 Reload $D _ { t }$   
+15 end   
+16 for $i \gets 1$ to $b$ do   
+17 Compute: $\mathcal { U } _ { t , i }  \mathcal { H } _ { t , i } / H _ { t } ^ { \prime }$ . Uncertainty normalized with max entropy   
+18 end   
+19 end   
+20 Compute $\hat { \mathcal { H } } \gets \operatorname* { m a x } _ { i \in \{ 1 , \dots , T \} } [ \bar { \mathcal { H } } _ { t } ]$ . Entropy of task with highest average entropy   
+21 Update ${ \mathcal { U } } _ { t , i } \gets { \mathcal { U } } _ { t , i } / { \hat { \mathcal { H } } }$ . Normalize each sample’s uncertainty measure   
+22 $\mathcal { B } ^ { \prime }  \mathrm { t o p \_ b } ( \{ \mathcal { U } _ { t , i } | t \in [ 1 , \ldots , T ] , i \in [ 1 , \ldots , b ] \} )$ . b samples w/ highest uncertainty   
+Return: With $B ^ { \prime }$ , solve eq. 1 with gradient descent; updated model $f$
+
+An advantage of our MT-Uncertainty Sampling approach is its ability to manage task difficulty. This is highlighted in Figure 8. In this experiment, we estimated task difficulty using the Evolutionary Data Measures $( \mathrm { E D M } ) ^ { 4 }$ proposed by Collins et al. (2018). The task difficulty estimate relies on multiple dataset statistics such as the data size, class diversity, class balance and class interference. Interestingly, estimated task difficulty correlates with the first instance that the selection of a specific task occurs. Supposing that QNLI is an outlier, we notice that peaks in the data occur whenever tasks are first selected by MT Uncertainty sampling. This process follows the following order: 1. MNLI 2. CoLA 3. RTE 4. QQP 5. MRPC 6.SST-2, which is the order from highest task difficulty to lowest task difficulty using EDM. As opposed to Curriculum Learning (Bengio et al., 2009), MT-Uncertainty dynamically prioritizes the most difficult tasks. As also discovered in MTL vision work (Guo et al., 2018), this type of prioritization on more difficult tasks may explain MT-Uncertainty’s improved performance over other task selection methods. In MTL, heuristics to balance tasks during training is typically done by weighting each task’s loss differently. We see here how MT-Uncertainty is able to prioritize task difficulty.
+
+![](images/1b7fec7c0382e63cbec403254adce395f5f10038e3fddb0fead9e8305978713c.jpg)  
+Figure 8: Task composition of MT-Uncertainty sampling and estimated task difficulty using EDM: number of training samples per task at each iteration for batch size of 32. The occurrence of first peaks and estimated difficulty follow the same order: From highest to lowest: $\mathrm { M N L I } > \mathrm { C o L A } > \mathrm { R T E } > \mathrm { Q Q P } = \mathrm { M R P C } > \mathrm { S S T - 2 } .$
+
+While the EDM difficulty measure is shown to correlate well with model performance, it lacks precision. As reported in Collins et al. (2018), the average score achieved on the Yahoo Answers dataset is $6 9 . 9 \%$ and its difficulty is 4.51. The average score achieved on Yelp Full is $5 6 . 8 \%$ , $1 3 . 1 \%$ less than Yahoo Answers and its difficulty is 4.42. The authors mention that “This indicates that the difficulty measure in its current incarnation may be more effective at assigning a class of difficulty to datasets, rather than a regression-like value”.
+
+# A.3 OTHER RELATED WORK
+
+Multi-Tasking in NLP and other fields. MTL weight sharing algorithms such as Mixture-of-Experts (MoE) have found success in NLP (Lepikhin et al., 2020). CA-MTL can complement MoE since the Transformers multi-headed attention can be seen as a form of MoE (Peng et al., 2020). In Vision, MTL can also improve with optimization (Sener & Koltun, 2018) or gradient-based approaches (Chen et al., 2017; Yu et al., 2020).
+
+Active Learning, Task Selection and Sampling. Ikhwantri et al. (2018) examined multi-task active learning for neural semantic role labeling in a low resource setting, using entity recognition as the sole auxiliary task. They used uncertainty sampling for active learning and found that $12 \%$ less data could be used compared to passive learning. Reichart et al. (2008) has examined different active learning techniques for the two task annotation scenario, focusing on named entity recognition and syntactic parse tree annotations. In contrast, here we examine the larger scale data regime, the modularization of a multi-task neural architecture, and the many task $\left( \gg 2 \right)$ ) setting among other differences. Other than MTAL (Reichart et al., 2008; Ikhwantri et al., 2018), Kendall et al. (2017) leveraged model uncertainty to balance MTL losses but not to select tasks as is proposed here.
+
+# A.4 ZERO-SHOT RESULTS ON SCITAIL AND SNLI
+
+Before testing models on domain adaptation in section 4.4, we ran zero-shot evaluations on the development set of SciTail and SNLI. Table 8 outlines 8-task CA-MTLBERT-BASE’s zero-shot transfer abilities when pretrained on GLUE with our MTL approach. We expand the task embedding layer to accommodate an extra task and explore various embedding initialization. We found that reusing STS-B and MRPC task embeddings worked best for SciTail and SNLI respectively.
+
+Table 8: CA-MTL is flexible and extensible to new tasks. However, CA-MTL is sensitive to the new task’s embedding. We tested multiple task embeddings that worked best on either SciTail or SNLI by checking performance in a zero shot setting or using $0 \%$ of the data.   
+
+<table><tr><td rowspan=1 colspan=1>Initializationof newtask embedding layer</td><td rowspan=1 colspan=1>SciTail0% of data</td><td rowspan=1 colspan=1>SNLI0% of data</td></tr><tr><td rowspan=12 colspan=1>CoLA&#x27;sembeddingsMNLI&#x27;s embeddingsMRPC&#x27;s embeddingsSTS-B&#x27;s embeddingsSST-2&#x27;s embeddingsQQP&#x27;s embeddingsQNLI&#x27;s embeddingsRTE&#x27;s embeddingsWNLI&#x27;s embeddingsAverageRandom initializationXavier initialization</td><td rowspan=1 colspan=1>43.0</td><td rowspan=1 colspan=1>34.0</td></tr><tr><td rowspan=1 colspan=1>24.2</td><td rowspan=1 colspan=1>33.0</td></tr><tr><td rowspan=1 colspan=1>34.5</td><td rowspan=1 colspan=1>45.5</td></tr><tr><td rowspan=1 colspan=1>46.9</td><td rowspan=1 colspan=1>33.2</td></tr><tr><td rowspan=1 colspan=1>25.8</td><td rowspan=1 colspan=1>34.2</td></tr><tr><td rowspan=1 colspan=1>31.7</td><td rowspan=2 colspan=1>37.338.0</td></tr><tr><td rowspan=1 colspan=1>32.0</td></tr><tr><td rowspan=1 colspan=1>32.3</td><td rowspan=3 colspan=1>40.630.437.7</td></tr><tr><td rowspan=1 colspan=1>29.0</td></tr><tr><td rowspan=1 colspan=1>28.7</td></tr><tr><td rowspan=1 colspan=1>46.8</td><td rowspan=1 colspan=1>34.0</td></tr><tr><td rowspan=1 colspan=1>29.8</td><td rowspan=1 colspan=1>37.6</td></tr></table>
+
+# A.5 MORE EXPERIMENTAL DETAILS
+
+We used a batch size of 32 and a seed of 12 in all experiments. We used Adam (Kingma & Ba, 2015) as the optimizer with a learning rate of 2e-5. We applied a learning rate decay with warm up over the first $10 \%$ of the training steps. Unless otherwise specified, we used 5 epochs, a seed of 12 and a sequence length of 128. Additional details are outlined in section . Our data prepossessing and linear decoder heads are the same as in Devlin et al. (2018). We used the same dropout rate of 0.1 in all layers. To run our experiments, we used either four NVIDIA P100 GPU for base models or four NVIDIA V100 GPU for larger ones. We did not perform parameter search. We do not use ensemble of models or task-specific tricks (Devlin et al., 2018; Liu et al., 2019b; Clark et al., 2019c). All models are either 12 Transformer layers for BASE and 24 Transformer layers for LARGE. Apart from CA-MTL, models trained in multi-task learning (BERT or RoBERTa without adapters) used random task sampling. For Table 1 and Figure 7, all BERT-based model have half their layers frozen (untrained) for a fair comparison of ablation results. For the 24-task MTL and CA-MTL models in Tables 4 and 5, we increased the input sequence length to 256 and used 8 epochs.
+
+# A.6 THE DIRECT SUM OPERATOR
+
+In section 2.1.1, we used the direct sum operator $\oplus$ . This operation allows us to create a block diagonal matrix. The direct sum of a matrix $A \in \mathbb { R } ^ { n \times m }$ and $\bar { \boldsymbol { B } } \in \mathbb { R } ^ { p \times q }$ results in a matrix of size $( m + p ) \times ( n + q )$ , defined as:
+
+$$
+\mathbf { A } \oplus \mathbf { B } = { \left[ \begin{array} { l l } { \mathbf { A } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { B } } \end{array} \right] } = { \left[ \begin{array} { l l l l l l } { a _ { 1 1 } } & { \cdots } & { a _ { 1 n } } & { 0 } & { \cdots } & { 0 } \\ { \vdots } & { \ddots } & { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { a _ { m 1 } } & { \cdots } & { a _ { m n } } & { 0 } & { \cdots } & { 0 } \\ { 0 } & { \cdots } & { 0 } & { b _ { 1 1 } } & { \cdots } & { b _ { 1 q } } \\ { \vdots } & { \ddots } & { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { 0 } & { \cdots } & { 0 } & { b _ { p 1 } } & { \cdots } & { b _ { p q } } \end{array} \right] }
+$$
+
+# A.7 BASELINES AND OTHER EXPERIMENTAL RESULTS
+
+In this section, we present our baseline results for BERT, RoBERTa, CA-MTL as well as other models. Our single task results (ST) that we ran ourselves surpass other paper’s reported scores in Table 9. Liu et al. (2019c) reports random seed median scores for RoBERTa. However, our RoBERTa ST baseline matches or surpasses the original paper’s scores 4 out 7 times on the development set when scores are comparable (QQP F1 and STS-B spearman are not reported).
+
+Table 9: F1 scores are reported for QQP/MRPC, Spearman’s correlation for STS-B, accuracy on the matched/mismatch sets for MNLI, Matthew’s correlation for CoLA and accuracy for other tasks. ${ \mathrm { S T } } { = } { \mathrm { : } }$ Single Task, MTL $\vartriangle$ Multitask. $^ { * } \mathrm { Q N L I }$ v1 (we report v2) $^ { * * }$ score or Spearman’s correlation is not reported. $^ { \ast \ast \ast }$ Unknown random seeds. Results from: 1Stickland et al. (2019) 2Liu et al. (2019b) 3Phang et al. (2018) 4Liu et al. (2019c).   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Total params</td><td rowspan="2">Trained params/task</td><td colspan="9">GLUE</td></tr><tr><td>CoLA</td><td>MNLI</td><td>MRPC</td><td></td><td></td><td></td><td></td><td>QNLI QQP RTE SST-2 STS-B</td><td>Avg</td></tr><tr><td colspan="10">Base Models-Dev set Results</td></tr><tr><td>PALs+Anneal Samp.1</td><td>1.13×</td><td>12.5%</td><td></td><td></td><td></td><td>一</td><td>一</td><td>1</td><td>一</td><td></td><td>81.70</td></tr><tr><td>8-task CA-MTLBERT-BASE (Ours)</td><td>1.12×</td><td>5.6%</td><td>60.9</td><td>82.7/83.1</td><td>88.9</td><td>90.7</td><td></td><td>90.379.1</td><td>91.9</td><td>88.8</td><td>84.03</td></tr><tr><td colspan="10">BERTLARGEModels DevsetResults</td></tr><tr><td>STBERT-LARGE2</td><td>9×</td><td>100%</td><td>60.5</td><td>86.7/85.9</td><td>89.3</td><td>92.7*</td><td>89.3</td><td>70.1</td><td>94.9</td><td>86.5</td><td>84.0</td></tr><tr><td>ST BERT-LARGE</td><td>9×</td><td>100%</td><td>62.1</td><td>86.2/86.2</td><td>92.3</td><td>89.4</td><td>88.5</td><td>70.0</td><td>92.5</td><td>90.1</td><td>84.1</td></tr><tr><td>ST BERT-LARGE (ours)</td><td>9×</td><td>100%</td><td>63.6</td><td>86.5/86.0</td><td>91.4</td><td>91.0</td><td>88.5</td><td>70.2</td><td>94.7</td><td>88.2</td><td>84.5</td></tr><tr><td>24-task CA-MTLBERT-LARGE (Ours)</td><td>1.12×</td><td>5.6%</td><td>63.8</td><td>86.3/86.0</td><td>92.9</td><td>93.4</td><td>88.1</td><td>84.5</td><td>94.5</td><td>90.3</td><td>86.6</td></tr><tr><td colspan="10">RoBERTaLARGEModels-DevsetResults</td></tr><tr><td>RoBERTa-LARGE4</td><td>9×</td><td></td><td></td><td></td><td></td><td></td><td>**</td><td></td><td></td><td></td><td></td></tr><tr><td>(Median 5 runs)***</td><td></td><td>100%</td><td>68.0</td><td>90.2</td><td>90.9</td><td>94.7</td><td></td><td>86.6</td><td>96.4</td><td>**</td><td></td></tr><tr><td>STRoBERTa-LARGE(ours)</td><td>9× 1.12×</td><td>100%</td><td>68.3</td><td>89.2/88.9</td><td>92.6</td><td>94.8</td><td>84.6</td><td>87.0 88.8 91.0</td><td>96.4</td><td>91.7</td><td>88.2</td></tr><tr><td>24-task CA-MTLRoBERTa-LARGE (ours)</td><td></td><td>5.6%</td><td>69.7</td><td>89.4/89.3</td><td>93.9</td><td>94.9</td><td></td><td></td><td>96.2</td><td>91.0</td><td>89.4</td></tr></table>
+
+A.8 SOME RESULTS ON LAYER FREEZING AND WITH FULL BLOCK ATTENTION.
+
+All experiments in this section were run for only 5 epochs, exclusively on the GLUE dataset for the large BERT-based 8-task CA-MTL model. Results in Table 10 reveal that as we freeze more layers, performance tends to decrease. However, since we wanted to preserve as much pretrained knowledge as possible, we chose to keep at least $50 \%$ of layers frozen. While this has slightly lowered our performance on 9 GLUE tasks, we believe that keeping as much of the original pretrained weights is beneficial when increasing the total number of tasks in MTL to 24 or more tasks. However, we did not explore this hypothesis more.
+
+Table 10: 8-task CA-MTLBERT-LARGE (see section 4.3) for various layer freezing configurations. F1 scores are reported for QQP/MRPC, Spearman’s correlation for STS-B, accuracy on the matched/mismatch sets for MNLI, Matthew’s correlation for CoLA and accuracy for other tasks. $\mathrm { F B A } = \mathrm { F u l l }$ Block Attention   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">%frozen layers</td><td rowspan="2">#tasks g.eST CoLA</td><td colspan="8">GLUE MRPC QNLI QQP RTE SST-2 STS-B Avg</td></tr><tr><td>MNLI</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="10">LARGEModels-Dev set Results</td></tr><tr><td>STBERT-LARGE (ours)</td><td>0%</td><td>一</td><td>63.6 86.5/86.0</td><td>91.4</td><td>91.0</td><td></td><td>88.570.2</td><td>93.1</td><td>88.2</td><td>84.3</td></tr><tr><td>CA-MTL</td><td>0%</td><td>7</td><td>60.2 86.2/86.0</td><td>92.0</td><td>91.5</td><td>88.7</td><td>76.3</td><td>93.3</td><td>89.5</td><td>84.9</td></tr><tr><td>CA-MTL</td><td>25%</td><td>6</td><td>63.7 86.1/85.8</td><td>89.1</td><td>91.2</td><td>88.6</td><td>79.7</td><td>92.9</td><td>88.5</td><td>85.1</td></tr><tr><td>CA-MTL</td><td>50%</td><td>3</td><td>63.2 85.5/85.5</td><td>91.8</td><td>90.9</td><td>88.3</td><td>81.4</td><td>93.0</td><td>90.1</td><td>85.5</td></tr><tr><td>CA-MTL FBA</td><td>50%</td><td>0</td><td>60.2 81.7/81.1</td><td>88.0</td><td>85.8</td><td></td><td>85.7 78.7</td><td>88.6</td><td>87.1</td><td>81.8</td></tr></table>
+
+# A.9 DATASET DESCRIPTION
+
+The datasets that were used for the domain adaptation experiments were SciTail5 and SNLI6. We jointly trained a CA-MTLRoBERTa-LARGE model on 9 GLUE tasks, 8 Super-GLUE7 tasks, 6 MRQA8 tasks, and on $\mathrm { W N U T 2 0 1 7 ^ { 9 } }$ (Derczynski et al., 2017).
+
+All GLUE tasks are binary classification, except STS-B (regression) and MNLI (three classes). We used the same GLUE data preprocessing as in Devlin et al. (2018).
+
+Table 11: GLUE (Wang et al., 2018) dataset description. References: 1Warstadt et al. (2018), 2Socher et al. (2013), 3Dolan & Brockett (2005), 4Cer et al. (2017), 5Williams et al. (2018), 6Wang et al. (2018), 7Levesque (2011)   
+
+<table><tr><td rowspan=1 colspan=1>Acronym</td><td rowspan=1 colspan=1>Corpus</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Task</td><td rowspan=1 colspan=1>Domain</td></tr><tr><td rowspan=4 colspan=1>CoLA1SST-22MRPC3STS-B4QQPMNLI5RTE6WNLI7</td><td rowspan=4 colspan=1>Corpus of Linguistic AcceptabilityStanford Sentiment TreebankMicrosoft Research Paraphrase CorpusSemantic Textual Similarity BenchmarkQuora Question PairsMulti-Genre NLIRecognition Textual EntailmentWinograd NLI</td><td rowspan=1 colspan=2>8.5K67K3.7K</td><td rowspan=4 colspan=1>acceptabilitysentiment detectionparaphrase detectiontextual similarityparaphrase detectioninferenceinference/entailmentcoreference</td><td rowspan=4 colspan=1>miscellaneousmovie reviewsnewsmiscellaneousonline QAmiscellaneousnews,Wikipediafiction books</td></tr><tr><td rowspan=1 colspan=2>7K364K</td></tr><tr><td rowspan=1 colspan=2>393K</td></tr><tr><td rowspan=1 colspan=2>2.5K634</td></tr></table>
+
+Table 12: Super-GLUE (Wang et al., 2019b) dataset description. References: 1Clark et al. (2019a), 2de Marneffe et al. (2019), 3Gordon et al. (2012), 4Khashabi et al. (2018), 5Zhang et al. (2018), 6Wang et al. (2019b), 7Poliak et al. (2018), 8Levesque (2011)   
+
+<table><tr><td>Acronym</td><td>Corpus</td><td>Train</td><td>Task</td><td>Domain</td></tr><tr><td>BoolQ1</td><td>Boolean Questions</td><td>9.4K</td><td>acceptability</td><td>Google queries,Wikipedia</td></tr><tr><td>CB² COPA3</td><td>CommitmentBank</td><td>250</td><td>sentiment detection</td><td>miscellaneous</td></tr><tr><td>MultiRC4</td><td>Choice of Plausible Alternatives Multi-Sentence Reading Comprehension</td><td>400 5.1K</td><td>paraphrase detection textual similarity</td><td>blogs,encyclopedia miscellaneous</td></tr><tr><td>ReCoRD5</td><td>Reading Comprehension</td><td>101K</td><td>paraphrase detection</td><td>news</td></tr><tr><td></td><td>and Commonsense Reasoning</td><td></td><td></td><td></td></tr><tr><td>RTE6 WiC7</td><td>Recognition Textual Entailment</td><td>2.5K 6K</td><td>inference</td><td>news,Wikipedia</td></tr><tr><td>WSC8</td><td>Word-in-Context</td><td>554</td><td>word sense disambiguation</td><td>WordNet,VerbNet</td></tr><tr><td></td><td>Winograd Schema Challenge</td><td></td><td>coreference resolution</td><td>fiction books</td></tr></table>
+
+Table 13: MRQA (Fisch et al., 2019) dataset description. References: 1Rajpurkar et al. (2016a), 2Trischler et al. (2017), 3Joshi et al. (2017), 4Dunn et al. (2017), 5Yang et al. (2018), $^ 6 \mathrm { K }$ wiatkowski et al. (2019)   
+
+<table><tr><td>Acronym</td><td>Corpus</td><td>Train</td><td>Task</td><td>Domain</td></tr><tr><td>SQuAD1</td><td>Stanford QADataset</td><td>86.6K</td><td>crowdsourced questions</td><td>Wikipedia</td></tr><tr><td>NewsQA²</td><td>NewsQA</td><td>74.2K</td><td>crowdsourced questions</td><td>news</td></tr><tr><td>TriviaQA3</td><td>TriviaQA</td><td>61.7K</td><td>trivia QA</td><td>web snippets</td></tr><tr><td>SearchQA4</td><td>SearchQA</td><td>117.4K</td><td>Jeopardy QA</td><td>web snippets</td></tr><tr><td>HotpotQA5</td><td>HotpotQA</td><td>72.9K</td><td>crowdsourced questions</td><td>Wikipedia</td></tr><tr><td>Natural Questions6</td><td>Natural Questions</td><td>104.7K</td><td>search logs</td><td>Wikipedia</td></tr></table>
+
+SuperGLUE has a more diverse task format than GLUE, which is mostly limited to sentence and sentence-pair classification. We follow the same preprocessing procedure as in Wang et al. (2019b). All tasks are binary classification tasks, except CB (three classes). Also, WiC and WSC are span based classification tasks. We used the same modified MRQA dataset and preprocessing steps that were used in Joshi et al. (2019). All MRQA tasks are span prediction tasks which seeks to identify start and end tokens of an answer span in the input text.
+
+Table 14: SNLI (Bowman et al., 2015) and SciTail (Khot et al., 2018) datasets description.   
+
+<table><tr><td>Acronym Corpus</td><td></td><td>|Train]</td><td>Task</td><td>Domain</td></tr><tr><td>SNLIT</td><td>Stanford Natural Language Inference</td><td>550.2k</td><td>inference</td><td>human-written English sentence pairs</td></tr><tr><td>SciTail²</td><td>Science and Entailment</td><td>23.5K</td><td>entailment</td><td>Science question answering</td></tr></table>
+
+SNLI is a natural inference task where we predict three classes. Examples of three target labels are: Entailment, Contradiction, and Neutral (irrelevant). SciTail is a textual entailment dataset. The hypotheses in SciTail are created from multiple-choice science exams and the answer candidates (premise) are extracted from the web using information retrieval tools. SciTail is a binary true/false classification tasks that seeks to predict whether the premise entails the hypothesis. The two datasets are used only for domain adaptation in this study (see section A.4 for the details of our approach).

parse/train/fUxqIofPPi/fUxqIofPPi.md

@@ -0,0 +1,223 @@
+# Projected GANs Converge Faster
+
+Axel Sauer1,2 Kashyap Chitta1,2 Jens Müller3 Andreas Geiger1,2 1University of Tübingen 2Max Planck Institute for Intelligent Systems, Tübingen 3Computer Vision and Learning Lab, University Heidelberg 2{firstname.lastname}@tue.mpg.de 3{firstname.lastname}@iwr.uni-heidelberg.de
+
+# Abstract
+
+Generative Adversarial Networks (GANs) produce high-quality images but are challenging to train. They need careful regularization, vast amounts of compute, and expensive hyper-parameter sweeps. We make significant headway on these issues by projecting generated and real samples into a fixed, pretrained feature space. Motivated by the finding that the discriminator cannot fully exploit features from deeper layers of the pretrained model, we propose a more effective strategy that mixes features across channels and resolutions. Our Projected GAN improves image quality, sample efficiency, and convergence speed. It is further compatible with resolutions of up to one Megapixel and advances the state-of-the-art Fréchet Inception Distance (FID) on twenty-two benchmark datasets. Importantly, Projected GANs match the previously lowest FIDs up to 40 times faster, cutting the wall-clock time from 5 days to less than 3 hours given the same computational resources.
+
+![](images/c883e6a958f6034a0b565bab650b45781fb0ee5d5d9d6b3449ca0d3672c443e5.jpg)  
+Figure 1: Convergence with Projected GANs. Evolution of samples for a fixed latent code during training on the AFHQ-Dog dataset [5]. We find that discriminating features in the projected feature space speeds up convergence and yields lower FIDs. This finding is consistent across many datasets.
+
+# 1 Introduction
+
+A Generative Adversarial Network (GAN) consists of a generator and a discriminator. For image synthesis, the generator’s task is to generate an RGB image; the discriminator aims to distinguish real from fake samples. On closer inspection, the discriminator’s task is two-fold: First, it projects the real and fake samples into a meaningful space, i.e., it learns a representation of the input space. Second, it discriminates based on this representation. Unfortunately, training the discriminator jointly with the generator is a notoriously hard task. While discriminator regularization techniques help to balance the adversarial game [31], standard regularization methods like gradient penalties [36] are susceptible to hyperparameter choices [26] and can lead to a substantial decrease in performance [4].
+
+In this paper, we explore the utility of pretrained representations to improve and stabilize GAN training. Using pretrained representations has become ubiquitous in computer vision [29, 30, 48] and natural language processing [18, 45, 47]. While combining pretrained perceptual networks [58] with GANs for image-to-image translation has led to impressive results [14, 49, 59, 64], this idea has not yet materialized for unconditional noise-to-image synthesis. Indeed, we confirm that a naïve application of this idea does not lead to state-of-the-art results (Section 4) as strong pretrained features enable the discriminator to dominate the two-player game, resulting in vanishing gradients for the generator [2]. In this work, we demonstrate how these challenges can be overcome and identify two key components for exploiting the full potential of pretrained perceptual feature spaces for GAN training: feature pyramids to enable multi-scale feedback with multiple discriminators and random projections to better utilize deeper layers of the pretrained network.
+
+We conduct extensive experiments on small and large datasets with a resolution of up to $1 0 2 4 ^ { 2 }$ pixels. Across all datasets, we demonstrate state-of-the-art image synthesis results at significantly reduced training time (Fig. 1). We also find that Projected GANs increase data efficiency and avoid the need for additional regularization, rendering expensive hyperparameter sweeps unnecessary. Code, models, and supplementary videos can be found on the project page https://sites.google.com/view/ projected-gan.
+
+# 2 Related Work
+
+We categorize related work into two main areas: pretraining for GANs and discriminator design.
+
+Pretrained Models for GAN Training. Work on leveraging pretrained representations for GANs can be divided into two categories: First, transferring parts of a GAN to a new dataset [15, 38, 65, 71] and, second, using pretrained models to control and improve GANs. The latter is advantageous as pretraining does not need to be adversarial. Our work falls into this second category. Pretrained models can be used as a guiding mechanism to disentangle causal generative factors [54], for text-driven image manipulation [44], matching the generator activations to inverted classifiers [19, 56], or to generate images via gradient ascent in the latent space of a generator [41]. The non-adversarial approach of [53] learns generative models with moment matching in pretrained models; however, the results remain far from competitive to standard GANs. An established method is the combination of adversarial and perceptual losses [21]. Commonly, the losses are combined additively [10, 14, 32, 52, 64]. Additive combination, however, is only possible if a reconstruction target is available, e.g., in paired image-toimage translation settings [74]. Instead of providing the pretrained network with a reconstruction target, Sungatullina et al. [59] propose to optimize an adversarial loss on frozen VGG features [58]. They show that their approach improves CycleGAN [74] on image translation tasks. In a similar vein, [49] recently proposed a different perceptual discriminator. They utilize a pretrained VGG and connect its features with the prediction of a pretrained segmentation network. The combined features are fed into multiple discriminators at different scales. The two last approaches are specific to the image-toimage translation task. We demonstrate that these methods do not work well for the more challenging unconditional setting where the entire image content is synthesized from a random latent code.
+
+Discriminator Design. Much work on GANs focuses on novel generator architectures [4, 26, 27, 69], while the discriminator often remains close to a vanilla convolutional neural network or mirrors the generator. Notable exceptions are [55,70] which utilize an encoder-decoder discriminator architecture. However, in contrast to us, they neither use pretrained features nor random projections. A different line of work considers a setup with multiple discriminators, applied to either the generated RGB image [8, 13] or low-dimensional projections thereof [1, 40]. The use of several discriminators promises improved sample diversity, training speed, and training stability. However, these approaches are not utilized in current state-of-the-art systems because of diminishing returns compared to the increased computational effort. Providing multi-scale feedback with one or multiple discriminators has been helpful for both image synthesis [23, 24] and image-to-image translation [43, 64]. While these works interpolate the RGB image at different resolutions, our findings indicate the importance of multi-scale feature maps, showing parallels to the success of pyramid networks for object detection [34]. Lastly, to prevent overfitting of the discriminator, differentiable augmentation methods have recently been proposed [25, 63, 72, 73]. We find that adopting these strategies helps exploit the full potential of pretrained representations for GAN training.
+
+# 3 Projected GANs
+
+GANs aim to model the distribution of a given training dataset. A generator $G$ maps latent vectors $\mathbf { z }$ sampled from a simple distribution $\mathbb { P } _ { \mathbf { z } }$ (typically a normal distribution) to corresponding generated samples $G ( \mathbf { z } )$ . The discriminator $D$ then aims to distinguish real samples $\mathbf { x } \sim \mathbb { P } _ { \mathbf { x } }$ from the generated samples $G ( \mathbf { z } ) \sim \mathbb { P } _ { G ( \mathbf { z } ) }$ . This basic idea results in the following minimax objective
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } \Big ( \mathbb { E } _ { \mathbf { x } } [ \log D ( \mathbf { x } ) ] + \mathbb { E } _ { \mathbf { z } } [ \log ( 1 - D ( G ( \mathbf { z } ) ) ) ] \Big )
+$$
+
+We introduce a set of feature projectors $\{ P _ { l } \}$ which map real and generated images to the discriminator’s input space. Projected GAN training can thus be formulated as follows
+
+$$
+\underset { G } { \mathrm { m i n } } \underset { \{ D _ { l } \} } { \mathrm { m a x } } \sum _ { l \in \mathcal { L } } \left( \mathbb { E } _ { \mathbf { x } } [ \log D _ { l } ( P _ { l } ( \mathbf { x } ) ) ] + \mathbb { E } _ { \mathbf { z } } [ \log ( 1 - D _ { l } ( P _ { l } ( G ( \mathbf { z } ) ) ) ) ] \right)
+$$
+
+where $\{ D _ { l } \}$ is a set of independent discriminators operating on different feature projections. Note that we keep $\{ P _ { l } \}$ fixed in (2) and only optimize the parameters of $G$ and $\{ D _ { l } \}$ . The feature projectors $\{ P _ { l } \}$ should satisfy two necessary conditions: they should be differentiable and provide sufficient statistics of their inputs, i.e., they should preserve important information. Moreover, we aim to find feature projectors $\{ P _ { l } \}$ which turn the (difficult to optimize) objective in (1) into an objective more amenable to gradient-based optimization. We now show that a projected GAN indeed matches the distribution in the projected feature space, before specifying the details of our feature projectors.
+
+# 3.1 Consistency
+
+The projected GAN objective in (2) no longer optimizes directly to match the true distribution $\mathbb { P } _ { T }$ . To understand the training properties under ideal conditions, we consider a more generalized form of the consistency theorem of [40]:
+
+Theorem 1. Let $\mathbb { P } _ { T }$ denote the density of the true data distribution and $\mathbb { P } _ { G }$ the density of the distribution the Generator $G$ produces. Let $P _ { l } \circ T$ and $P _ { l } \circ G$ be the functional composition of the differentiable and fixed function $P _ { l }$ and the true/generated data distribution, and y be the transformed input to the discriminator. For a fixed $G$ , the optimal discriminators are given by
+
+$$
+D _ { l , G } ^ { * } ( \mathbf { y } ) = \frac { \mathbb { P } _ { P _ { l } \circ T } ( \mathbf { y } ) } { \mathbb { P } _ { P _ { l } \circ T } ( \mathbf { y } ) + \mathbb { P } _ { P _ { l } \circ G } ( \mathbf { y } ) }
+$$
+
+for all $l \in \mathcal L$ . In this case, the optimal $G$ under (2) is achieved iff $\mathbb { P } _ { P _ { l } \circ T } = \mathbb { P } _ { P _ { l } \circ G }$ for all $l \in \mathcal L$
+
+A proof of the theorem is provided in the appendix. From the theorem, we conclude that a feature projector $P _ { l }$ with its associated discriminator $D _ { l }$ encourages the generator to match the true distribution along the marginal through $P _ { l }$ . Therefore, at convergence, $G$ matches the generated and true distributions in feature space. The theorem also holds when using stochastic data augmentations [25] before the deterministic projections $P _ { l }$ .
+
+# 3.2 Model Overview
+
+Projecting to and training in pretrained feature spaces opens up a realm of new questions which we address below. This section will provide an overview of the general system and is followed by extensive ablations of each design choice. As our feature projections affect the discriminator, we focus on $P _ { l }$ and $D _ { l }$ in this section and postpone the discussion of generator architectures to Section 5.
+
+Multi-Scale Discriminators. We obtain features from four layers $L _ { l }$ of a pretrained feature network $F$ at resolutions $( L _ { 1 } = 6 4 ^ { 2 } , L _ { 2 } = 3 2 ^ { 2 } , L _ { 3 } = 1 6 ^ { 2 } , L _ { 4 } = 8 ^ { 2 } )$ . We associate a separate discriminator $D _ { l }$ with the features at layer $L _ { l }$ , respectively. Each discriminator $D _ { l }$ uses a simple convolutional architecture with spectral normalization [37] at each convolutional layer. We observe better performance if all discriminators output logits at the same resolution $( 4 ^ { 2 } )$ . Accordingly, we use fewer down-sampling blocks for lower resolution inputs. Following common practice, we sum all logits for computing the overall loss. For the generator pass, we sum the losses of all discriminators. More complex strategies [1, 13] did not improve performance in our experiments.
+
+Random Projections. We observe that features at deeper layers are significantly harder to cover, as evidenced by our experiments in Section 4. We hypothesize that a discriminator can focus on a subset of the feature space while wholly disregarding other parts. This problem might be especially prominent in the deeper, more semantic layers. Therefore, we propose two different strategies to dilute prominent features, encouraging the discriminator to utilize all available information equally. Common to both strategies is that they mix features using differentiable random projections which are fixed, i.e., after random initialization, the parameters of these layers are not trained.
+
+Cross-Channel Mixing (CCM). Empirically, we found two properties to be desirable: (i) the random projection should be information preserving to leverage the full representational power of $F$ , and (ii) it should not be trivially invertible. The easiest way to mix across channels is a $1 \times 1$ convolution. A $1 \times 1$ convolution with an equal number of output and input channels is a generalization of a permutation [28] and consequently preserves information about its input. In practice, we find that more output channels lead to better performance as the mapping remains injective and therefore information preserving. Kingma et al. [28] initialize their convolutional layers as a random rotation matrix as a good starting point for optimization. We do not find this to improve GAN performance (see Appendix), arguably since it violates (ii). We therefore randomly initialize the weights of the convolutional layer via Kaiming initialization [16]. Note that we do not add any activation functions. We apply this random projection at each of the four scales and feed the transformed feature to the discriminator as depicted in Fig. 2.
+
+Cross-Scale Mixing (CSM). To encourage feature mixing across scales, CSM extends CCM with random $3 \times 3$ convolutions and bilinear upsampling, yielding a U-Net [50] architecture, see Fig. 3. However, our CSM block is simpler than a vanilla U-Net [50]: we only use a single convolutional layer at each scale. As for CCM, we utilize Kaiming initialization for all weights.
+
+![](images/aaa199bcd852ba250a50b0e4fa19c774e05bf5db00029dfe9e012a6cc4efec6b.jpg)
+
+Pretrained Feature Networks. We ablate over varying feature networks. First, we investigate different versions of EfficientNets, which allow for direct control over model size versus performance. EfficientNets are image classification models trained on ImageNet [7] and designed to provide favorable accuracy-compute tradeoffs. Second, we use ResNets of varying sizes. To analyze the dependency on ImageNet features (Section 4.3), we also consider R50-CLIP [46], a ResNet optimized with a contrastive language-image objective on a dataset of 400 million (image, text) pairs. Lastly, we utilize a vision transformer architecture (ViTBase) [9] and its efficient follow-up (DeiT-small distilled) [62]. We do not choose an inception network [60] to avoid strong correlations with the evaluation metric FID [17]. In the appendix, we also evaluate several other neural and non-neural metrics to rule out correlations. These additional metrics reflect the rankings obtained by FID.
+
+![](images/49ee5915d7fd475a516ce08c9ee80d2bc56371dfdca42948cc5b77f9b446b8fb.jpg)  
+Figure 2: CCM (dashed blue arrows) employs $1 \times 1$ convolutions with random weights.   
+Figure 3: CSM (dashed red arrows) adds random $3 \times 3$ convolutions and bilinear upsampling, yielding a U-Network.
+
+In the following, we conduct a systematic ablation study to analyze the importance and best configuration of each component in our Projected GAN model, before comparing it to the state-of-the-art.
+
+# 4 Ablation Study
+
+To determine the best configuration of discriminators, mixing strategy, and pretrained feature network, we conduct experiments on LSUN-Church [67], which is medium-sized (126k images) and reasonably visually complex, using a resolution of $2 5 6 ^ { 2 }$ pixels. For the generator $G$ we use the generator architecture of FastGAN [35], consisting of several upsampling blocks, with additional skip-layerexcitation blocks. Using a hinge loss [33], we train with a batch size of 64 until 1 million real images have been shown to the discriminator, a sufficient amount for $G$ to reach values close to convergence. If not specified otherwise, we use an EfficientNet-Lite1 [61] feature network in this section. We found that discriminator augmentation [25, 63, 72, 73] consistently improves the performance of all methods, and is required to reach state-of-the-art performance. We leverage differentiable data-augmentation [72] which we found to yield the best results in combination with the FastGAN generator.
+
+<table><tr><td>Discriminator(s)</td><td>rel-FD1↓</td><td>rel-FD2↓</td><td>rel-FD3↓</td><td></td><td>rel-FD4↓rel-FID↓</td></tr><tr><td colspan="6">No Projection</td></tr><tr><td>onL1</td><td>0.56</td><td>0.32</td><td>0.31</td><td>0.55</td><td>0.66</td></tr><tr><td>on Li,L2</td><td>0.35</td><td>0.21</td><td>0.23</td><td>0.47</td><td>0.53</td></tr><tr><td>on L1,L2,L3</td><td>0.42</td><td>0.26</td><td>0.28</td><td>0.64</td><td>0.90</td></tr><tr><td>On L1,L2,L3,L4</td><td>0.46</td><td>0.34</td><td>0.38</td><td>0.79</td><td>1.15</td></tr><tr><td>on L2,L3,L4</td><td>0.95</td><td>0.67</td><td>0.71</td><td>1.19</td><td>1.99</td></tr><tr><td>onL3,L4</td><td>2.14</td><td>1.41</td><td>1.18</td><td>1.99</td><td>3.46</td></tr><tr><td>onL4</td><td>10.92</td><td>5.74</td><td>2.56</td><td>2.79</td><td>5.08</td></tr><tr><td>Perceptual D</td><td>2.98</td><td>1.76</td><td>1.20</td><td>1.89</td><td>2.73</td></tr><tr><td colspan="6">CCM</td></tr><tr><td>on L1</td><td>0.27</td><td>0.21</td><td>0.26</td><td>0.50</td><td>0.59</td></tr><tr><td>on L1,L2</td><td>0.27</td><td>0.18</td><td>0.21</td><td>0.41</td><td>0.48</td></tr><tr><td>on L1,L2,L3</td><td>0.31</td><td>0.25</td><td>0.24</td><td>0.54</td><td>0.67</td></tr><tr><td>on L1,L2,L3,L4</td><td>0.53</td><td>0.34</td><td>0.34</td><td>0.59</td><td>0.77</td></tr><tr><td>Perceptual D</td><td>5.33</td><td>3.06</td><td>2.14</td><td>1.09</td><td>4.77</td></tr><tr><td colspan="6">CCM + CSM</td></tr><tr><td>onL1</td><td>0.34</td><td>0.25</td><td>0.19</td><td>0.35</td><td>0.44</td></tr><tr><td>on L1,L2</td><td>0.21</td><td>0.18</td><td>0.16</td><td>0.27</td><td>0.31</td></tr><tr><td>on Li,L2,L3</td><td>0.41</td><td>0.26</td><td>0.17</td><td>0.23</td><td>0.29</td></tr><tr><td>on L1,L2,L3,L4</td><td>0.26</td><td>0.16</td><td>0.13</td><td>0.16</td><td>0.24</td></tr><tr><td>Perceptual D</td><td>2.53</td><td>1.37</td><td>0.89</td><td>0.43</td><td>2.13</td></tr></table>
+
+Table 1: Feature Space Fréchet Distances. We aim to find the best combination of discriminators and random projections to fit the distributions in feature network $F$ . We show the relative FD at different layers of $F$ $( r e l – F D _ { i } )$ between $5 0 \mathrm { k }$ generated and real images on LSUN-Church. rel- $F D _ { i }$ is normalized using the baseline Fréchet Distances for a model with a standard single RGB image discriminator. Hence, values $> 1$ indicate worse performance than the RGB baseline. We report rel- $. F D$ for four layers of an EfficientNet $( L _ { 1 } , L _ { 2 } , L _ { 3 }$ and $L _ { 4 }$ from shallow to deep), as well as relative Fréchet Inception Distance (FID) [17]. Note that $r e l – F D _ { i }$ should not be compared between different feature spaces, i.e., only within-column comparisons are meaningful. Blue boxes highlight the layers which we supervise via independent discriminators. The green box corresponds to a perceptual discriminator [59], which takes in all feature maps at once.
+
+# 4.1 Which feature network layers are most informative?
+
+We first investigate the relevance of independent multi-scale discriminators. For this experiment, we do not use feature mixing. To measure how well $G$ fits a particular feature space, we employ the Fréchet Distance (FD) [12] on the spatially pooled features denoted as $F D _ { i }$ for layer $i$ . FDs across different feature spaces are not directly comparable. Therefore, we train a GAN baseline with a standard RGB discriminator, record $F \bar { D _ { i } ^ { R G B } }$ at each layer and quantify the relative improvement via the fraction rel- $F D _ { i } = F D _ { i } / F D _ { i } ^ { R G B }$ . We also investigate a perceptual discriminator [59], where feature maps are fed into different layers of the same discriminator to predict a single logit.
+
+The results in Table 1 (No Projection) show that two discriminators are better than one and improve over the vanilla RGB baseline. Surprisingly, adding discriminators at deep layers hurts performance. We conclude that these more semantic features do not respond well to direct adversarial losses. We also experimented with discriminators at resized versions of the original image, but could not find a setting of hyperparameters and architectures that improves over the single image baseline. Omitting the discriminators on the shallow features decreases performance, which is anticipated, as these layers contain most of the information about the original image. A similar effect has been observed for feature inversion [11] – the deeper the layer, the harder it is to reconstruct its input. Lastly, we observe that independent discriminators outperform the perceptual discriminator by a significant margin.
+
+<table><tr><td rowspan="2"></td><td colspan="5">EfficientNet</td><td colspan="3">ResNet</td><td colspan="2">Transformer</td></tr><tr><td>lite0</td><td>lite1</td><td>lite2</td><td>lite3</td><td>lite4</td><td>R18</td><td>R50</td><td>R50-CLIP</td><td>DeiT</td><td>ViT</td></tr><tr><td>Params (M)↓</td><td>2.96</td><td>3.72</td><td>4.36</td><td>6.42</td><td>11.15</td><td>11.18</td><td>23.51</td><td>23.53</td><td>92.36</td><td>317.52</td></tr><tr><td>IN top-1个</td><td>75.48</td><td>76.64</td><td>77.47</td><td>79.82</td><td>81.54</td><td>69.75</td><td>79.04</td><td>N/A</td><td>85.42</td><td>85.16</td></tr><tr><td>FID↓</td><td>2.53</td><td>1.65</td><td>1.69</td><td>1.79</td><td>2.35</td><td>4.16</td><td>4.40</td><td>3.80</td><td>2.46</td><td>12.38</td></tr></table>
+
+Table 2: Pretrained Feature Networks Study. We train the projected GAN with different pretrained feature networks. We find that compact EfficientNets outperform both ResNets and Transformers.
+
+# 4.2 How can we best utilize the pretrained features?
+
+Given the insights from the previous section, we aim to improve the utilization of deep features. For this experiment, we only investigate configurations that include discriminators at high resolutions. Table 1 (CCM and $\mathbf { C C M } + \mathbf { C S M } )$ presents the results for both mixing strategies. CCM moderately decreases the FDs across all settings, confirming our hypothesis that mixing channels results in better feedback for the generator. When adding CSM, we achieve another notable improvement across all configurations. Especially rel- $F D _ { i }$ at deeper layers are significantly decreased, demonstrating CSM’s usefulness to leverage deep semantic features. Interestingly, we observe that the best performance is now obtained by combining all four discriminators. A perceptual discriminator is again inferior to multiple discriminators. We remark that integrating the original image, via an independent discriminator or CCM or CSM always resulted in worse performance. This failure suggests that naïvely combining non-projected with projected adversarial optimization impairs training dynamics.
+
+# 4.3 Which feature network architecture is most effective?
+
+Using the best setting determined by the experiments above $\mathbf { \mathrm { C C M } } + \mathbf { \mathrm { C S M } }$ with four discriminators), we study the effectiveness of various perceptual feature network architectures for Projected GAN training. To ensure convergence, also for larger architectures, we train for 10 million images. Table 2 reports the FIDs achieved on LSUN-Church. Surprisingly, we find that there is no correlation with ImageNet accuracy. On the contrary, we observe lower FIDs for smaller models (e.g., EfficientNetslite). This observation indicates that a more compact representation is beneficial while at the same time reducing computational overhead and consequently training time. R50-CLIP slightly outperforms its R50 counterpart, indicating that ImageNet features are not required to achieve low FID. For the sake of completeness, we also train with randomly initialized feature networks, which, however, converge to much higher FID values (see Appendix). In the following, we thus use EfficientNet-Lite1 as our feature network.
+
+# 5 Comparison to State-of-the-Art
+
+This section conducts a comprehensive analysis demonstrating the advantages of Projected GANs with respect to state-of-the-art models. Our experiments are structured into three sections: evaluation of convergence speed and data efficiency (5.1), and comparisons on large (5.2) and small (5.3) benchmark datasets. We cover a wide variety of datasets in terms of size (hundreds to millions of samples), resolution $2 5 6 ^ { 2 }$ to $1 0 2 4 ^ { 2 }$ ), and visual complexity (clip-art, paintings, and photographs).
+
+Evaluation Protocol. We measure image quality using the Fréchet Inception Distance (FID) [17]. Following [26, 27], we report the FID between 50k generated and all real images. We select the snapshot with the best FID for each method. In addition to image quality, we include a metric to evaluate convergence. As in [25], we measure training progress based on the number of real images shown to the discriminator (Imgs). We report the number of images required by the model for the FID to reach values within $5 \%$ of the best FID over training. In the appendix, we also report other metrics that are less benchmarked in GAN literature: KID [3], SwAV-FID [39], precision and recall [51]. Unless otherwise specified, we follow the evaluation protocol of [20] to facilitate fair comparisons. Specifically, we compare all approaches given the same fixed number of images (10 million). With this setting, each experiment takes roughly 100-200 GPU hours on a NVIDIA V100, for more details we refer to the appendix.
+
+Baselines. We use StyleGAN2-ADA [25] and FastGAN [35] as baselines. StyleGAN2-ADA is the strongest model on most datasets in terms of sample quality, whereas FastGAN excels in training speed. We implement these baselines and our Projected GANs within the codebase provided by the authors of StyleGAN2-ADA [25]. For each model, we ran two kinds of data augmentation:
+
+![](images/b0daf437d42b8753ba9cb84d4e91753512629b3119522e23e9b1f8198557046d.jpg)  
+Figure 4: Training Properties. Left: Projected FastGAN surpasses the best FID of StyleGAN2 (at $\mathbf { 8 8 \ M }$ images) after just $1 . 1 \textbf { M }$ images on LSUN-Church. Right: Projected FastGAN yields significantly improved FID scores, even when using subsets of CLEVR with 1k and $1 0 \mathrm { k }$ samples.
+
+![](images/286fbe8b3c1b202601efc7433e4086776d4341d3681a0692ed6230be2e0e1844.jpg)  
+Figure 5: Training progress on LSUN church at $2 5 6 ^ { 2 }$ pixels. Shown are samples for a fixed noise vector z over k images. From top to bottom: FastGAN, StyleGAN2-ADA, Projected GAN.
+
+differentiable data-augmentation [72] and adaptive discriminator augmentation [25]. We select the better performing augmentation strategy per model. For all baselines and datasets, we perform data amplification through $\mathbf { X }$ -flips. Projected GANs use the same generator and discriminator architecture and training hyperparameters (learning rate and batch size) for all experiments. For high-resolution image generation, additional upsampling blocks are included in the generator to match the desired output resolution. We carefully tune all hyper-parameters for both baselines for best results: we find that FastGAN is sensitive to the choice of batch size, and StyleGAN2-ADA to the learning rate and R1 penalty. The appendix documents additional implementation details used in each of our experiments.
+
+# 5.1 Convergence Speed and Data Efficiency
+
+Following [20] and [68], we analyze the training properties of Projected GANs on LSUN-Church at an image resolution of $2 5 6 ^ { 2 }$ pixels and on the 70k CLEVR dataset [22]. In this section, we also train longer than $1 0 \mathbf { M }$ images if necessary, as we are interested in convergence properties.
+
+Convergence Speed. We apply projected GAN training for both the style-based generator of StyleGAN2 and the standard generator with a single input noise vector of FastGAN. As shown in Fig. 4 (left), FastGAN converges quickly but saturates at a high FID. StyleGAN2 converges more slowly $\mathbf { \delta } ^ { \mathrm { 8 8 \mathbf { \delta } M } }$ images) but reaches a lower FID. Projected GAN training improves both generators. Particularly for FastGAN, improvements in both convergence speed and final FID are significant while improvements for StyleGAN2 are less pronounced. Remarkably, Projected FastGAN reaches the previously best FID of StyleGAN2 after experiencing only $1 . 1 \mathbf { M }$ images as compared to $\mathbf { 8 8  M }$ of StyleGAN2. In wall clock time, this corresponds to less than 3 hours instead of 5 days. Hence, from now on, we utilize the FastGAN generator and refer to this model simply as Projected GAN.
+
+Fig. 5 shows samples for a fixed noise vector z during training on LSUN-Church. For both FastGAN and StyleGAN, patches of texture gradually morph into a global structure. For Projected GAN, we directly observe the emergence of structure which becomes more detailed over time. Interestingly, the Projected GAN latent space appears to be very volatile, i.e., for fixed $\mathbf { z }$ the images undergo significant perceptual changes during training. In the non-projected cases, these changes are more gradual. We hypothesize that this induced volatility might be due to the discriminator providing more semantic feedback compared to conventional RGB losses. Such semantic feedback could introduce more stochasticity during training which in turn improves convergence and performance. We also observed that the signed real logits of the discriminator remain at the same level throughout training (see Appendix). Stable signed logits indicate that the discriminator does not suffer from overfitting.
+
+Sample Efficiency. The use of pretrained models is generally linked to improved sample efficiency. To evaluate this property, we also created two subsets of the 70k CLEVR dataset by randomly subsampling $1 0 \mathrm { k }$ and 1k images from it, respectively. As depicted in Fig. 4 (right), our Projected GAN significantly improves over both baselines across all dataset splits.
+
+# 5.2 Large Datasets
+
+Besides CLEVR and LSUN-Church, we benchmark Projected GANs against various state-of-the-art models on three other large datasets: LSUN-Bedroom [67] (3M indoor bedroom scenes), FFHQ [26] $7 0 \mathrm { k }$ images of faces) and Cityscapes [6] ( $2 5 \mathrm { k }$ driving scenes captured from a vehicle). For all datasets, we use an image resolution of $2 5 6 ^ { 2 }$ pixels. As Cityscapes and CLEVR images are not of aspect ratio 1:1 we resize them to $2 5 6 ^ { 2 }$ for training. Besides StyleGAN2-ADA and FastGAN, we compare against SAGAN [69] and GANsformers [20]. All models were trained for $1 0 \mathbf { M }$ images. For the large datasets, we also report numbers for StyleGAN2 trained for more than $1 0 \mathbf { M }$ images to report the lowest FID values achieved in previous literature (denoted as StyleGAN2\*). In the appendix, we report results on nine more large datasets.
+
+Table 3 shows that the Projected GAN outperforms all state-of-the-art models in terms of FID values on all datasets by a large margin. For example, on LSUN-Bedroom, it achieves an FID value of 1.52 compared to 6.15 by GANsformer, the previously best model in this setting. Projected GAN achieves state-of-the-art FID values remarkably fast, e.g., on LSUN-church, it achieves an FID value of 3.18 after 1.1 M Imgs. StyleGAN2 has obtained the previously lowest FID value of 3.39 after 88 M Imgs, 80 times as many as needed by Projected GAN. Similar speed-ups are also realized for all other large datasets as shown in Table 3. Interestingly, when training longer on FFHQ (39 M Imgs), we observe further improvements of Projected GAN to an FID of 2.2. Note that all five datasets represent very different objects in various scenes. This demonstrates that the performance gain is robust to the choice of the dataset, although the feature network is trained only on ImageNet. It is important to note that the main improvements are based on improved sample diversity as indicated by recall which we report in the appendix. The improvement in diversity is most notable on large datasets, e.g., LSUN church, where the image fidelity appears to be similar to StyleGAN.
+
+# 5.3 Small Datasets
+
+To further evaluate our method in the few-shot setting, we compare against StyleGAN2-ADA and FastGAN on art paintings from WikiArt (1000 images; wikiart.org), Oxford Flowers (1360 images) [42], photographs of landscapes (4319 images; flickr.com), AnimalFace-Dog (389 images) [57] and Pokemon (833 images; pokemon.com). Further, we report results on high-resolution versions of Pokemon and Art-Painting $( 1 0 2 4 ^ { 2 } )$ . Lastly, we evaluate on AFHQ-Cat, -Dog and -Wild at $5 1 2 ^ { 2 }$ [5]. The AFHQ datasets contain ${ \sim } 5 \mathrm { k }$ closeups per category cat, dog, or wildlife. We do not have a license to re-distribute these datasets, but we provide the URLs to enable reproducibility, similar to [35].
+
+Projected GAN outperforms all baselines in terms of FID values by a significant margin on all datasets and all resolutions as shown in Table 3. Remarkably, our model beats the prior state-of-the-art on all datasets $( 2 5 6 ^ { 2 } )$ after observing fewer than $6 0 0 \mathrm { k }$ images. For AnimalFace-Dog, the Projected GAN surpasses the previously best FID after only 20k images. One might argue that the EfficientNet used as feature network facilitates data generation for the animal datasets as EfficientNet is trained on ImageNet which contains many animal classes (e.g., 120 classes for dog breeds). However, it is interesting to observe that Projected GANs also achieve state-of-the-art FID on Pokemon and Art Painting though these datasets differ significantly from ImageNet. This evidences the generality of ImageNet features. For the high-resolution datasets, Projected GANs achieve the same FID value many times faster than the best baselines, e.g., ten times faster than StyleGAN2-ADA on AFHQCat or four times faster than FastGAN on Pokemon. We remark that $F$ and $D _ { l }$ generalize to any resolution as they are fully convolutional.
+
+![](images/c613d56c378d9adb5ccd64519c6837c7f6c69eb4fe98ed6515aeeb136758ad4e.jpg)  
+Figure 6: Real samples (top rows) vs. samples by Projected GAN (bottom rows). Datasets (top left to bottom right): CLEVR $( 2 5 6 ^ { 2 } )$ ), LSUN church $( 2 5 6 ^ { 2 } )$ , Art Painting $( 2 5 6 ^ { 2 } )$ , Landscapes $( 2 5 6 ^ { 2 } )$ , AFHQ-wild $( 5 1 \bar { 2 } ^ { 2 } )$ , Pokemon $( 2 5 6 ^ { 2 } )$ , AFHQ-dog $( 5 1 2 ^ { 2 } )$ , AFHQ-cat $( \bar { 5 1 } 2 ^ { 2 } )$ ).
+
+<table><tr><td></td><td colspan="9">FID Imgs FID Imgs FID</td><td>FID</td><td></td></tr><tr><td></td><td colspan="9">Large Datasets (2562)</td></tr><tr><td></td><td colspan="2">CLEVR</td><td colspan="2">FFHQ</td><td colspan="2">Cityscapes</td><td colspan="2">Bedroom</td><td colspan="2">Church</td></tr><tr><td>SAGAN [69]</td><td>26.04</td><td>10M</td><td>16.21</td><td>10 M</td><td>12.81</td><td>10M</td><td>14.06</td><td>10 M</td><td>6.15</td><td>10M</td></tr><tr><td>STYLEGAN2-ADA [25]</td><td>10.17</td><td>10M</td><td>7.32</td><td>10M</td><td>8.35</td><td>10M</td><td>11.53</td><td>10M</td><td>5.85</td><td>10M</td></tr><tr><td>GANSFORMERS [20]</td><td>9.24</td><td>10M</td><td>7.42</td><td>10M</td><td>5.23</td><td>10M</td><td>6.15</td><td>10M</td><td>5.47</td><td>10M</td></tr><tr><td>FASTGAN [35]</td><td>3.24</td><td>10M</td><td>12.69</td><td>10M</td><td>8.78</td><td>1.8M</td><td>8.24</td><td>4.8M</td><td>8.43</td><td>8.9M</td></tr><tr><td>PROJECTED GAN</td><td>0.89</td><td>4.5M</td><td>3.39</td><td>7.1M</td><td>3.41</td><td>1.7 M</td><td>1.52</td><td>5.2 M</td><td>1.59</td><td>9.2 M</td></tr><tr><td>PROJECTED GAN*</td><td>3.39 5.05</td><td>0.5M 25M</td><td>3.56</td><td>7.0 M</td><td>4.60</td><td>1.1M</td><td>2.58</td><td>1.5 M</td><td>3.18</td><td>1.1 M</td></tr><tr><td>STYLEGAN2* [25,26,68]</td><td></td><td></td><td>3.62</td><td>25M</td><td>1</td><td>-</td><td>2.65</td><td>70M</td><td>3.39</td><td>88M</td></tr><tr><td></td><td colspan="10">Small Datasets (2562)</td></tr><tr><td>STYLEGAN2-ADA [25]</td><td colspan="2">Art Painting</td><td colspan="2">Landscape</td><td colspan="2">AnimalFace</td><td colspan="2">Flowers</td><td colspan="2">Pokemon</td></tr><tr><td>FASTGAN [35]</td><td>43.07</td><td>3.2M</td><td>15.99</td><td>6.3M</td><td>60.90</td><td>2.2M</td><td>21.66</td><td>3.8M</td><td>40.38</td><td>3.4M</td></tr><tr><td>PROJECTED GAN</td><td>44.02</td><td>0.7M</td><td>16.44</td><td>1.8 M</td><td>62.11</td><td>0.2M</td><td>26.23</td><td>0.8M</td><td>81.86</td><td>2.5M</td></tr><tr><td></td><td>27.96</td><td>0.8M</td><td>6.92</td><td>3.5M</td><td>17.88</td><td>10M</td><td>13.86</td><td>1.8 M</td><td>26.36</td><td>0.8M</td></tr><tr><td>PROJECTED GAN*</td><td>40.22</td><td>0.2M</td><td>14.99</td><td>0.6M</td><td>58.07</td><td>0.02 M</td><td>21.60</td><td>0.2M</td><td>36.57</td><td>0.3M</td></tr><tr><td></td><td colspan="4">1024²</td><td colspan="7">5122</td></tr><tr><td>STYLEGAN2-ADA [25]</td><td colspan="2"> Art Painting</td><td colspan="2">Pokemon</td><td colspan="2">AFHQ-Cat</td><td colspan="2">AFHQ-Dog</td><td colspan="2">AFHQ-Wild</td></tr><tr><td></td><td>41.69</td><td>1.0M</td><td>56.76</td><td>0.6M</td><td>3.55</td><td>10M</td><td>7.40</td><td>10M</td><td>3.05</td><td>10M</td></tr><tr><td>FASTGAN [35]</td><td>46.71</td><td>0.8M</td><td>56.46</td><td>0.8M</td><td>4.69</td><td>1.1 M</td><td>13.09</td><td>1.6 M</td><td>3.14</td><td>1.6 M</td></tr><tr><td>PROJECTED GAN</td><td>32.07</td><td>0.9 M</td><td>33.96</td><td>1.3M</td><td>2.16</td><td>3.7M</td><td>4.52</td><td>3.8M</td><td>2.17</td><td>5.4M</td></tr><tr><td>PROJECTED GAN*</td><td>40.33</td><td>0.2M</td><td>53.74</td><td>0.2 M</td><td>3.53</td><td>1.0 M</td><td>7.10</td><td>0.9 M</td><td>3.03</td><td>1.6 M</td></tr></table>
+
+Table 3: Quantitative Results. Projected $\mathrm { G A N ^ { * } }$ reports the point where our approach surpasses the state-of-the-art. StyleGAN2\* obtains the lowest FID in previous literature if trained long enough.
+
+# 6 Discussion and Future Work
+
+While we achieve low FID on all datasets, we also identify two systematic failure cases: As depicted in Fig. 7, we sometimes observe “floating heads” on AFHQ. In a few samples, the animals appear in high quality but resemble cutouts on blurry or bland backgrounds. We hypothesize that generating a realistic background and image composition is less critical when a prominent object is already depicted. This hypothesis follows from the fact that we used image classification models for the projection, which have been shown to only marginally reduce in accuracy when applied on images of objects with removed background [66]. On FFHQ, projected GAN sometimes produces poor-quality samples with wrong proportions and artifacts, even at state-of-the-art FID, see Fig. 8.
+
+![](images/ae0c53cd1b97844fa8180802230f7d2ece451efea963ce834374ad854fe23c5c.jpg)  
+Figure 7: "Floating Heads"
+
+![](images/a0fe6abaf21c6270a3c608c0c9511d9a5e1da6365adf60b259000b133b070663.jpg)  
+Figure 8: Artifacts on FFHQ.
+
+In terms of generators, StyleGAN is more challenging to tune and does not profit as much from projected training. The FastGAN generator is fast to optimize but simultaneously produced unrealistic samples in some parts of the latent space – a problem that could be solved by a mapping network similar to StyleGAN. Hence, we speculate that unifying the strengths of both architectures in combination with projected training might improve performance further. Moreover, our study of different pretrained networks indicates that efficient models are especially suitable for projected GAN training. Exploring this connection in-depth, and in general, determining desirable feature space properties opens up exciting new research opportunities. Lastly, our work advances efficiency for generative models. More efficient models lower the barrier of computational effort needed for generating realistic images. A lower barrier facilitates malignant use of generative models (e.g., “deep fakes”) while simultaneously also democratizing research in this area.
+
+# Acknowledgments and Disclosure of Funding
+
+We acknowledge the financial support by the BMWi in the project KI Delta Learning (project number 19A19013O). Andreas Geiger was supported by the ERC Starting Grant LEGO-3D (850533). Kashyap Chitta was supported by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ: 01IS18039B and the International Max Planck Research School for Intelligent Systems (IMPRS-IS). Jens Müller received funding by the Heidelberg Collaboratory for Image Processing (HCI). We thank the Center for Information Services and High Performance Computing (ZIH) at Dresden University of Technology for generous allocations of computation time. Lastly, we would like to thank Vanessa Sauer for her general support.
+
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parse/train/fUxqIofPPi/fUxqIofPPi_content_list.json

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+    {
+        "type": "text",
+        "text": "Projected GANs Converge Faster ",
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+        "text": "Axel Sauer1,2 Kashyap Chitta1,2 Jens Müller3 Andreas Geiger1,2 1University of Tübingen 2Max Planck Institute for Intelligent Systems, Tübingen 3Computer Vision and Learning Lab, University Heidelberg 2{firstname.lastname}@tue.mpg.de 3{firstname.lastname}@iwr.uni-heidelberg.de ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
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+        "text": "Generative Adversarial Networks (GANs) produce high-quality images but are challenging to train. They need careful regularization, vast amounts of compute, and expensive hyper-parameter sweeps. We make significant headway on these issues by projecting generated and real samples into a fixed, pretrained feature space. Motivated by the finding that the discriminator cannot fully exploit features from deeper layers of the pretrained model, we propose a more effective strategy that mixes features across channels and resolutions. Our Projected GAN improves image quality, sample efficiency, and convergence speed. It is further compatible with resolutions of up to one Megapixel and advances the state-of-the-art Fréchet Inception Distance (FID) on twenty-two benchmark datasets. Importantly, Projected GANs match the previously lowest FIDs up to 40 times faster, cutting the wall-clock time from 5 days to less than 3 hours given the same computational resources. ",
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+        "img_path": "images/c883e6a958f6034a0b565bab650b45781fb0ee5d5d9d6b3449ca0d3672c443e5.jpg",
+        "image_caption": [
+            "Figure 1: Convergence with Projected GANs. Evolution of samples for a fixed latent code during training on the AFHQ-Dog dataset [5]. We find that discriminating features in the projected feature space speeds up convergence and yields lower FIDs. This finding is consistent across many datasets. "
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "A Generative Adversarial Network (GAN) consists of a generator and a discriminator. For image synthesis, the generator’s task is to generate an RGB image; the discriminator aims to distinguish real from fake samples. On closer inspection, the discriminator’s task is two-fold: First, it projects the real and fake samples into a meaningful space, i.e., it learns a representation of the input space. Second, it discriminates based on this representation. Unfortunately, training the discriminator jointly with the generator is a notoriously hard task. While discriminator regularization techniques help to balance the adversarial game [31], standard regularization methods like gradient penalties [36] are susceptible to hyperparameter choices [26] and can lead to a substantial decrease in performance [4]. ",
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+        "type": "text",
+        "text": "In this paper, we explore the utility of pretrained representations to improve and stabilize GAN training. Using pretrained representations has become ubiquitous in computer vision [29, 30, 48] and natural language processing [18, 45, 47]. While combining pretrained perceptual networks [58] with GANs for image-to-image translation has led to impressive results [14, 49, 59, 64], this idea has not yet materialized for unconditional noise-to-image synthesis. Indeed, we confirm that a naïve application of this idea does not lead to state-of-the-art results (Section 4) as strong pretrained features enable the discriminator to dominate the two-player game, resulting in vanishing gradients for the generator [2]. In this work, we demonstrate how these challenges can be overcome and identify two key components for exploiting the full potential of pretrained perceptual feature spaces for GAN training: feature pyramids to enable multi-scale feedback with multiple discriminators and random projections to better utilize deeper layers of the pretrained network. ",
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+        "text": "We conduct extensive experiments on small and large datasets with a resolution of up to $1 0 2 4 ^ { 2 }$ pixels. Across all datasets, we demonstrate state-of-the-art image synthesis results at significantly reduced training time (Fig. 1). We also find that Projected GANs increase data efficiency and avoid the need for additional regularization, rendering expensive hyperparameter sweeps unnecessary. Code, models, and supplementary videos can be found on the project page https://sites.google.com/view/ projected-gan. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "text": "We categorize related work into two main areas: pretraining for GANs and discriminator design. ",
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+        "type": "text",
+        "text": "Pretrained Models for GAN Training. Work on leveraging pretrained representations for GANs can be divided into two categories: First, transferring parts of a GAN to a new dataset [15, 38, 65, 71] and, second, using pretrained models to control and improve GANs. The latter is advantageous as pretraining does not need to be adversarial. Our work falls into this second category. Pretrained models can be used as a guiding mechanism to disentangle causal generative factors [54], for text-driven image manipulation [44], matching the generator activations to inverted classifiers [19, 56], or to generate images via gradient ascent in the latent space of a generator [41]. The non-adversarial approach of [53] learns generative models with moment matching in pretrained models; however, the results remain far from competitive to standard GANs. An established method is the combination of adversarial and perceptual losses [21]. Commonly, the losses are combined additively [10, 14, 32, 52, 64]. Additive combination, however, is only possible if a reconstruction target is available, e.g., in paired image-toimage translation settings [74]. Instead of providing the pretrained network with a reconstruction target, Sungatullina et al. [59] propose to optimize an adversarial loss on frozen VGG features [58]. They show that their approach improves CycleGAN [74] on image translation tasks. In a similar vein, [49] recently proposed a different perceptual discriminator. They utilize a pretrained VGG and connect its features with the prediction of a pretrained segmentation network. The combined features are fed into multiple discriminators at different scales. The two last approaches are specific to the image-toimage translation task. We demonstrate that these methods do not work well for the more challenging unconditional setting where the entire image content is synthesized from a random latent code. ",
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+        "type": "text",
+        "text": "Discriminator Design. Much work on GANs focuses on novel generator architectures [4, 26, 27, 69], while the discriminator often remains close to a vanilla convolutional neural network or mirrors the generator. Notable exceptions are [55,70] which utilize an encoder-decoder discriminator architecture. However, in contrast to us, they neither use pretrained features nor random projections. A different line of work considers a setup with multiple discriminators, applied to either the generated RGB image [8, 13] or low-dimensional projections thereof [1, 40]. The use of several discriminators promises improved sample diversity, training speed, and training stability. However, these approaches are not utilized in current state-of-the-art systems because of diminishing returns compared to the increased computational effort. Providing multi-scale feedback with one or multiple discriminators has been helpful for both image synthesis [23, 24] and image-to-image translation [43, 64]. While these works interpolate the RGB image at different resolutions, our findings indicate the importance of multi-scale feature maps, showing parallels to the success of pyramid networks for object detection [34]. Lastly, to prevent overfitting of the discriminator, differentiable augmentation methods have recently been proposed [25, 63, 72, 73]. We find that adopting these strategies helps exploit the full potential of pretrained representations for GAN training. ",
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+        "type": "text",
+        "text": "3 Projected GANs ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "GANs aim to model the distribution of a given training dataset. A generator $G$ maps latent vectors $\\mathbf { z }$ sampled from a simple distribution $\\mathbb { P } _ { \\mathbf { z } }$ (typically a normal distribution) to corresponding generated samples $G ( \\mathbf { z } )$ . The discriminator $D$ then aims to distinguish real samples $\\mathbf { x } \\sim \\mathbb { P } _ { \\mathbf { x } }$ from the generated samples $G ( \\mathbf { z } ) \\sim \\mathbb { P } _ { G ( \\mathbf { z } ) }$ . This basic idea results in the following minimax objective ",
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+        "type": "equation",
+        "img_path": "images/6aeb5b35114e0ffcb27b889c8f6efda7b1e03c75bc828526b043c80699da6067.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D } \\Big ( \\mathbb { E } _ { \\mathbf { x } } [ \\log D ( \\mathbf { x } ) ] + \\mathbb { E } _ { \\mathbf { z } } [ \\log ( 1 - D ( G ( \\mathbf { z } ) ) ) ] \\Big )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We introduce a set of feature projectors $\\{ P _ { l } \\}$ which map real and generated images to the discriminator’s input space. Projected GAN training can thus be formulated as follows ",
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+    {
+        "type": "equation",
+        "img_path": "images/ca0b8c7f7b5eca23b1fa27490dd4df4336e34dbf7b8ec8321ed5e9c4d23557e3.jpg",
+        "text": "$$\n\\underset { G } { \\mathrm { m i n } } \\underset { \\{ D _ { l } \\} } { \\mathrm { m a x } } \\sum _ { l \\in \\mathcal { L } } \\left( \\mathbb { E } _ { \\mathbf { x } } [ \\log D _ { l } ( P _ { l } ( \\mathbf { x } ) ) ] + \\mathbb { E } _ { \\mathbf { z } } [ \\log ( 1 - D _ { l } ( P _ { l } ( G ( \\mathbf { z } ) ) ) ) ] \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\{ D _ { l } \\}$ is a set of independent discriminators operating on different feature projections. Note that we keep $\\{ P _ { l } \\}$ fixed in (2) and only optimize the parameters of $G$ and $\\{ D _ { l } \\}$ . The feature projectors $\\{ P _ { l } \\}$ should satisfy two necessary conditions: they should be differentiable and provide sufficient statistics of their inputs, i.e., they should preserve important information. Moreover, we aim to find feature projectors $\\{ P _ { l } \\}$ which turn the (difficult to optimize) objective in (1) into an objective more amenable to gradient-based optimization. We now show that a projected GAN indeed matches the distribution in the projected feature space, before specifying the details of our feature projectors. ",
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+        "type": "text",
+        "text": "3.1 Consistency ",
+        "text_level": 1,
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+        "text": "The projected GAN objective in (2) no longer optimizes directly to match the true distribution $\\mathbb { P } _ { T }$ . To understand the training properties under ideal conditions, we consider a more generalized form of the consistency theorem of [40]: ",
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+        "text": "Theorem 1. Let $\\mathbb { P } _ { T }$ denote the density of the true data distribution and $\\mathbb { P } _ { G }$ the density of the distribution the Generator $G$ produces. Let $P _ { l } \\circ T$ and $P _ { l } \\circ G$ be the functional composition of the differentiable and fixed function $P _ { l }$ and the true/generated data distribution, and y be the transformed input to the discriminator. For a fixed $G$ , the optimal discriminators are given by ",
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+        "text": "$$\nD _ { l , G } ^ { * } ( \\mathbf { y } ) = \\frac { \\mathbb { P } _ { P _ { l } \\circ T } ( \\mathbf { y } ) } { \\mathbb { P } _ { P _ { l } \\circ T } ( \\mathbf { y } ) + \\mathbb { P } _ { P _ { l } \\circ G } ( \\mathbf { y } ) }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "for all $l \\in \\mathcal L$ . In this case, the optimal $G$ under (2) is achieved iff $\\mathbb { P } _ { P _ { l } \\circ T } = \\mathbb { P } _ { P _ { l } \\circ G }$ for all $l \\in \\mathcal L$ ",
+        "bbox": [
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+        "type": "text",
+        "text": "A proof of the theorem is provided in the appendix. From the theorem, we conclude that a feature projector $P _ { l }$ with its associated discriminator $D _ { l }$ encourages the generator to match the true distribution along the marginal through $P _ { l }$ . Therefore, at convergence, $G$ matches the generated and true distributions in feature space. The theorem also holds when using stochastic data augmentations [25] before the deterministic projections $P _ { l }$ . ",
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+    {
+        "type": "text",
+        "text": "3.2 Model Overview ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Projecting to and training in pretrained feature spaces opens up a realm of new questions which we address below. This section will provide an overview of the general system and is followed by extensive ablations of each design choice. As our feature projections affect the discriminator, we focus on $P _ { l }$ and $D _ { l }$ in this section and postpone the discussion of generator architectures to Section 5. ",
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+        "type": "text",
+        "text": "Multi-Scale Discriminators. We obtain features from four layers $L _ { l }$ of a pretrained feature network $F$ at resolutions $( L _ { 1 } = 6 4 ^ { 2 } , L _ { 2 } = 3 2 ^ { 2 } , L _ { 3 } = 1 6 ^ { 2 } , L _ { 4 } = 8 ^ { 2 } )$ . We associate a separate discriminator $D _ { l }$ with the features at layer $L _ { l }$ , respectively. Each discriminator $D _ { l }$ uses a simple convolutional architecture with spectral normalization [37] at each convolutional layer. We observe better performance if all discriminators output logits at the same resolution $( 4 ^ { 2 } )$ . Accordingly, we use fewer down-sampling blocks for lower resolution inputs. Following common practice, we sum all logits for computing the overall loss. For the generator pass, we sum the losses of all discriminators. More complex strategies [1, 13] did not improve performance in our experiments. ",
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+        "type": "text",
+        "text": "Random Projections. We observe that features at deeper layers are significantly harder to cover, as evidenced by our experiments in Section 4. We hypothesize that a discriminator can focus on a subset of the feature space while wholly disregarding other parts. This problem might be especially prominent in the deeper, more semantic layers. Therefore, we propose two different strategies to dilute prominent features, encouraging the discriminator to utilize all available information equally. Common to both strategies is that they mix features using differentiable random projections which are fixed, i.e., after random initialization, the parameters of these layers are not trained. ",
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+        "type": "text",
+        "text": "Cross-Channel Mixing (CCM). Empirically, we found two properties to be desirable: (i) the random projection should be information preserving to leverage the full representational power of $F$ , and (ii) it should not be trivially invertible. The easiest way to mix across channels is a $1 \\times 1$ convolution. A $1 \\times 1$ convolution with an equal number of output and input channels is a generalization of a permutation [28] and consequently preserves information about its input. In practice, we find that more output channels lead to better performance as the mapping remains injective and therefore information preserving. Kingma et al. [28] initialize their convolutional layers as a random rotation matrix as a good starting point for optimization. We do not find this to improve GAN performance (see Appendix), arguably since it violates (ii). We therefore randomly initialize the weights of the convolutional layer via Kaiming initialization [16]. Note that we do not add any activation functions. We apply this random projection at each of the four scales and feed the transformed feature to the discriminator as depicted in Fig. 2. ",
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+    {
+        "type": "text",
+        "text": "Cross-Scale Mixing (CSM). To encourage feature mixing across scales, CSM extends CCM with random $3 \\times 3$ convolutions and bilinear upsampling, yielding a U-Net [50] architecture, see Fig. 3. However, our CSM block is simpler than a vanilla U-Net [50]: we only use a single convolutional layer at each scale. As for CCM, we utilize Kaiming initialization for all weights. ",
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+    {
+        "type": "image",
+        "img_path": "images/aaa199bcd852ba250a50b0e4fa19c774e05bf5db00029dfe9e012a6cc4efec6b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Pretrained Feature Networks. We ablate over varying feature networks. First, we investigate different versions of EfficientNets, which allow for direct control over model size versus performance. EfficientNets are image classification models trained on ImageNet [7] and designed to provide favorable accuracy-compute tradeoffs. Second, we use ResNets of varying sizes. To analyze the dependency on ImageNet features (Section 4.3), we also consider R50-CLIP [46], a ResNet optimized with a contrastive language-image objective on a dataset of 400 million (image, text) pairs. Lastly, we utilize a vision transformer architecture (ViTBase) [9] and its efficient follow-up (DeiT-small distilled) [62]. We do not choose an inception network [60] to avoid strong correlations with the evaluation metric FID [17]. In the appendix, we also evaluate several other neural and non-neural metrics to rule out correlations. These additional metrics reflect the rankings obtained by FID. ",
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+        "image_caption": [
+            "Figure 2: CCM (dashed blue arrows) employs $1 \\times 1$ convolutions with random weights. ",
+            "Figure 3: CSM (dashed red arrows) adds random $3 \\times 3$ convolutions and bilinear upsampling, yielding a U-Network. "
+        ],
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+        "type": "text",
+        "text": "In the following, we conduct a systematic ablation study to analyze the importance and best configuration of each component in our Projected GAN model, before comparing it to the state-of-the-art. ",
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+    {
+        "type": "text",
+        "text": "4 Ablation Study ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To determine the best configuration of discriminators, mixing strategy, and pretrained feature network, we conduct experiments on LSUN-Church [67], which is medium-sized (126k images) and reasonably visually complex, using a resolution of $2 5 6 ^ { 2 }$ pixels. For the generator $G$ we use the generator architecture of FastGAN [35], consisting of several upsampling blocks, with additional skip-layerexcitation blocks. Using a hinge loss [33], we train with a batch size of 64 until 1 million real images have been shown to the discriminator, a sufficient amount for $G$ to reach values close to convergence. If not specified otherwise, we use an EfficientNet-Lite1 [61] feature network in this section. We found that discriminator augmentation [25, 63, 72, 73] consistently improves the performance of all methods, and is required to reach state-of-the-art performance. We leverage differentiable data-augmentation [72] which we found to yield the best results in combination with the FastGAN generator. ",
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+    {
+        "type": "table",
+        "img_path": "images/58f8014f5496a2b69ee499d788bfdffd0879232dd58fdda16a76fdeb47fbd9fd.jpg",
+        "table_caption": [],
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+        "table_body": "<table><tr><td>Discriminator(s)</td><td>rel-FD1↓</td><td>rel-FD2↓</td><td>rel-FD3↓</td><td></td><td>rel-FD4↓rel-FID↓</td></tr><tr><td colspan=\"6\">No Projection</td></tr><tr><td>onL1</td><td>0.56</td><td>0.32</td><td>0.31</td><td>0.55</td><td>0.66</td></tr><tr><td>on Li,L2</td><td>0.35</td><td>0.21</td><td>0.23</td><td>0.47</td><td>0.53</td></tr><tr><td>on L1,L2,L3</td><td>0.42</td><td>0.26</td><td>0.28</td><td>0.64</td><td>0.90</td></tr><tr><td>On L1,L2,L3,L4</td><td>0.46</td><td>0.34</td><td>0.38</td><td>0.79</td><td>1.15</td></tr><tr><td>on L2,L3,L4</td><td>0.95</td><td>0.67</td><td>0.71</td><td>1.19</td><td>1.99</td></tr><tr><td>onL3,L4</td><td>2.14</td><td>1.41</td><td>1.18</td><td>1.99</td><td>3.46</td></tr><tr><td>onL4</td><td>10.92</td><td>5.74</td><td>2.56</td><td>2.79</td><td>5.08</td></tr><tr><td>Perceptual D</td><td>2.98</td><td>1.76</td><td>1.20</td><td>1.89</td><td>2.73</td></tr><tr><td colspan=\"6\">CCM</td></tr><tr><td>on L1</td><td>0.27</td><td>0.21</td><td>0.26</td><td>0.50</td><td>0.59</td></tr><tr><td>on L1,L2</td><td>0.27</td><td>0.18</td><td>0.21</td><td>0.41</td><td>0.48</td></tr><tr><td>on L1,L2,L3</td><td>0.31</td><td>0.25</td><td>0.24</td><td>0.54</td><td>0.67</td></tr><tr><td>on L1,L2,L3,L4</td><td>0.53</td><td>0.34</td><td>0.34</td><td>0.59</td><td>0.77</td></tr><tr><td>Perceptual D</td><td>5.33</td><td>3.06</td><td>2.14</td><td>1.09</td><td>4.77</td></tr><tr><td colspan=\"6\">CCM + CSM</td></tr><tr><td>onL1</td><td>0.34</td><td>0.25</td><td>0.19</td><td>0.35</td><td>0.44</td></tr><tr><td>on L1,L2</td><td>0.21</td><td>0.18</td><td>0.16</td><td>0.27</td><td>0.31</td></tr><tr><td>on Li,L2,L3</td><td>0.41</td><td>0.26</td><td>0.17</td><td>0.23</td><td>0.29</td></tr><tr><td>on L1,L2,L3,L4</td><td>0.26</td><td>0.16</td><td>0.13</td><td>0.16</td><td>0.24</td></tr><tr><td>Perceptual D</td><td>2.53</td><td>1.37</td><td>0.89</td><td>0.43</td><td>2.13</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Feature Space Fréchet Distances. We aim to find the best combination of discriminators and random projections to fit the distributions in feature network $F$ . We show the relative FD at different layers of $F$ $( r e l – F D _ { i } )$ between $5 0 \\mathrm { k }$ generated and real images on LSUN-Church. rel- $F D _ { i }$ is normalized using the baseline Fréchet Distances for a model with a standard single RGB image discriminator. Hence, values $> 1$ indicate worse performance than the RGB baseline. We report rel- $. F D$ for four layers of an EfficientNet $( L _ { 1 } , L _ { 2 } , L _ { 3 }$ and $L _ { 4 }$ from shallow to deep), as well as relative Fréchet Inception Distance (FID) [17]. Note that $r e l – F D _ { i }$ should not be compared between different feature spaces, i.e., only within-column comparisons are meaningful. Blue boxes highlight the layers which we supervise via independent discriminators. The green box corresponds to a perceptual discriminator [59], which takes in all feature maps at once. ",
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+    {
+        "type": "text",
+        "text": "4.1 Which feature network layers are most informative? ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We first investigate the relevance of independent multi-scale discriminators. For this experiment, we do not use feature mixing. To measure how well $G$ fits a particular feature space, we employ the Fréchet Distance (FD) [12] on the spatially pooled features denoted as $F D _ { i }$ for layer $i$ . FDs across different feature spaces are not directly comparable. Therefore, we train a GAN baseline with a standard RGB discriminator, record $F \\bar { D _ { i } ^ { R G B } }$ at each layer and quantify the relative improvement via the fraction rel- $F D _ { i } = F D _ { i } / F D _ { i } ^ { R G B }$ . We also investigate a perceptual discriminator [59], where feature maps are fed into different layers of the same discriminator to predict a single logit. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The results in Table 1 (No Projection) show that two discriminators are better than one and improve over the vanilla RGB baseline. Surprisingly, adding discriminators at deep layers hurts performance. We conclude that these more semantic features do not respond well to direct adversarial losses. We also experimented with discriminators at resized versions of the original image, but could not find a setting of hyperparameters and architectures that improves over the single image baseline. Omitting the discriminators on the shallow features decreases performance, which is anticipated, as these layers contain most of the information about the original image. A similar effect has been observed for feature inversion [11] – the deeper the layer, the harder it is to reconstruct its input. Lastly, we observe that independent discriminators outperform the perceptual discriminator by a significant margin. ",
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+    {
+        "type": "table",
+        "img_path": "images/262d3179ec27dff6ff9840b60352189736cabac3c2b780c63656ebff3f792930.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Pretrained Feature Networks Study. We train the projected GAN with different pretrained feature networks. We find that compact EfficientNets outperform both ResNets and Transformers. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"5\">EfficientNet</td><td colspan=\"3\">ResNet</td><td colspan=\"2\">Transformer</td></tr><tr><td>lite0</td><td>lite1</td><td>lite2</td><td>lite3</td><td>lite4</td><td>R18</td><td>R50</td><td>R50-CLIP</td><td>DeiT</td><td>ViT</td></tr><tr><td>Params (M)↓</td><td>2.96</td><td>3.72</td><td>4.36</td><td>6.42</td><td>11.15</td><td>11.18</td><td>23.51</td><td>23.53</td><td>92.36</td><td>317.52</td></tr><tr><td>IN top-1个</td><td>75.48</td><td>76.64</td><td>77.47</td><td>79.82</td><td>81.54</td><td>69.75</td><td>79.04</td><td>N/A</td><td>85.42</td><td>85.16</td></tr><tr><td>FID↓</td><td>2.53</td><td>1.65</td><td>1.69</td><td>1.79</td><td>2.35</td><td>4.16</td><td>4.40</td><td>3.80</td><td>2.46</td><td>12.38</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 How can we best utilize the pretrained features? ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given the insights from the previous section, we aim to improve the utilization of deep features. For this experiment, we only investigate configurations that include discriminators at high resolutions. Table 1 (CCM and $\\mathbf { C C M } + \\mathbf { C S M } )$ presents the results for both mixing strategies. CCM moderately decreases the FDs across all settings, confirming our hypothesis that mixing channels results in better feedback for the generator. When adding CSM, we achieve another notable improvement across all configurations. Especially rel- $F D _ { i }$ at deeper layers are significantly decreased, demonstrating CSM’s usefulness to leverage deep semantic features. Interestingly, we observe that the best performance is now obtained by combining all four discriminators. A perceptual discriminator is again inferior to multiple discriminators. We remark that integrating the original image, via an independent discriminator or CCM or CSM always resulted in worse performance. This failure suggests that naïvely combining non-projected with projected adversarial optimization impairs training dynamics. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Which feature network architecture is most effective? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Using the best setting determined by the experiments above $\\mathbf { \\mathrm { C C M } } + \\mathbf { \\mathrm { C S M } }$ with four discriminators), we study the effectiveness of various perceptual feature network architectures for Projected GAN training. To ensure convergence, also for larger architectures, we train for 10 million images. Table 2 reports the FIDs achieved on LSUN-Church. Surprisingly, we find that there is no correlation with ImageNet accuracy. On the contrary, we observe lower FIDs for smaller models (e.g., EfficientNetslite). This observation indicates that a more compact representation is beneficial while at the same time reducing computational overhead and consequently training time. R50-CLIP slightly outperforms its R50 counterpart, indicating that ImageNet features are not required to achieve low FID. For the sake of completeness, we also train with randomly initialized feature networks, which, however, converge to much higher FID values (see Appendix). In the following, we thus use EfficientNet-Lite1 as our feature network. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 Comparison to State-of-the-Art ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This section conducts a comprehensive analysis demonstrating the advantages of Projected GANs with respect to state-of-the-art models. Our experiments are structured into three sections: evaluation of convergence speed and data efficiency (5.1), and comparisons on large (5.2) and small (5.3) benchmark datasets. We cover a wide variety of datasets in terms of size (hundreds to millions of samples), resolution $2 5 6 ^ { 2 }$ to $1 0 2 4 ^ { 2 }$ ), and visual complexity (clip-art, paintings, and photographs). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evaluation Protocol. We measure image quality using the Fréchet Inception Distance (FID) [17]. Following [26, 27], we report the FID between 50k generated and all real images. We select the snapshot with the best FID for each method. In addition to image quality, we include a metric to evaluate convergence. As in [25], we measure training progress based on the number of real images shown to the discriminator (Imgs). We report the number of images required by the model for the FID to reach values within $5 \\%$ of the best FID over training. In the appendix, we also report other metrics that are less benchmarked in GAN literature: KID [3], SwAV-FID [39], precision and recall [51]. Unless otherwise specified, we follow the evaluation protocol of [20] to facilitate fair comparisons. Specifically, we compare all approaches given the same fixed number of images (10 million). With this setting, each experiment takes roughly 100-200 GPU hours on a NVIDIA V100, for more details we refer to the appendix. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Baselines. We use StyleGAN2-ADA [25] and FastGAN [35] as baselines. StyleGAN2-ADA is the strongest model on most datasets in terms of sample quality, whereas FastGAN excels in training speed. We implement these baselines and our Projected GANs within the codebase provided by the authors of StyleGAN2-ADA [25]. For each model, we ran two kinds of data augmentation: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/b0daf437d42b8753ba9cb84d4e91753512629b3119522e23e9b1f8198557046d.jpg",
+        "image_caption": [
+            "Figure 4: Training Properties. Left: Projected FastGAN surpasses the best FID of StyleGAN2 (at $\\mathbf { 8 8 \\ M }$ images) after just $1 . 1 \\textbf { M }$ images on LSUN-Church. Right: Projected FastGAN yields significantly improved FID scores, even when using subsets of CLEVR with 1k and $1 0 \\mathrm { k }$ samples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/286fbe8b3c1b202601efc7433e4086776d4341d3681a0692ed6230be2e0e1844.jpg",
+        "image_caption": [
+            "Figure 5: Training progress on LSUN church at $2 5 6 ^ { 2 }$ pixels. Shown are samples for a fixed noise vector z over k images. From top to bottom: FastGAN, StyleGAN2-ADA, Projected GAN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "differentiable data-augmentation [72] and adaptive discriminator augmentation [25]. We select the better performing augmentation strategy per model. For all baselines and datasets, we perform data amplification through $\\mathbf { X }$ -flips. Projected GANs use the same generator and discriminator architecture and training hyperparameters (learning rate and batch size) for all experiments. For high-resolution image generation, additional upsampling blocks are included in the generator to match the desired output resolution. We carefully tune all hyper-parameters for both baselines for best results: we find that FastGAN is sensitive to the choice of batch size, and StyleGAN2-ADA to the learning rate and R1 penalty. The appendix documents additional implementation details used in each of our experiments. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Convergence Speed and Data Efficiency ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following [20] and [68], we analyze the training properties of Projected GANs on LSUN-Church at an image resolution of $2 5 6 ^ { 2 }$ pixels and on the 70k CLEVR dataset [22]. In this section, we also train longer than $1 0 \\mathbf { M }$ images if necessary, as we are interested in convergence properties. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Convergence Speed. We apply projected GAN training for both the style-based generator of StyleGAN2 and the standard generator with a single input noise vector of FastGAN. As shown in Fig. 4 (left), FastGAN converges quickly but saturates at a high FID. StyleGAN2 converges more slowly $\\mathbf { \\delta } ^ { \\mathrm { 8 8 \\mathbf { \\delta } M } }$ images) but reaches a lower FID. Projected GAN training improves both generators. Particularly for FastGAN, improvements in both convergence speed and final FID are significant while improvements for StyleGAN2 are less pronounced. Remarkably, Projected FastGAN reaches the previously best FID of StyleGAN2 after experiencing only $1 . 1 \\mathbf { M }$ images as compared to $\\mathbf { 8 8  M }$ of StyleGAN2. In wall clock time, this corresponds to less than 3 hours instead of 5 days. Hence, from now on, we utilize the FastGAN generator and refer to this model simply as Projected GAN. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Fig. 5 shows samples for a fixed noise vector z during training on LSUN-Church. For both FastGAN and StyleGAN, patches of texture gradually morph into a global structure. For Projected GAN, we directly observe the emergence of structure which becomes more detailed over time. Interestingly, the Projected GAN latent space appears to be very volatile, i.e., for fixed $\\mathbf { z }$ the images undergo significant perceptual changes during training. In the non-projected cases, these changes are more gradual. We hypothesize that this induced volatility might be due to the discriminator providing more semantic feedback compared to conventional RGB losses. Such semantic feedback could introduce more stochasticity during training which in turn improves convergence and performance. We also observed that the signed real logits of the discriminator remain at the same level throughout training (see Appendix). Stable signed logits indicate that the discriminator does not suffer from overfitting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Sample Efficiency. The use of pretrained models is generally linked to improved sample efficiency. To evaluate this property, we also created two subsets of the 70k CLEVR dataset by randomly subsampling $1 0 \\mathrm { k }$ and 1k images from it, respectively. As depicted in Fig. 4 (right), our Projected GAN significantly improves over both baselines across all dataset splits. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.2 Large Datasets ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Besides CLEVR and LSUN-Church, we benchmark Projected GANs against various state-of-the-art models on three other large datasets: LSUN-Bedroom [67] (3M indoor bedroom scenes), FFHQ [26] $7 0 \\mathrm { k }$ images of faces) and Cityscapes [6] ( $2 5 \\mathrm { k }$ driving scenes captured from a vehicle). For all datasets, we use an image resolution of $2 5 6 ^ { 2 }$ pixels. As Cityscapes and CLEVR images are not of aspect ratio 1:1 we resize them to $2 5 6 ^ { 2 }$ for training. Besides StyleGAN2-ADA and FastGAN, we compare against SAGAN [69] and GANsformers [20]. All models were trained for $1 0 \\mathbf { M }$ images. For the large datasets, we also report numbers for StyleGAN2 trained for more than $1 0 \\mathbf { M }$ images to report the lowest FID values achieved in previous literature (denoted as StyleGAN2\\*). In the appendix, we report results on nine more large datasets. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 3 shows that the Projected GAN outperforms all state-of-the-art models in terms of FID values on all datasets by a large margin. For example, on LSUN-Bedroom, it achieves an FID value of 1.52 compared to 6.15 by GANsformer, the previously best model in this setting. Projected GAN achieves state-of-the-art FID values remarkably fast, e.g., on LSUN-church, it achieves an FID value of 3.18 after 1.1 M Imgs. StyleGAN2 has obtained the previously lowest FID value of 3.39 after 88 M Imgs, 80 times as many as needed by Projected GAN. Similar speed-ups are also realized for all other large datasets as shown in Table 3. Interestingly, when training longer on FFHQ (39 M Imgs), we observe further improvements of Projected GAN to an FID of 2.2. Note that all five datasets represent very different objects in various scenes. This demonstrates that the performance gain is robust to the choice of the dataset, although the feature network is trained only on ImageNet. It is important to note that the main improvements are based on improved sample diversity as indicated by recall which we report in the appendix. The improvement in diversity is most notable on large datasets, e.g., LSUN church, where the image fidelity appears to be similar to StyleGAN. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.3 Small Datasets ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To further evaluate our method in the few-shot setting, we compare against StyleGAN2-ADA and FastGAN on art paintings from WikiArt (1000 images; wikiart.org), Oxford Flowers (1360 images) [42], photographs of landscapes (4319 images; flickr.com), AnimalFace-Dog (389 images) [57] and Pokemon (833 images; pokemon.com). Further, we report results on high-resolution versions of Pokemon and Art-Painting $( 1 0 2 4 ^ { 2 } )$ . Lastly, we evaluate on AFHQ-Cat, -Dog and -Wild at $5 1 2 ^ { 2 }$ [5]. The AFHQ datasets contain ${ \\sim } 5 \\mathrm { k }$ closeups per category cat, dog, or wildlife. We do not have a license to re-distribute these datasets, but we provide the URLs to enable reproducibility, similar to [35]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Projected GAN outperforms all baselines in terms of FID values by a significant margin on all datasets and all resolutions as shown in Table 3. Remarkably, our model beats the prior state-of-the-art on all datasets $( 2 5 6 ^ { 2 } )$ after observing fewer than $6 0 0 \\mathrm { k }$ images. For AnimalFace-Dog, the Projected GAN surpasses the previously best FID after only 20k images. One might argue that the EfficientNet used as feature network facilitates data generation for the animal datasets as EfficientNet is trained on ImageNet which contains many animal classes (e.g., 120 classes for dog breeds). However, it is interesting to observe that Projected GANs also achieve state-of-the-art FID on Pokemon and Art Painting though these datasets differ significantly from ImageNet. This evidences the generality of ImageNet features. For the high-resolution datasets, Projected GANs achieve the same FID value many times faster than the best baselines, e.g., ten times faster than StyleGAN2-ADA on AFHQCat or four times faster than FastGAN on Pokemon. We remark that $F$ and $D _ { l }$ generalize to any resolution as they are fully convolutional. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c613d56c378d9adb5ccd64519c6837c7f6c69eb4fe98ed6515aeeb136758ad4e.jpg",
+        "image_caption": [
+            "Figure 6: Real samples (top rows) vs. samples by Projected GAN (bottom rows). Datasets (top left to bottom right): CLEVR $( 2 5 6 ^ { 2 } )$ ), LSUN church $( 2 5 6 ^ { 2 } )$ , Art Painting $( 2 5 6 ^ { 2 } )$ , Landscapes $( 2 5 6 ^ { 2 } )$ , AFHQ-wild $( 5 1 \\bar { 2 } ^ { 2 } )$ , Pokemon $( 2 5 6 ^ { 2 } )$ , AFHQ-dog $( 5 1 2 ^ { 2 } )$ , AFHQ-cat $( \\bar { 5 1 } 2 ^ { 2 } )$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/0783840e719a4067baf0886192f65b93cbc8c6b8ce9cb284f4f4aa9ca3f1b6b4.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Quantitative Results. Projected $\\mathrm { G A N ^ { * } }$ reports the point where our approach surpasses the state-of-the-art. StyleGAN2\\* obtains the lowest FID in previous literature if trained long enough. "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"9\">FID Imgs FID Imgs FID</td><td>FID</td><td></td></tr><tr><td></td><td colspan=\"9\">Large Datasets (2562)</td></tr><tr><td></td><td colspan=\"2\">CLEVR</td><td colspan=\"2\">FFHQ</td><td colspan=\"2\">Cityscapes</td><td colspan=\"2\">Bedroom</td><td colspan=\"2\">Church</td></tr><tr><td>SAGAN [69]</td><td>26.04</td><td>10M</td><td>16.21</td><td>10 M</td><td>12.81</td><td>10M</td><td>14.06</td><td>10 M</td><td>6.15</td><td>10M</td></tr><tr><td>STYLEGAN2-ADA [25]</td><td>10.17</td><td>10M</td><td>7.32</td><td>10M</td><td>8.35</td><td>10M</td><td>11.53</td><td>10M</td><td>5.85</td><td>10M</td></tr><tr><td>GANSFORMERS [20]</td><td>9.24</td><td>10M</td><td>7.42</td><td>10M</td><td>5.23</td><td>10M</td><td>6.15</td><td>10M</td><td>5.47</td><td>10M</td></tr><tr><td>FASTGAN [35]</td><td>3.24</td><td>10M</td><td>12.69</td><td>10M</td><td>8.78</td><td>1.8M</td><td>8.24</td><td>4.8M</td><td>8.43</td><td>8.9M</td></tr><tr><td>PROJECTED GAN</td><td>0.89</td><td>4.5M</td><td>3.39</td><td>7.1M</td><td>3.41</td><td>1.7 M</td><td>1.52</td><td>5.2 M</td><td>1.59</td><td>9.2 M</td></tr><tr><td>PROJECTED GAN*</td><td>3.39 5.05</td><td>0.5M 25M</td><td>3.56</td><td>7.0 M</td><td>4.60</td><td>1.1M</td><td>2.58</td><td>1.5 M</td><td>3.18</td><td>1.1 M</td></tr><tr><td>STYLEGAN2* [25,26,68]</td><td></td><td></td><td>3.62</td><td>25M</td><td>1</td><td>-</td><td>2.65</td><td>70M</td><td>3.39</td><td>88M</td></tr><tr><td></td><td colspan=\"10\">Small Datasets (2562)</td></tr><tr><td>STYLEGAN2-ADA [25]</td><td colspan=\"2\">Art Painting</td><td colspan=\"2\">Landscape</td><td colspan=\"2\">AnimalFace</td><td colspan=\"2\">Flowers</td><td colspan=\"2\">Pokemon</td></tr><tr><td>FASTGAN [35]</td><td>43.07</td><td>3.2M</td><td>15.99</td><td>6.3M</td><td>60.90</td><td>2.2M</td><td>21.66</td><td>3.8M</td><td>40.38</td><td>3.4M</td></tr><tr><td>PROJECTED GAN</td><td>44.02</td><td>0.7M</td><td>16.44</td><td>1.8 M</td><td>62.11</td><td>0.2M</td><td>26.23</td><td>0.8M</td><td>81.86</td><td>2.5M</td></tr><tr><td></td><td>27.96</td><td>0.8M</td><td>6.92</td><td>3.5M</td><td>17.88</td><td>10M</td><td>13.86</td><td>1.8 M</td><td>26.36</td><td>0.8M</td></tr><tr><td>PROJECTED GAN*</td><td>40.22</td><td>0.2M</td><td>14.99</td><td>0.6M</td><td>58.07</td><td>0.02 M</td><td>21.60</td><td>0.2M</td><td>36.57</td><td>0.3M</td></tr><tr><td></td><td colspan=\"4\">1024²</td><td colspan=\"7\">5122</td></tr><tr><td>STYLEGAN2-ADA [25]</td><td colspan=\"2\"> Art Painting</td><td colspan=\"2\">Pokemon</td><td colspan=\"2\">AFHQ-Cat</td><td colspan=\"2\">AFHQ-Dog</td><td colspan=\"2\">AFHQ-Wild</td></tr><tr><td></td><td>41.69</td><td>1.0M</td><td>56.76</td><td>0.6M</td><td>3.55</td><td>10M</td><td>7.40</td><td>10M</td><td>3.05</td><td>10M</td></tr><tr><td>FASTGAN [35]</td><td>46.71</td><td>0.8M</td><td>56.46</td><td>0.8M</td><td>4.69</td><td>1.1 M</td><td>13.09</td><td>1.6 M</td><td>3.14</td><td>1.6 M</td></tr><tr><td>PROJECTED GAN</td><td>32.07</td><td>0.9 M</td><td>33.96</td><td>1.3M</td><td>2.16</td><td>3.7M</td><td>4.52</td><td>3.8M</td><td>2.17</td><td>5.4M</td></tr><tr><td>PROJECTED GAN*</td><td>40.33</td><td>0.2M</td><td>53.74</td><td>0.2 M</td><td>3.53</td><td>1.0 M</td><td>7.10</td><td>0.9 M</td><td>3.03</td><td>1.6 M</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "6 Discussion and Future Work ",
+        "text_level": 1,
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+        "text": "While we achieve low FID on all datasets, we also identify two systematic failure cases: As depicted in Fig. 7, we sometimes observe “floating heads” on AFHQ. In a few samples, the animals appear in high quality but resemble cutouts on blurry or bland backgrounds. We hypothesize that generating a realistic background and image composition is less critical when a prominent object is already depicted. This hypothesis follows from the fact that we used image classification models for the projection, which have been shown to only marginally reduce in accuracy when applied on images of objects with removed background [66]. On FFHQ, projected GAN sometimes produces poor-quality samples with wrong proportions and artifacts, even at state-of-the-art FID, see Fig. 8. ",
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+        "img_path": "images/ae0c53cd1b97844fa8180802230f7d2ece451efea963ce834374ad854fe23c5c.jpg",
+        "image_caption": [
+            "Figure 7: \"Floating Heads\" "
+        ],
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+        "img_path": "images/a0fe6abaf21c6270a3c608c0c9511d9a5e1da6365adf60b259000b133b070663.jpg",
+        "image_caption": [
+            "Figure 8: Artifacts on FFHQ. "
+        ],
+        "image_footnote": [],
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+    {
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+        "text": "In terms of generators, StyleGAN is more challenging to tune and does not profit as much from projected training. The FastGAN generator is fast to optimize but simultaneously produced unrealistic samples in some parts of the latent space – a problem that could be solved by a mapping network similar to StyleGAN. Hence, we speculate that unifying the strengths of both architectures in combination with projected training might improve performance further. Moreover, our study of different pretrained networks indicates that efficient models are especially suitable for projected GAN training. Exploring this connection in-depth, and in general, determining desirable feature space properties opens up exciting new research opportunities. Lastly, our work advances efficiency for generative models. More efficient models lower the barrier of computational effort needed for generating realistic images. A lower barrier facilitates malignant use of generative models (e.g., “deep fakes”) while simultaneously also democratizing research in this area. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We acknowledge the financial support by the BMWi in the project KI Delta Learning (project number 19A19013O). Andreas Geiger was supported by the ERC Starting Grant LEGO-3D (850533). Kashyap Chitta was supported by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ: 01IS18039B and the International Max Planck Research School for Intelligent Systems (IMPRS-IS). Jens Müller received funding by the Heidelberg Collaboratory for Image Processing (HCI). We thank the Center for Information Services and High Performance Computing (ZIH) at Dresden University of Technology for generous allocations of computation time. Lastly, we would like to thank Vanessa Sauer for her general support. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+            174,
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parse/train/r1glDpNYwS/r1glDpNYwS.md

@@ -0,0 +1,260 @@
+# LABELFOOL: A TRICK IN THE LABEL SPACE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+It is widely known that well-designed perturbations can cause state-of-the-art machine learning classifiers to mis-label an image, with sufficiently small perturbations that are imperceptible to the human eyes. However, by detecting the inconsistency between the image and wrong label, the human observer would be alerted of the attack. In this paper, we aim to design attacks that not only make classifiers generate wrong labels, but also make the wrong labels imperceptible to human observers. To achieve this, we propose an algorithm called LabelFool which identifies a target label similar to the ground truth label and finds a perturbation of the image for this target label. We first find the target label for an input image by a probability model, then move the input in the feature space towards the target label. Subjective studies on ImageNet show that in the label space, our attack is much less recognizable by human observers, while objective experimental results on ImageNet show that we maintain similar performance in the image space as well as attack rates to state-of-the-art attack algorithms.
+
+# 1 INTRODUCTION
+
+Deep neural networks are powerful learning models that achieve state-of-the-art pattern recognition performance in classification tasks (Krizhevsky et al., 2012b; LeCun et al., 2010; He et al., 2016). Nevertheless, it is found that adding well-designed perturbations to original samples can make classifiers of deep neural networks fail (Szegedy et al., 2013). These kinds of samples are called adversarial samples. Techniques for generating adversarial samples are called attackers.
+
+We think the ideal attacker should satisfy three levels of requirements. The first requirement is fooling networks which means making classifiers fail to classify an image successfully. For example, a dog image can be classified as a cat after added some well-designed perturbations. There are a number of methods for achieving a high attack rate (Goodfellow et al., 2015; Carlini & Wagner, 2017; Dong et al., 2018).
+
+![](images/d27b18d28b4fd6123cce7fb3d178e723891357a0599bc37a964b48ff63dab500.jpg)  
+Figure 1: This graph illustrates the importance of the imperceptibility of adversarial samples in both image space and label space. Triangles are three attackers. Circles represent human observers.
+
+The second requirement for the ideal attacker is the imperceptibility in the image space. This means the magnitude of perturbations in the pixel level needs to be as tiny as possible so that it is imperceptible to human eyes. For example, additive perturbations are minimized with $l _ { p }$ norm to generate imperceptible adversarial samples (Seyed-Mohsen et al., 2016). Extreme cases also exist where only changing one or a few pixels (Su et al., 2019; Modas et al., 2019) can make classifiers fail. Moosavi-Dezfooli et al. (2017) even show the existence of universal (image-agnostic) perturbations.
+
+The third requirement for the ideal attacker, which is newly proposed in this paper, is the imperceptibility of the error made by the classifier in the label space. It means making the classifier to mis-classify an image as the label which is similar to its ground truth, so that people won’t notice the misclassification. For example, in Figure 1, a human user will probably ignore the mis-classification if an attacker caused a “church” to be mis-classified as a “monastery” as the third attacker does. However, a human user will easily notice the mistake if an attacker caused a “church” to be misclassified as a “dome” as the second attacker does or caused an apparent perturbation in the image space as the first attacker does. In real applications, a human user will take defensive measures as soon as he notices the attack. Therefore making the whole attack process imperceptible is crucial for letting observers’ guard down. Tiny perturbations in the image space but large perturbations in the label space can muddle through on the input terminal. But as soon as observers check on the output terminal and see the obviously-incorrect label for an input, they will realize that the classifier fail due to some attacks and take defensive measures immediately, just as Figure 1 shows. This justifies the power of attacks which also confuse people in the label space. So the imperceptibility in the label space is quite important. However, to our best knowledge, few attackers have realized this point.
+
+In this paper, we propose an untargeted-attack algorithm called LabelFool, to perturb an image to be mis-classified as the label which is similar to its ground truth, so that people won’t notice the misclassification. In the meantime, LabelFool also guarantees the imperceptibility in the image space as well as maintaining a high attack rate in fooling classifiers. There are two steps by which we accomplish our goal. The first step is to choose a target label which is similar to the input image’s ground truth. The second step is to perturb the input to be classified as this target label. The way is finding the classification boundary between the current label and the target label, and then moving the input towards this boundary until it is classified as the target label. We conduct a subjective experiment on ImageNet (Deng et al., 2009) which shows that adversarial samples generated by our method are indeed much less recognizable in the label space by human observers than other attacks. We also perform objective experiments on ImageNet to demonstrate that adversarial samples generated by LabelFool still guarantee the imperceptibility in the image space as well as maintaining a high attack rate in fooling classifiers.
+
+# 2 RELATED WORK
+
+The phenomenon that neural networks are sensitive to adversarial samples was proposed by Szegedy et al. (2013). Since then, many researchers have studied how to generat adversarial samples. FGSM (Goodfellow et al., 2015) was proposed to maximize the classification error subject to $l _ { \infty }$ -norm based distortion constraints. CW attack (Carlini & Wagner, 2017) generates adversarial samples by solving an optimization problem based on $l _ { 0 } / l _ { 2 } / l _ { \infty }$ constraint, and $l _ { 0 }$ CW attack is the first proposed method that can cause targeted misclassification on the ImageNet dataset, meaning that we can specify the label of adversarial samples. But this designation of the target label is arbitrary. Until this paper, there has no guide about how to choose a target label such that it is difficult for a person to notice that the network has failed.
+
+Besides achieving the goal of misclassification, many researchers realize the importance of imperceptibility in the image space (Xu et al., 2019). One-pixel attack (Su et al., 2019) and SparseFool (Modas et al., 2019) attack networks in a scenario where perturbing only one/a few pixels can make a big difference. Moosavi-Dezfooli et al. (2017) show the existence of universal image-agnostic perturbations for state-of-the-art deep neural networks. DeepFool (Seyed-Mohsen et al., 2016) seeks the minimum image-level distortion. And for generating adversarial samples, it directly moves the input sample to the nearest class in the feature space. This is the most closely related work to ours, because features extracted from classification models can reflect images’ perceptual information and the classes which are close in the feature space are often perceptually similar. However, DeepFool approximates the multi-dimensional classification boundaries in two dimensions and this might make big errors on finding the nearest class. All these attacks generate adversarial samples by iteration and the algorithm stops as soon as an adversarial sample is born no matter what label it belongs to. This will lead to an apparent misclassification so that observers will sound the defensive alarm quickly.
+
+In this paper, we will compare our method with three attacks: FGSM, DeepFool and SparseFool to show the advantage of our method in the imperceptibility in the label space. We will also demonstrate that the performance gain in the label space is not at the expense of the loss in the image space or attack rate.
+
+# 3 LABELFOOL
+
+In this section, we will introduce our method about how to choose a target label which is undetectable by human observers and how we can perturb the input image so that the classifier assigns this specific label. The whole pipeline is shown in Figure 2. All the symbols and notations used in this paper are summarized in Table 1. We use the same notation $i ( i = 1 , 2 , \dots )$ for “class” and “label”, because “class” and “label” are interchangeable in this paper. LabelFool contains two steps. The first step is to choose a target label for the input image which is similar to its ground truth. The second step is to perturb the input image to be classified as this label. Inspired by DeepFool (Seyed-Mohsen et al., 2016), we make modifications at the feature level. We keep moving the input towards this chosen class at the feature level until it is classified as the label we want.
+
+# 3.1 CHOOSE A TARGET LABEL
+
+The first step of our method is choosing the target label $t _ { x }$ for an input image $x$ . As we want the target label to be imperceptible in the label space to human observers, we need to find the most “similar” label to the input image’s ground truth $l _ { x }$ where the most “similar” means the nearest in the perceptual distance metric. However, $l _ { x }$ is usually unknown when an input image is given. So it is important to estimate the probability distribution $P$ of an input’s ground truth $l _ { x }$ , based on which, we can compute the distance between each class in the dataset and $l _ { x }$ , then choose the nearest one as the target class. We propose a weighted distance model to achieve this goal. Before introducing the model, there are some preparations.
+
+Given two image $x , y$ , we choose pre-trained image classification models to extract features $\phi _ { x } , \phi _ { y }$ because these features can reflect some perceptual information. As we want to calculate the distance in the perceptual distance metric and cosine distance has been used to measure perceptual similarity in many works (Lin et al., 2016; Wang et al., 2019), we compute the distance between $x$ and $y$ as $d ( x , y ) = 1 - \cos { ( \phi _ { x } , \phi _ { y } ) }$ . After having the distance between two images, we can compute the distance between classes. Each class is a set of images. To measure the distance between two sets, we choose Hausdorff distance (Henrikson, 1999). The distance between class $i$ and class $j$ is denoted as $D _ { i , j }$ . Suppose a dataset has $n$ classes. Then, we can construct a matrix $\pmb { { \cal D } } \in \mathbb { R } ^ { n \times n }$ by calculating the distance between all pairs of classes in the dataset, and it will be used in the following probability model to provide the distance we need. After these preparations, we can start to decide the target label for an input image.
+
+As introduced before, we need to estimate the probability distribution $P$ of the ground truth $l _ { x }$ because we want to find the nearest label to $l _ { x }$ which is unknown in the beginning. When an image $x$ is put into a classifier $f$ , state-of-the-art machine learning classifiers usually output a predicted label $\hat { l } _ { x }$ and a probability vector $\hat { p }$ whose elements mean $P ( x \in c l a s s \ i ) = \hat { p } _ { i }$ . For simplicity, we suppose the elements in $\hat { p }$ are sorted in the descending order. Meanwhile, $\hat { p }$ can be thought as $P$ ’s approximation. Furthermore, we define a distance function between $l _ { x }$ and the class $i$ in a $\mathbf { n }$ -classes dataset as $D _ { i } ( l _ { x } )$ . In order to choose the nearest label to $l _ { x }$ as the target label $t _ { x }$ , we need to estimate the expectation of $D _ { i } ( l _ { x } )$ which is denoted as $\mathbb { E } _ { l _ { x } \sim P } [ D _ { i } ( l _ { x } ) ] ( i = \bar { 1 } , \dots , n )$ . In general, our target function is Eq. (1).
+
+$$
+t _ { x } = \underset { i = 1 , \ldots , n } { \arg \operatorname* { m i n } } \mathbb { E } _ { l _ { x } \sim P } [ D _ { i } ( l _ { x } ) ]
+$$
+
+Specifically, when $\hat { p } _ { 1 }$ is larger than some threshold $\delta _ { 1 }$ , we use Maximum Likelihood Estimation (MLE) (Pfanzagl, 2011) which means we believe the classifier and take the predicted label $\hat { l } _ { x }$ as
+
+# Meaning
+
+$x$ Input image   
+$\phi _ { x }$ The feature of image $x$   
+$_ { D }$ The perceptual distance matrix where $D _ { i , j }$ represents the distance between class $i$ and class $j$ $f$ Classifier   
+$\hat { p }$ Probability vector (elements are in descending order)   
+$l _ { x }$ The ground truth of the input image $x$   
+$\hat { l } _ { x }$ The predicted class of the input image $x$ by the classifier   
+$D _ { i } ( l _ { x } )$ A function calculating the distance between class $i$ and the ground truth $l _ { x }$   
+$t _ { x }$ Target label for input $x$   
+$\delta _ { 1 } , \delta _ { 2 }$ Two thresholds, in this paper, $\delta _ { 1 } = 0 . 8 , \delta _ { 2 } = 0 . 0 1$   
+$M$ The number of elements larger than $\delta _ { 2 }$ in $\hat { p } _ { : }$ , i.e. $M = \mathrm { m a x } _ { j = 1 , \dots , n } \{ j : { \hat { p } } _ { j } > \delta _ { 2 } \}$   
+${ \mathcal { F } } _ { j }$ The classification boundary between class $\hat { l } _ { x }$ and class $j$ of an image
+
+the ground truth $l _ { x }$ , then choose the label (except $\hat { l } _ { x }$ ) nearest to $\hat { l } _ { x }$ as the target label $t _ { x }$ . So in this circumstance, we assume $l _ { x } = \hat { l } _ { x }$ and $\mathbb { E } _ { l _ { x } \sim P } [ D _ { i } ( l _ { x } ) ] = D _ { i , \hat { l } _ { x } }$ . Therefore, $t _ { x }$ is
+
+$$
+t _ { x } = \operatorname * { a r g m i n } _ { i \neq \hat { l } _ { x } , i = 1 , \ldots , n } D _ { i , \hat { l } _ { x } } \quad \mathrm { i f } \hat { p } _ { 1 } > \delta _ { 1 } .
+$$
+
+When $\hat { p } _ { 1 }$ is smaller than the threshold $\delta _ { 1 }$ , we are not sure whether $l _ { x }$ is equal to $\hat { l } _ { x }$ . Instead, we sample some labels and compute the weighted distance between each label $i$ and these labels. We sample all labels whose probability are larger than a threshold $\delta _ { 2 }$ and we use $M$ to represent the number of sampled labels. We think the input image might belong to one of these $M$ labels. The labels whose probability are smaller than $\delta _ { 2 }$ will be abandoned because we think the input image can hardly fall into these categories. The weight and the distance is provided by the vector $\hat { p }$ and matrix $_ D$ respectively. So in this circumstance, as we are not sure which label is the ground truth, we want to find a target label which has the minimum expected distance with all these possible labels. Therefore, the value of $\mathbb { E } _ { l _ { x } \sim P } [ D _ { i } ( l _ { x } ) ]$ can be approximated as X j=1 pˆj · Di,j and the target label $t _ { x }$ is shown in Eq. (3). This can be explained by Importance Sampling (Owen & Zhou, 2000) because it is hard to sample from the real probability distribution $P$ . We can only use the probability distribution $\hat { p }$ which is an approximation of $P$ to estimate the value of $\mathbb { E } _ { l _ { x } \sim P } [ D _ { i } ( l _ { x } ) ]$ .
+
+$$
+t _ { x } = \underset { i = 1 , \ldots , n } { \arg \operatorname* { m i n } } \sum _ { j = 1 } ^ { M } \hat { p } _ { j } \cdot D _ { i , j } \quad \mathrm { i f } \hat { p } _ { 1 } \leq \delta _ { 1 }
+$$
+
+In conclusion, the whole strategy for choosing the target label $t _ { x }$ of an input image $x$ is computed as Eq. (4). The target label $t _ { x }$ minimizes $\mathbb { E } _ { l _ { x } \sim P } [ D _ { i } ( l _ { x } ) ]$ just as Figure 2 shows.
+
+$$
+t _ { x } = \left\{ \begin{array} { l l } { \displaystyle \arg \operatorname* { m i n } _ { i , \hat { l } _ { x } } D _ { i , \hat { l } _ { x } } } & { \mathrm { i f } \hat { p } _ { 1 } > \delta _ { 1 } } \\ { \displaystyle i \neq \hat { l } _ { x } , i = 1 , . . . , n } \\ { \displaystyle \arg \operatorname* { m i n } _ { i = 1 , . . . , n } \sum _ { j = 1 } ^ { M } \hat { p } _ { j } \cdot D _ { i , j } } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+# 3.2 GENERATE ADVERSARIAL SAMPLES
+
+After having the target label, the second step is to attack the input image to be mis-classified as this target label. It’s easy to achieve by taking the target label as a parameter and putting it into the targeted-attack algorithm such as targeted-FGSM (Goodfellow et al., 2015) and targeted-CW (Carlini & Wagner, 2017), but this operation may suffer huge loss in the image space because of large perturbations.
+
+![](images/130ca587a82453324ed51da5a452a8791633884559a83a5803a3898b7eeaaf9b.jpg)  
+Figure 2: Pipeline of our method. There are two steps. In the first step, we first compute the distance $D _ { i , j }$ between every two class $i , j$ in a n-classes dataset. Then we choose the target label $t _ { x }$ for an input image $x$ by two strategies according to the value of $\hat { p } _ { 1 }$ . The second step is to attack the input into this target label. Solid lines are the real boundaries between the current label and the indicated label and dashed lines with notes $\mathcal { F }$ are the approximate two-dimensional boundaries. Red indicates the target label while blue indicates other labels. Our method moves the input towards the boundary $\mathcal { F } _ { t _ { x } }$ until it is classified as $t _ { x }$ .
+
+Inspired by DeepFool (Seyed-Mohsen et al., 2016), we propose a method which can not only attack an input image to be mis-classified as the target label successfully, but also ensure tiny perturbations in the image space. The mathematical derivation in this step is similar to DeepFool (Seyed-Mohsen et al., 2016) and the only difference is that, we have a target label chosen in the first step while DeepFool doesn’t. As introduced in DeepFool (Seyed-Mohsen et al., 2016), a high dimensional classification boundary can be approximated by a line in two dimensions. As shown in Figure 2, for an image $x _ { 0 }$ , ${ \mathcal { F } } _ { j }$ represents the 2D approximated boundary between its current predicted class and class $j$ and $t _ { x }$ is the target class we choose in the first step. In the first iteration, we move $x _ { 0 }$ towards $\mathcal { F } _ { t _ { x } }$ and get a new point $x _ { 1 }$ . The direction of movement is perpendicular to $\mathcal { F } _ { t _ { x } }$ . The distance of the movement is the vertical distance from $x _ { 0 }$ to $\mathcal { F } _ { t _ { x } }$ . If the predicted label $\hat { l } _ { x _ { 1 } }$ of the new point equals $\hat { l } _ { x _ { 0 } }$ , the classification boundaries are the same as those before moving $x _ { 0 }$ . Otherwise, the classification boundaries change. No matter whether the boundaries change or not, we repeatedly move the current point towards $\mathcal { F } _ { t _ { x } }$ until it is classified as label $t _ { x }$ or the maximum number of iterations has been reached. A pseudocode of the second step is shown in Algorithm 1.
+
+# Algorithm 1 : Generate Adversarial Samples
+
+Input: image $x$ , classifier $f$ , target label $t _ { x }$ Output: Adversarial image $\hat { x }$ 1: initialize $x _ { 0 } \gets x , i \gets 0$ 2: while $\hat { l } _ { x _ { i } } \neq t _ { x }$ and i < max iter do 3: $w \gets \nabla f _ { \hat { l } _ { x _ { i } } } ( x _ { i } ) - \nabla f _ { t _ { x } } ( x _ { i } )$ 4: $g \gets f _ { \hat { l } _ { x _ { i } } } ( \bar { x } _ { i } ) - f _ { t _ { x } } ( x _ { i } )$
+
+5: ri ← kw k22 |g| w   
+6: xi+1 ← xi + ri, i ← i + 1   
+7: end while   
+8: return $\hat { x } = x _ { i + 1 }$
+
+# 4 EXPERIMENTS
+
+In this paper, all experiments are conducted on ImageNet. ImageNet provides the CLS-LOC dataset for classification tasks. Its train split contains about 1300 thousand images. There are 50 thousand validation images and 100 thousand test images. Our experiments are conducted on the train split of CLS-LOC dataset which will be noted as ImageNet-train split in the following part. We perform extensive experiments to show LabelFool can satisfy all three levels of requirements as an attacker. First we demonstrate the deceptiveness of samples generated by LabelFool to humans in the label space through a subjective experiment. Then we calculate the perceptibility and image quality of adversarial samples to show there is not much loss in the image space even compared to DeepFool (Seyed-Mohsen et al., 2016), which is the state-of-the-art method in the image space. Finally, we conduct attacks on several models to prove the exceptional ability of our method on fooling neural networks which is the first requirement for an ideal attack.
+
+# 4.1 IMPERCEPTIBILITY IN THE LABEL SPACE
+
+Setup. In this part, we will compare LabelFool with three attack methods: DeepFool, FGSM and SparseFool. We first sample 600 source images from ImageNet-train split randomly, and these images are in different classes. Each source image will derive four adversarial images and a baseline image, namely DeepFool-attacked image, LabelFool-attacked image, FGSM-attacked image, SparseFool-attacked image and clean image. Each adversarial image has its mis-classified label and each baseline image has the truth label. We then use term “puzzles” to describe the combination of an image and its label, for $5 \times 6 0 0 = 3 0 0 0$ puzzles. A human observer needs to determine whether the label is correct for the image, answering “True” or “False” for each puzzle. To eliminate observers’ memory effects, we split 3000 puzzles into five groups, ensuring that 600 images in one group come from different source images. An interface presentation of our subjective experiment is shown in Appendix A. We have 10 observers (3 females and 7 males, age between 20-29) to do our subjective experiment.
+
+Evaluation. In this paper, we define an index named Performance Gain $( P G )$ as the evaluation index. Human observers answer “True” or “False” for each puzzle, and the rate with which they answer incorrectly is called Confusion Rate $( C R )$ . So every observer has a $C R$ for each attack method or baseline. It is an absolute indicator demonstrating how much observers are confused by a set of puzzles. But doing arithmetic on $C R$ of different observers is meaningless, as different observers have different baseline results. So we define a relative indicator called Performance Gain $( P G )$ , which demonstrates how much improvement in the confusion rate after attacking comparing with baseline. It is a kind of normalization. The formula for PG is shown in Eq. (5), where $C R _ { A }$ means the confusion rate of an attacker and $C R _ { B }$ means the confusion rate of baseline. The higher $P G$ an attacker has, the better it confuses people in the label space.
+
+$$
+P G _ { A } = \frac { C R _ { A } - C R _ { B } } { C R _ { B } } \ ( A i s \ a n \ a t t a c k e r , B = B a s e l i n e )
+$$
+
+Results. We report the average performance gain of 10 observers in total images in the left of Figure 3. As a whole, there is a huge improvement compared with FGSM and SparseFool, about 25 percent improvement and 30 percent improvement respectively. Compared with DeepFool, the gap in $P G$ is a little smaller because DeepFool finds the nearest class in the feature level and features usually reflect images’ perceptual information as we introduced in Section 2. But there is still 3 percent improvement in performance gain.
+
+As animal classes are more fine-grained, the effects of the imperceptibility in the label space become more pronounced. We attack an animal image so that its true animal label changed into a similar animal label, it is difficult for humans to notice that our attack is taking place. Meanwhile, other attacks change the label into an obviously-incorrect label such as a non-animal category or another species (Some examples in Appendix B). In our subjective experiment, there are 247 animal images out of 600 source images. The right graph in Figure 3 shows the average performance gain of 10 observers in animal images. The accurate data for both graphs in Figure 3 is shown in Appendix C.
+
+As for animal images, the improvement is very obvious comparing with all three attack methods. There are about nearly 90 percent improvement in performance gain comparing with FGSM and
+
+![](images/5b15f8f4e910472b4d11cf21cf87b2c22d63c70a811de85807b897e853634270.jpg)  
+Figure 3: A line chart for average performance gain of 10 observers. The horizontal axis represents four attack methods. The vertical axis represents the mean value of 10 human observers’ performance gain. The graph in the left is for total results, and the right one is for animal images.
+
+SparseFool. A significant improvement can also be seen when comparing with DeepFool, there are about 50 percent improvement in average performance gain.
+
+# 4.2 IMPERCEPTIBILITY IN THE IMAGE SPACE
+
+In this subsection, we will show our performance in the image space to demonstrate that our improvement in the label space is not at the cost of huge loss in the image space. We use three metrics to evaluate performance in the image space. One is perceptibility which is similar to the definition in previous works (Szegedy et al., 2013; Seyed-Mohsen et al., 2016) : $p =$ $\frac { 1 } { W _ { N } \times H _ { N } } \sum _ { w = 1 } ^ { W _ { N } } \sum _ { h = 1 } ^ { \bar { H } _ { N } } { \lVert { \Delta y _ { w , h } } \rVert ^ { 2 } }$ , where $y _ { w , h }$ is a 3-dimensional vector representing the RGB intensities (normalized in [0, 1]) of a pixel. The other two are perceptual similarity (Zhang et al., 2018) and PieAPP (Prashnani et al., 2018). These two are metrics for image quality. Perceptual similarity measures the perceptual distance between an image and its reference image while PieAPP measures the perceptual error. In this paper, the reference image is the clean image. And the smaller these three metrics are, the better the adversarial samples are.
+
+![](images/ed9e42db6e82a9808f050163a4330bc20b03303d3bf3ddc677dfb578e01c720e.jpg)  
+Figure 4: Mean value of perceptibility, perceptual similarity and PieAPP for adversarial samples generated by different attack methods on different models.
+
+We randomly choose 1000 images from ImageNet and attack the classifier to generate 1000 adversarial samples. Then we compute mean value for these adversarial samples of three metrics. In this experiment, we test four classifiers: ResNet-34, ResNet-50, VGG-19 (with batch normalization) (Simonyan & Zisserman, 2014) and AlexNet (Krizhevsky et al., 2012a). The results are shown in Figure 4 whose original data are reported in Appendix C. We can see although LabelFool is significantly better than FGSM and SparseFool, it is still a little worse than DeepFool in all three metrics. However, visual results (Figure 5) indicate that human observers can not notice the difference between LabelFool and DeepFool in the image space as the metric value is on such a small scale.
+
+# 4.3 FOOL NETWORKS
+
+We will show attack rate in the last experiment which is the most fundamental requirement for an attacker. Results are shown in Table 2. The results are the average value of three groups of
+
+![](images/a86925c24d1da2a9bbbd5f719bdd968d50e528b34f4526c191a2cfa41244e9f9.jpg)
+
+Figure 5: Two visual results of adversarial samples generated from AlexNet in the image space. In each result, the left image is the clean image, the middle one is DeepFool-attacked adversarial sample and the right one is LabelFool-attacked adversarial sample. Above the images are the true label/ the label after attacked. The three metrics of adversarial samples are reported in the table below them.
+
+Table 2: Attack rate of different methods on different models.   
+
+<table><tr><td>Model</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>ResNet-34</td><td>92.67%</td><td>97.50%</td><td>95.03%</td><td>92.60%</td></tr><tr><td>ResNet-50</td><td>93.08%</td><td>97.88%</td><td>95.09%</td><td>92.53%</td></tr><tr><td>VGG-19(bn)</td><td>92.03%</td><td>97.48%</td><td>94.59%</td><td>83.70%</td></tr><tr><td>AlexNet</td><td>90.35%</td><td>97.38%</td><td>96.44%</td><td>89.11%</td></tr></table>
+
+experiments. Each group has 1000 original images from ImageNet, we use these original images to generate adversarial images for four models respectively. We surprisingly find that LabelFool has the highest attack rate on all models comparing with other methods. This might benefit from our probability model which is used to choose the target label. Because in our strategy, when $\hat { p } _ { 1 } \leq \delta _ { 1 }$ , we do not use the predicted label as the ground truth like other methods do. Instead, we consider all labels whose probability are larger than $\delta _ { 2 }$ and choose the label nearest to all these labels as the target label. This operation can avoid some mistakes and improve the attack rate when the classifier doesn’t give a correct classification result. An example is shown in Appendix D.
+
+# 5 CONCLUSION AND FURTHER DISCUSSION
+
+Conclusion. In this study, we pay attention to tiny perturbations in the label space. To our best knowledge, we are the first one who points out the importance of the imperceptibility in the label space for adversarial samples. Furthermore, we explore a feasible method named LabelFool to identify a target label “similar” with an input image’s ground truth and perturb the input image to be mis-classified as this target label so that a human observer will overlook the misclassification and lower the vigilance of defenses. Our experiments show that, while LabelFool is a little behind DeepFool in the image space, it is much imperceptible in the label space to human observers. Since we adopt Importance Sampling instead of MLE only in traditional method, the success rate of attack also get gains.
+
+Further discussion. In this paper, we just propose a feasible way to generate adversarial samples which can confuse people in the label space. However, there is room for improvement in our approach. Our results provide the following avenues for future research.
+
+• The perceptual features can be optimized by a well-designed loss function which can improve the accuracy rate in finding nearest label ulteriorly.   
+• We only consider perceptual distance in this paper, but semantic distance also has its significance for reference of confusing people in the label space. We may take the semantic tree into consideration and make a trade off between perceptual distance and semantic distance in future research.
+
+# REFERENCES
+
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+
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+
+# A AN INTERFACE PRESENTATION OF THE SUBJECTIVE EXPERIMENT
+
+Figure 6 shows the interface of our subjective experiments.
+
+![](images/9712671a42dd9bc7ad66310ebe725ff1fb9b7026e368d207f4430fbfbb04b916.jpg)  
+Note:Please determine whether the label given is the correct label for the picture.   
+Figure 6: Interface presentation of our subjective experiment.
+
+# B EXAMPLES FOR ANIMAL CLASSES
+
+Figure 7 shows three examples for animal classes to demonstrate that LabelFool makes fine-grained changes but other methods make some ridiculous changes instead.
+
+![](images/641ad54c33971d3bc7cf120270597da0706328bfe4d90f2dd5946013e1f160d8.jpg)
+
+Figure 7: Three examples for animal images. The first column shows the clean image. The second column shows the ground truth label and other columns show the label after attacked.
+
+# C ORIGINAL DATA FOR FIGURE 3 AND 4
+
+Table 3 is the original data for Figure 3. The original data of perceptibility, perceptual similarity, PieAPP in Figure 4 is reported in Table 4, 5, 6 respectively. It is provided for the sake of convince if anyone wants to rewrite Figure 3 or 4.
+
+Table 3: Data for Figure 3   
+
+<table><tr><td colspan="2">Index</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>Total</td><td>Performance Gain</td><td>1.68</td><td>1.71</td><td>1.39</td><td>1.34</td></tr><tr><td>Animal</td><td>Performance Gain</td><td>3.55</td><td>4.04</td><td>3.11</td><td>3.00</td></tr></table>
+
+Table 4: Perceptibility Data for Figure 4   
+
+<table><tr><td>Model</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>ResNet-50</td><td>3.36E-05</td><td>4.75E-05</td><td>6.55E-04</td><td>2.56E-03</td></tr><tr><td>ResNet-34</td><td>3.35E-05</td><td>4.84E-05</td><td>6.59E-04</td><td>2.56E-03</td></tr><tr><td>VGG-19bn</td><td>3.18E-05</td><td>3.80E-05</td><td>6.44E-04</td><td>2.56E-03</td></tr><tr><td>AlexNet</td><td>3.59E-05</td><td>7.24E-05</td><td>6.68E-04</td><td>2.56E-03</td></tr></table>
+
+Table 5: Perceptual Similarity Data for Figure 4   
+
+<table><tr><td>Model</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>ResNet-50</td><td>1.48E-3</td><td>5.64E-3</td><td>0.20</td><td>0.42</td></tr><tr><td>ResNet-34</td><td>1.39E-3</td><td>5.91E-3</td><td>0.20</td><td>0.41</td></tr><tr><td>VGG-19bn</td><td>6.86E-4</td><td>2.54E-3</td><td>0.18</td><td>0.41</td></tr><tr><td>AlexNet</td><td>1.34E-2</td><td>3.50E-2</td><td>0.30</td><td>0.40</td></tr></table>
+
+Table 6: PieAPP Data for Figure 4   
+
+<table><tr><td>Model</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>ResNet-50</td><td>0.07</td><td>0.14</td><td>1.35</td><td>1.57</td></tr><tr><td>ResNet-34</td><td>0.07</td><td>0.15</td><td>1.31</td><td>1.46</td></tr><tr><td>VGG-19bn</td><td>0.05</td><td>0.08</td><td>1.27</td><td>1.59</td></tr><tr><td>AlexNet</td><td>0.07</td><td>0.21</td><td>1.28</td><td>1.21</td></tr></table>
+
+# D AN EXAMPLE FOR SECTION 4.3
+
+This is an example to illustrate why our method has the highest attack rate. We only give an example of DeepFool and LabelFool. SparseFool and FGSM have similar effects with DeepFool.
+
+Figure 8 is an example where the classifier fail to give a correct classification for the input image $x$ . The ground truth of $x$ is class 2 while the predicted class is class 3. In this example, DeepFool takes class 3 as the true class. Then DeepFool finds the nearest class to class 3 in the feature space which is class 2 in this example, and moves the input image towards class 2. When the perturbed image is classified as class 2 which is different from the predicted class, DeepFool considers the attack succeed and stops the algorithm. However, it fails to attack actually because class 2 is the true class of $x$ .
+
+Different from DeepFool, LabelFool sample top 3 classes in this example because their probabilities are larger than 0.01 and compute the expected distance between each class in the dataset and these 3 classes (Eq.(3)). Finally, LabelFool choose class 394 as the target class because it has the minimum expected distance with top 3 classes. By moving $x$ towards class 394, LabelFool attacks successfully.
+
+![](images/68ba82957f5f95e3dcd977744f4aaa68e10413b4b341ab50c91d806ce43d34af.jpg)  
+Figure 8: The line chart in the right part shows the top 10 elements in $\hat { p }$ , when an image $x$ whose ground truth class is class 2 is given into the classifier VGG 19bn. The horizontal axis represents the class and the vertical axis represents the probability that $x$ belongs to this class. The predicted class $\hat { l } _ { x }$ is class 3 and it means the classifier fail to give a correct classification.
+
+# E SUPPLEMENTARY EXAMPLES FOR LABELFOOL
+
+![](images/eccefb207082d4ff210ccb586ffbb616b97e0347e3d512eb9ef7f7ccd18ebd5c.jpg)  
+Figure 9: Some other examples for illustrating what LabelFool does. The first column shows the clean image. The second column shows the ground truth label and other columns show the label after attacked.
+
+# F AN APPLICATION: FACE RECOGNITION
+
+Figure 10 is an application to show it is necessary to generate imperceptible adversarial examples in the label space even for image classification tasks. We take face recognition system for entrance as an example. In Figure 10, A is the person who is using the face system to go into the gate. LabelFool aims to let the system misclassify A and B who is the one looks like A, but other untargeted attacks may let the system misclassify A an C who looks totally different from A. The attack is easy to be detected by the guard if the system misclassifies A and C, but it is hard to detect if the system misclassifies A and B. Letting a fake B in will bring great potential risks to security and safety.
+
+![](images/549416db53bf719da6cae0d5ce3460fa6fa29dd8b3f41f41d2c957b2530bf131.jpg)  
+Figure 10: A is the person who is using a face system to enter the gate. Green lines represent what LabelFool aims to do, that is to miscalssify A and B who looks like A. Red lines represent what other untargeted attacks do. They misclassify A and C who looks totally different from A and this error is easy to be detected by the guard.

parse/train/r1glDpNYwS/r1glDpNYwS_content_list.json

@@ -0,0 +1,1338 @@
+[
+    {
+        "type": "text",
+        "text": "LABELFOOL: A TRICK IN THE LABEL SPACE ",
+        "text_level": 1,
+        "bbox": [
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+            99,
+            542,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            170,
+            400,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "It is widely known that well-designed perturbations can cause state-of-the-art machine learning classifiers to mis-label an image, with sufficiently small perturbations that are imperceptible to the human eyes. However, by detecting the inconsistency between the image and wrong label, the human observer would be alerted of the attack. In this paper, we aim to design attacks that not only make classifiers generate wrong labels, but also make the wrong labels imperceptible to human observers. To achieve this, we propose an algorithm called LabelFool which identifies a target label similar to the ground truth label and finds a perturbation of the image for this target label. We first find the target label for an input image by a probability model, then move the input in the feature space towards the target label. Subjective studies on ImageNet show that in the label space, our attack is much less recognizable by human observers, while objective experimental results on ImageNet show that we maintain similar performance in the image space as well as attack rates to state-of-the-art attack algorithms. ",
+        "bbox": [
+            233,
+            266,
+            764,
+            460
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            488,
+            336,
+            503
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks are powerful learning models that achieve state-of-the-art pattern recognition performance in classification tasks (Krizhevsky et al., 2012b; LeCun et al., 2010; He et al., 2016). Nevertheless, it is found that adding well-designed perturbations to original samples can make classifiers of deep neural networks fail (Szegedy et al., 2013). These kinds of samples are called adversarial samples. Techniques for generating adversarial samples are called attackers. ",
+        "bbox": [
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+            520,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We think the ideal attacker should satisfy three levels of requirements. The first requirement is fooling networks which means making classifiers fail to classify an image successfully. For example, a dog image can be classified as a cat after added some well-designed perturbations. There are a number of methods for achieving a high attack rate (Goodfellow et al., 2015; Carlini & Wagner, 2017; Dong et al., 2018). ",
+        "bbox": [
+            174,
+            597,
+            825,
+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/d27b18d28b4fd6123cce7fb3d178e723891357a0599bc37a964b48ff63dab500.jpg",
+        "image_caption": [
+            "Figure 1: This graph illustrates the importance of the imperceptibility of adversarial samples in both image space and label space. Triangles are three attackers. Circles represent human observers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            680,
+            794,
+            818
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The second requirement for the ideal attacker is the imperceptibility in the image space. This means the magnitude of perturbations in the pixel level needs to be as tiny as possible so that it is imperceptible to human eyes. For example, additive perturbations are minimized with $l _ { p }$ norm to generate imperceptible adversarial samples (Seyed-Mohsen et al., 2016). Extreme cases also exist where only changing one or a few pixels (Su et al., 2019; Modas et al., 2019) can make classifiers fail. Moosavi-Dezfooli et al. (2017) even show the existence of universal (image-agnostic) perturbations. ",
+        "bbox": [
+            176,
+            882,
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+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The third requirement for the ideal attacker, which is newly proposed in this paper, is the imperceptibility of the error made by the classifier in the label space. It means making the classifier to mis-classify an image as the label which is similar to its ground truth, so that people won’t notice the misclassification. For example, in Figure 1, a human user will probably ignore the mis-classification if an attacker caused a “church” to be mis-classified as a “monastery” as the third attacker does. However, a human user will easily notice the mistake if an attacker caused a “church” to be misclassified as a “dome” as the second attacker does or caused an apparent perturbation in the image space as the first attacker does. In real applications, a human user will take defensive measures as soon as he notices the attack. Therefore making the whole attack process imperceptible is crucial for letting observers’ guard down. Tiny perturbations in the image space but large perturbations in the label space can muddle through on the input terminal. But as soon as observers check on the output terminal and see the obviously-incorrect label for an input, they will realize that the classifier fail due to some attacks and take defensive measures immediately, just as Figure 1 shows. This justifies the power of attacks which also confuse people in the label space. So the imperceptibility in the label space is quite important. However, to our best knowledge, few attackers have realized this point. ",
+        "bbox": [
+            174,
+            154,
+            825,
+            375
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose an untargeted-attack algorithm called LabelFool, to perturb an image to be mis-classified as the label which is similar to its ground truth, so that people won’t notice the misclassification. In the meantime, LabelFool also guarantees the imperceptibility in the image space as well as maintaining a high attack rate in fooling classifiers. There are two steps by which we accomplish our goal. The first step is to choose a target label which is similar to the input image’s ground truth. The second step is to perturb the input to be classified as this target label. The way is finding the classification boundary between the current label and the target label, and then moving the input towards this boundary until it is classified as the target label. We conduct a subjective experiment on ImageNet (Deng et al., 2009) which shows that adversarial samples generated by our method are indeed much less recognizable in the label space by human observers than other attacks. We also perform objective experiments on ImageNet to demonstrate that adversarial samples generated by LabelFool still guarantee the imperceptibility in the image space as well as maintaining a high attack rate in fooling classifiers. ",
+        "bbox": [
+            174,
+            382,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            588,
+            343,
+            604
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The phenomenon that neural networks are sensitive to adversarial samples was proposed by Szegedy et al. (2013). Since then, many researchers have studied how to generat adversarial samples. FGSM (Goodfellow et al., 2015) was proposed to maximize the classification error subject to $l _ { \\infty }$ -norm based distortion constraints. CW attack (Carlini & Wagner, 2017) generates adversarial samples by solving an optimization problem based on $l _ { 0 } / l _ { 2 } / l _ { \\infty }$ constraint, and $l _ { 0 }$ CW attack is the first proposed method that can cause targeted misclassification on the ImageNet dataset, meaning that we can specify the label of adversarial samples. But this designation of the target label is arbitrary. Until this paper, there has no guide about how to choose a target label such that it is difficult for a person to notice that the network has failed. ",
+        "bbox": [
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+            625,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Besides achieving the goal of misclassification, many researchers realize the importance of imperceptibility in the image space (Xu et al., 2019). One-pixel attack (Su et al., 2019) and SparseFool (Modas et al., 2019) attack networks in a scenario where perturbing only one/a few pixels can make a big difference. Moosavi-Dezfooli et al. (2017) show the existence of universal image-agnostic perturbations for state-of-the-art deep neural networks. DeepFool (Seyed-Mohsen et al., 2016) seeks the minimum image-level distortion. And for generating adversarial samples, it directly moves the input sample to the nearest class in the feature space. This is the most closely related work to ours, because features extracted from classification models can reflect images’ perceptual information and the classes which are close in the feature space are often perceptually similar. However, DeepFool approximates the multi-dimensional classification boundaries in two dimensions and this might make big errors on finding the nearest class. All these attacks generate adversarial samples by iteration and the algorithm stops as soon as an adversarial sample is born no matter what label it belongs to. This will lead to an apparent misclassification so that observers will sound the defensive alarm quickly. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we will compare our method with three attacks: FGSM, DeepFool and SparseFool to show the advantage of our method in the imperceptibility in the label space. We will also demonstrate that the performance gain in the label space is not at the expense of the loss in the image space or attack rate. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 LABELFOOL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            217,
+            310,
+            234
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we will introduce our method about how to choose a target label which is undetectable by human observers and how we can perturb the input image so that the classifier assigns this specific label. The whole pipeline is shown in Figure 2. All the symbols and notations used in this paper are summarized in Table 1. We use the same notation $i ( i = 1 , 2 , \\dots )$ for “class” and “label”, because “class” and “label” are interchangeable in this paper. LabelFool contains two steps. The first step is to choose a target label for the input image which is similar to its ground truth. The second step is to perturb the input image to be classified as this label. Inspired by DeepFool (Seyed-Mohsen et al., 2016), we make modifications at the feature level. We keep moving the input towards this chosen class at the feature level until it is classified as the label we want. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 CHOOSE A TARGET LABEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            396,
+            401,
+            410
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The first step of our method is choosing the target label $t _ { x }$ for an input image $x$ . As we want the target label to be imperceptible in the label space to human observers, we need to find the most “similar” label to the input image’s ground truth $l _ { x }$ where the most “similar” means the nearest in the perceptual distance metric. However, $l _ { x }$ is usually unknown when an input image is given. So it is important to estimate the probability distribution $P$ of an input’s ground truth $l _ { x }$ , based on which, we can compute the distance between each class in the dataset and $l _ { x }$ , then choose the nearest one as the target class. We propose a weighted distance model to achieve this goal. Before introducing the model, there are some preparations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given two image $x , y$ , we choose pre-trained image classification models to extract features $\\phi _ { x } , \\phi _ { y }$ because these features can reflect some perceptual information. As we want to calculate the distance in the perceptual distance metric and cosine distance has been used to measure perceptual similarity in many works (Lin et al., 2016; Wang et al., 2019), we compute the distance between $x$ and $y$ as $d ( x , y ) = 1 - \\cos { ( \\phi _ { x } , \\phi _ { y } ) }$ . After having the distance between two images, we can compute the distance between classes. Each class is a set of images. To measure the distance between two sets, we choose Hausdorff distance (Henrikson, 1999). The distance between class $i$ and class $j$ is denoted as $D _ { i , j }$ . Suppose a dataset has $n$ classes. Then, we can construct a matrix $\\pmb { { \\cal D } } \\in \\mathbb { R } ^ { n \\times n }$ by calculating the distance between all pairs of classes in the dataset, and it will be used in the following probability model to provide the distance we need. After these preparations, we can start to decide the target label for an input image. ",
+        "bbox": [
+            174,
+            541,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As introduced before, we need to estimate the probability distribution $P$ of the ground truth $l _ { x }$ because we want to find the nearest label to $l _ { x }$ which is unknown in the beginning. When an image $x$ is put into a classifier $f$ , state-of-the-art machine learning classifiers usually output a predicted label $\\hat { l } _ { x }$ and a probability vector $\\hat { p }$ whose elements mean $P ( x \\in c l a s s \\ i ) = \\hat { p } _ { i }$ . For simplicity, we suppose the elements in $\\hat { p }$ are sorted in the descending order. Meanwhile, $\\hat { p }$ can be thought as $P$ ’s approximation. Furthermore, we define a distance function between $l _ { x }$ and the class $i$ in a $\\mathbf { n }$ -classes dataset as $D _ { i } ( l _ { x } )$ . In order to choose the nearest label to $l _ { x }$ as the target label $t _ { x }$ , we need to estimate the expectation of $D _ { i } ( l _ { x } )$ which is denoted as $\\mathbb { E } _ { l _ { x } \\sim P } [ D _ { i } ( l _ { x } ) ] ( i = \\bar { 1 } , \\dots , n )$ . In general, our target function is Eq. (1). ",
+        "bbox": [
+            174,
+            700,
+            825,
+            830
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/408de8d892d3acc28d50ae6ef93d25372c1a2e6ef6b7cc9a41e548f0726ad348.jpg",
+        "text": "$$\nt _ { x } = \\underset { i = 1 , \\ldots , n } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { l _ { x } \\sim P } [ D _ { i } ( l _ { x } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            852,
+            596,
+            878
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Specifically, when $\\hat { p } _ { 1 }$ is larger than some threshold $\\delta _ { 1 }$ , we use Maximum Likelihood Estimation (MLE) (Pfanzagl, 2011) which means we believe the classifier and take the predicted label $\\hat { l } _ { x }$ as ",
+        "bbox": [
+            173,
+            892,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Meaning ",
+        "text_level": 1,
+        "bbox": [
+            529,
+            143,
+            593,
+            157
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$x$ Input image   \n$\\phi _ { x }$ The feature of image $x$   \n$_ { D }$ The perceptual distance matrix where $D _ { i , j }$ represents the distance between class $i$ and class $j$ $f$ Classifier   \n$\\hat { p }$ Probability vector (elements are in descending order)   \n$l _ { x }$ The ground truth of the input image $x$   \n$\\hat { l } _ { x }$ The predicted class of the input image $x$ by the classifier   \n$D _ { i } ( l _ { x } )$ A function calculating the distance between class $i$ and the ground truth $l _ { x }$   \n$t _ { x }$ Target label for input $x$   \n$\\delta _ { 1 } , \\delta _ { 2 }$ Two thresholds, in this paper, $\\delta _ { 1 } = 0 . 8 , \\delta _ { 2 } = 0 . 0 1$   \n$M$ The number of elements larger than $\\delta _ { 2 }$ in $\\hat { p } _ { : }$ , i.e. $M = \\mathrm { m a x } _ { j = 1 , \\dots , n } \\{ j : { \\hat { p } } _ { j } > \\delta _ { 2 } \\}$   \n${ \\mathcal { F } } _ { j }$ The classification boundary between class $\\hat { l } _ { x }$ and class $j$ of an image ",
+        "bbox": [
+            179,
+            171,
+            867,
+            344
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "the ground truth $l _ { x }$ , then choose the label (except $\\hat { l } _ { x }$ ) nearest to $\\hat { l } _ { x }$ as the target label $t _ { x }$ . So in this circumstance, we assume $l _ { x } = \\hat { l } _ { x }$ and $\\mathbb { E } _ { l _ { x } \\sim P } [ D _ { i } ( l _ { x } ) ] = D _ { i , \\hat { l } _ { x } }$ . Therefore, $t _ { x }$ is ",
+        "bbox": [
+            174,
+            375,
+            825,
+            409
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/722f7ed9fcb97702dfd0d8b93124a553ccfd8eee983cd74823821e4bb8c821ab.jpg",
+        "text": "$$\nt _ { x } = \\operatorname * { a r g m i n } _ { i \\neq \\hat { l } _ { x } , i = 1 , \\ldots , n } D _ { i , \\hat { l } _ { x } } \\quad \\mathrm { i f } \\hat { p } _ { 1 } > \\delta _ { 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            426,
+            620,
+            455
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When $\\hat { p } _ { 1 }$ is smaller than the threshold $\\delta _ { 1 }$ , we are not sure whether $l _ { x }$ is equal to $\\hat { l } _ { x }$ . Instead, we sample some labels and compute the weighted distance between each label $i$ and these labels. We sample all labels whose probability are larger than a threshold $\\delta _ { 2 }$ and we use $M$ to represent the number of sampled labels. We think the input image might belong to one of these $M$ labels. The labels whose probability are smaller than $\\delta _ { 2 }$ will be abandoned because we think the input image can hardly fall into these categories. The weight and the distance is provided by the vector $\\hat { p }$ and matrix $_ D$ respectively. So in this circumstance, as we are not sure which label is the ground truth, we want to find a target label which has the minimum expected distance with all these possible labels. Therefore, the value of $\\mathbb { E } _ { l _ { x } \\sim P } [ D _ { i } ( l _ { x } ) ]$ can be approximated as X j=1 pˆj · Di,j and the target label $t _ { x }$ is shown in Eq. (3). This can be explained by Importance Sampling (Owen & Zhou, 2000) because it is hard to sample from the real probability distribution $P$ . We can only use the probability distribution $\\hat { p }$ which is an approximation of $P$ to estimate the value of $\\mathbb { E } _ { l _ { x } \\sim P } [ D _ { i } ( l _ { x } ) ]$ . ",
+        "bbox": [
+            173,
+            467,
+            826,
+            648
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/239ee6197e5924863c38e21f735796e29db234aeb91960074102f63a3db476b4.jpg",
+        "text": "$$\nt _ { x } = \\underset { i = 1 , \\ldots , n } { \\arg \\operatorname* { m i n } } \\sum _ { j = 1 } ^ { M } \\hat { p } _ { j } \\cdot D _ { i , j } \\quad \\mathrm { i f } \\hat { p } _ { 1 } \\leq \\delta _ { 1 }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In conclusion, the whole strategy for choosing the target label $t _ { x }$ of an input image $x$ is computed as Eq. (4). The target label $t _ { x }$ minimizes $\\mathbb { E } _ { l _ { x } \\sim P } [ D _ { i } ( l _ { x } ) ]$ just as Figure 2 shows. ",
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+        "type": "equation",
+        "img_path": "images/ca7df80dddb9a26126c80fe3080f88f177c7491f3cf13888cf85b195d5fa50e9.jpg",
+        "text": "$$\nt _ { x } = \\left\\{ \\begin{array} { l l } { \\displaystyle \\arg \\operatorname* { m i n } _ { i , \\hat { l } _ { x } } D _ { i , \\hat { l } _ { x } } } & { \\mathrm { i f } \\hat { p } _ { 1 } > \\delta _ { 1 } } \\\\ { \\displaystyle i \\neq \\hat { l } _ { x } , i = 1 , . . . , n } \\\\ { \\displaystyle \\arg \\operatorname* { m i n } _ { i = 1 , . . . , n } \\sum _ { j = 1 } ^ { M } \\hat { p } _ { j } \\cdot D _ { i , j } } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
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+    {
+        "type": "text",
+        "text": "3.2 GENERATE ADVERSARIAL SAMPLES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "After having the target label, the second step is to attack the input image to be mis-classified as this target label. It’s easy to achieve by taking the target label as a parameter and putting it into the targeted-attack algorithm such as targeted-FGSM (Goodfellow et al., 2015) and targeted-CW (Carlini & Wagner, 2017), but this operation may suffer huge loss in the image space because of large perturbations. ",
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+        "type": "image",
+        "img_path": "images/130ca587a82453324ed51da5a452a8791633884559a83a5803a3898b7eeaaf9b.jpg",
+        "image_caption": [
+            "Figure 2: Pipeline of our method. There are two steps. In the first step, we first compute the distance $D _ { i , j }$ between every two class $i , j$ in a n-classes dataset. Then we choose the target label $t _ { x }$ for an input image $x$ by two strategies according to the value of $\\hat { p } _ { 1 }$ . The second step is to attack the input into this target label. Solid lines are the real boundaries between the current label and the indicated label and dashed lines with notes $\\mathcal { F }$ are the approximate two-dimensional boundaries. Red indicates the target label while blue indicates other labels. Our method moves the input towards the boundary $\\mathcal { F } _ { t _ { x } }$ until it is classified as $t _ { x }$ . "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Inspired by DeepFool (Seyed-Mohsen et al., 2016), we propose a method which can not only attack an input image to be mis-classified as the target label successfully, but also ensure tiny perturbations in the image space. The mathematical derivation in this step is similar to DeepFool (Seyed-Mohsen et al., 2016) and the only difference is that, we have a target label chosen in the first step while DeepFool doesn’t. As introduced in DeepFool (Seyed-Mohsen et al., 2016), a high dimensional classification boundary can be approximated by a line in two dimensions. As shown in Figure 2, for an image $x _ { 0 }$ , ${ \\mathcal { F } } _ { j }$ represents the 2D approximated boundary between its current predicted class and class $j$ and $t _ { x }$ is the target class we choose in the first step. In the first iteration, we move $x _ { 0 }$ towards $\\mathcal { F } _ { t _ { x } }$ and get a new point $x _ { 1 }$ . The direction of movement is perpendicular to $\\mathcal { F } _ { t _ { x } }$ . The distance of the movement is the vertical distance from $x _ { 0 }$ to $\\mathcal { F } _ { t _ { x } }$ . If the predicted label $\\hat { l } _ { x _ { 1 } }$ of the new point equals $\\hat { l } _ { x _ { 0 } }$ , the classification boundaries are the same as those before moving $x _ { 0 }$ . Otherwise, the classification boundaries change. No matter whether the boundaries change or not, we repeatedly move the current point towards $\\mathcal { F } _ { t _ { x } }$ until it is classified as label $t _ { x }$ or the maximum number of iterations has been reached. A pseudocode of the second step is shown in Algorithm 1. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 : Generate Adversarial Samples ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Input: image $x$ , classifier $f$ , target label $t _ { x }$ Output: Adversarial image $\\hat { x }$ 1: initialize $x _ { 0 } \\gets x , i \\gets 0$ 2: while $\\hat { l } _ { x _ { i } } \\neq t _ { x }$ and i < max iter do 3: $w \\gets \\nabla f _ { \\hat { l } _ { x _ { i } } } ( x _ { i } ) - \\nabla f _ { t _ { x } } ( x _ { i } )$ 4: $g \\gets f _ { \\hat { l } _ { x _ { i } } } ( \\bar { x } _ { i } ) - f _ { t _ { x } } ( x _ { i } )$ ",
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+        "text": "5: ri ← kw k22 |g| w   \n6: xi+1 ← xi + ri, i ← i + 1   \n7: end while   \n8: return $\\hat { x } = x _ { i + 1 }$ ",
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+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "In this paper, all experiments are conducted on ImageNet. ImageNet provides the CLS-LOC dataset for classification tasks. Its train split contains about 1300 thousand images. There are 50 thousand validation images and 100 thousand test images. Our experiments are conducted on the train split of CLS-LOC dataset which will be noted as ImageNet-train split in the following part. We perform extensive experiments to show LabelFool can satisfy all three levels of requirements as an attacker. First we demonstrate the deceptiveness of samples generated by LabelFool to humans in the label space through a subjective experiment. Then we calculate the perceptibility and image quality of adversarial samples to show there is not much loss in the image space even compared to DeepFool (Seyed-Mohsen et al., 2016), which is the state-of-the-art method in the image space. Finally, we conduct attacks on several models to prove the exceptional ability of our method on fooling neural networks which is the first requirement for an ideal attack. ",
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+    {
+        "type": "text",
+        "text": "4.1 IMPERCEPTIBILITY IN THE LABEL SPACE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Setup. In this part, we will compare LabelFool with three attack methods: DeepFool, FGSM and SparseFool. We first sample 600 source images from ImageNet-train split randomly, and these images are in different classes. Each source image will derive four adversarial images and a baseline image, namely DeepFool-attacked image, LabelFool-attacked image, FGSM-attacked image, SparseFool-attacked image and clean image. Each adversarial image has its mis-classified label and each baseline image has the truth label. We then use term “puzzles” to describe the combination of an image and its label, for $5 \\times 6 0 0 = 3 0 0 0$ puzzles. A human observer needs to determine whether the label is correct for the image, answering “True” or “False” for each puzzle. To eliminate observers’ memory effects, we split 3000 puzzles into five groups, ensuring that 600 images in one group come from different source images. An interface presentation of our subjective experiment is shown in Appendix A. We have 10 observers (3 females and 7 males, age between 20-29) to do our subjective experiment. ",
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+        "text": "Evaluation. In this paper, we define an index named Performance Gain $( P G )$ as the evaluation index. Human observers answer “True” or “False” for each puzzle, and the rate with which they answer incorrectly is called Confusion Rate $( C R )$ . So every observer has a $C R$ for each attack method or baseline. It is an absolute indicator demonstrating how much observers are confused by a set of puzzles. But doing arithmetic on $C R$ of different observers is meaningless, as different observers have different baseline results. So we define a relative indicator called Performance Gain $( P G )$ , which demonstrates how much improvement in the confusion rate after attacking comparing with baseline. It is a kind of normalization. The formula for PG is shown in Eq. (5), where $C R _ { A }$ means the confusion rate of an attacker and $C R _ { B }$ means the confusion rate of baseline. The higher $P G$ an attacker has, the better it confuses people in the label space. ",
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+    {
+        "type": "equation",
+        "img_path": "images/8e2a1cacce0cf0456a3cd0e12b860e50c471b52d9a7cc871f309365bacdc287d.jpg",
+        "text": "$$\nP G _ { A } = \\frac { C R _ { A } - C R _ { B } } { C R _ { B } } \\ ( A i s \\ a n \\ a t t a c k e r , B = B a s e l i n e )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Results. We report the average performance gain of 10 observers in total images in the left of Figure 3. As a whole, there is a huge improvement compared with FGSM and SparseFool, about 25 percent improvement and 30 percent improvement respectively. Compared with DeepFool, the gap in $P G$ is a little smaller because DeepFool finds the nearest class in the feature level and features usually reflect images’ perceptual information as we introduced in Section 2. But there is still 3 percent improvement in performance gain. ",
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+        "text": "As animal classes are more fine-grained, the effects of the imperceptibility in the label space become more pronounced. We attack an animal image so that its true animal label changed into a similar animal label, it is difficult for humans to notice that our attack is taking place. Meanwhile, other attacks change the label into an obviously-incorrect label such as a non-animal category or another species (Some examples in Appendix B). In our subjective experiment, there are 247 animal images out of 600 source images. The right graph in Figure 3 shows the average performance gain of 10 observers in animal images. The accurate data for both graphs in Figure 3 is shown in Appendix C. ",
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+        "type": "text",
+        "text": "As for animal images, the improvement is very obvious comparing with all three attack methods. There are about nearly 90 percent improvement in performance gain comparing with FGSM and ",
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+    {
+        "type": "image",
+        "img_path": "images/5b15f8f4e910472b4d11cf21cf87b2c22d63c70a811de85807b897e853634270.jpg",
+        "image_caption": [
+            "Figure 3: A line chart for average performance gain of 10 observers. The horizontal axis represents four attack methods. The vertical axis represents the mean value of 10 human observers’ performance gain. The graph in the left is for total results, and the right one is for animal images. "
+        ],
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+        "type": "text",
+        "text": "SparseFool. A significant improvement can also be seen when comparing with DeepFool, there are about 50 percent improvement in average performance gain. ",
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+    {
+        "type": "text",
+        "text": "4.2 IMPERCEPTIBILITY IN THE IMAGE SPACE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this subsection, we will show our performance in the image space to demonstrate that our improvement in the label space is not at the cost of huge loss in the image space. We use three metrics to evaluate performance in the image space. One is perceptibility which is similar to the definition in previous works (Szegedy et al., 2013; Seyed-Mohsen et al., 2016) : $p =$ $\\frac { 1 } { W _ { N } \\times H _ { N } } \\sum _ { w = 1 } ^ { W _ { N } } \\sum _ { h = 1 } ^ { \\bar { H } _ { N } } { \\lVert { \\Delta y _ { w , h } } \\rVert ^ { 2 } }$ , where $y _ { w , h }$ is a 3-dimensional vector representing the RGB intensities (normalized in [0, 1]) of a pixel. The other two are perceptual similarity (Zhang et al., 2018) and PieAPP (Prashnani et al., 2018). These two are metrics for image quality. Perceptual similarity measures the perceptual distance between an image and its reference image while PieAPP measures the perceptual error. In this paper, the reference image is the clean image. And the smaller these three metrics are, the better the adversarial samples are. ",
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+    {
+        "type": "image",
+        "img_path": "images/ed9e42db6e82a9808f050163a4330bc20b03303d3bf3ddc677dfb578e01c720e.jpg",
+        "image_caption": [
+            "Figure 4: Mean value of perceptibility, perceptual similarity and PieAPP for adversarial samples generated by different attack methods on different models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We randomly choose 1000 images from ImageNet and attack the classifier to generate 1000 adversarial samples. Then we compute mean value for these adversarial samples of three metrics. In this experiment, we test four classifiers: ResNet-34, ResNet-50, VGG-19 (with batch normalization) (Simonyan & Zisserman, 2014) and AlexNet (Krizhevsky et al., 2012a). The results are shown in Figure 4 whose original data are reported in Appendix C. We can see although LabelFool is significantly better than FGSM and SparseFool, it is still a little worse than DeepFool in all three metrics. However, visual results (Figure 5) indicate that human observers can not notice the difference between LabelFool and DeepFool in the image space as the metric value is on such a small scale. ",
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+    {
+        "type": "text",
+        "text": "4.3 FOOL NETWORKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We will show attack rate in the last experiment which is the most fundamental requirement for an attacker. Results are shown in Table 2. The results are the average value of three groups of ",
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+        "img_path": "images/a86925c24d1da2a9bbbd5f719bdd968d50e528b34f4526c191a2cfa41244e9f9.jpg",
+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "Figure 5: Two visual results of adversarial samples generated from AlexNet in the image space. In each result, the left image is the clean image, the middle one is DeepFool-attacked adversarial sample and the right one is LabelFool-attacked adversarial sample. Above the images are the true label/ the label after attacked. The three metrics of adversarial samples are reported in the table below them. ",
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+    {
+        "type": "table",
+        "img_path": "images/e8f507b3f8da581fc1158f84ba28bf096f1f3bef694348e8e264e0fb9e3821df.jpg",
+        "table_caption": [
+            "Table 2: Attack rate of different methods on different models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>ResNet-34</td><td>92.67%</td><td>97.50%</td><td>95.03%</td><td>92.60%</td></tr><tr><td>ResNet-50</td><td>93.08%</td><td>97.88%</td><td>95.09%</td><td>92.53%</td></tr><tr><td>VGG-19(bn)</td><td>92.03%</td><td>97.48%</td><td>94.59%</td><td>83.70%</td></tr><tr><td>AlexNet</td><td>90.35%</td><td>97.38%</td><td>96.44%</td><td>89.11%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "experiments. Each group has 1000 original images from ImageNet, we use these original images to generate adversarial images for four models respectively. We surprisingly find that LabelFool has the highest attack rate on all models comparing with other methods. This might benefit from our probability model which is used to choose the target label. Because in our strategy, when $\\hat { p } _ { 1 } \\leq \\delta _ { 1 }$ , we do not use the predicted label as the ground truth like other methods do. Instead, we consider all labels whose probability are larger than $\\delta _ { 2 }$ and choose the label nearest to all these labels as the target label. This operation can avoid some mistakes and improve the attack rate when the classifier doesn’t give a correct classification result. An example is shown in Appendix D. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION AND FURTHER DISCUSSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Conclusion. In this study, we pay attention to tiny perturbations in the label space. To our best knowledge, we are the first one who points out the importance of the imperceptibility in the label space for adversarial samples. Furthermore, we explore a feasible method named LabelFool to identify a target label “similar” with an input image’s ground truth and perturb the input image to be mis-classified as this target label so that a human observer will overlook the misclassification and lower the vigilance of defenses. Our experiments show that, while LabelFool is a little behind DeepFool in the image space, it is much imperceptible in the label space to human observers. Since we adopt Importance Sampling instead of MLE only in traditional method, the success rate of attack also get gains. ",
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+    {
+        "type": "text",
+        "text": "Further discussion. In this paper, we just propose a feasible way to generate adversarial samples which can confuse people in the label space. However, there is room for improvement in our approach. Our results provide the following avenues for future research. ",
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+    {
+        "type": "text",
+        "text": "• The perceptual features can be optimized by a well-designed loss function which can improve the accuracy rate in finding nearest label ulteriorly.   \n• We only consider perceptual distance in this paper, but semantic distance also has its significance for reference of confusing people in the label space. We may take the semantic tree into consideration and make a trade off between perceptual distance and semantic distance in future research. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
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+    },
+    {
+        "type": "text",
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+        "text": "Ekta Prashnani, Hong Cai, Yasamin Mostofi, and Pradeep Sen. PieAPP: Perceptual image-error assessment through pairwise preference. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1808–1817, 2018. ",
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+        "text": "Seyed-Mohsen, Moosavi-Dezfooli, Pascal Frossard Alhussein Fawzi, and Ecole Polytechnique Federale de Lausanne. DeepFool: a simple and accurate method to fool deep neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016. ",
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+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaidi Xu, Sijia Liu, Pu Zhao, Pin Yu Chen, Huan Zhang, Quanfu Fan, Deniz Erdogmus, Yanzhi Wang, and Xue Lin. Structured adversarial attack: Towards general implementation and better interpretability. In International Conference of Learning Representation, 2019. ",
+        "bbox": [
+            176,
+            194,
+            821,
+            237
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 586–595, 2018. ",
+        "bbox": [
+            173,
+            246,
+            825,
+            289
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A AN INTERFACE PRESENTATION OF THE SUBJECTIVE EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            318,
+            750,
+            333
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 shows the interface of our subjective experiments. ",
+        "bbox": [
+            173,
+            348,
+            560,
+            363
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/9712671a42dd9bc7ad66310ebe725ff1fb9b7026e368d207f4430fbfbb04b916.jpg",
+        "image_caption": [
+            "Note:Please determine whether the label given is the correct label for the picture. ",
+            "Figure 6: Interface presentation of our subjective experiment. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            328,
+            401,
+            660,
+            570
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B EXAMPLES FOR ANIMAL CLASSES ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            637,
+            493,
+            651
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows three examples for animal classes to demonstrate that LabelFool makes fine-grained changes but other methods make some ridiculous changes instead. ",
+        "bbox": [
+            171,
+            667,
+            823,
+            696
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/641ad54c33971d3bc7cf120270597da0706328bfe4d90f2dd5946013e1f160d8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            218,
+            709,
+            779,
+            864
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Figure 7: Three examples for animal images. The first column shows the clean image. The second column shows the ground truth label and other columns show the label after attacked. ",
+        "bbox": [
+            173,
+            885,
+            825,
+            914
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "C ORIGINAL DATA FOR FIGURE 3 AND 4 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            524,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 3 is the original data for Figure 3. The original data of perceptibility, perceptual similarity, PieAPP in Figure 4 is reported in Table 4, 5, 6 respectively. It is provided for the sake of convince if anyone wants to rewrite Figure 3 or 4. ",
+        "bbox": [
+            174,
+            132,
+            825,
+            175
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/9e6c05a79367119d8a3b7be272d129a1544bbc734c74747da2b43d92367ae90f.jpg",
+        "table_caption": [
+            "Table 3: Data for Figure 3 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Index</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>Total</td><td>Performance Gain</td><td>1.68</td><td>1.71</td><td>1.39</td><td>1.34</td></tr><tr><td>Animal</td><td>Performance Gain</td><td>3.55</td><td>4.04</td><td>3.11</td><td>3.00</td></tr></table>",
+        "bbox": [
+            204,
+            220,
+            794,
+            282
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/e5922eb303b62b2ded923d7c28b2a4746f6e1fb1a8e5e62c4a780a35f97110db.jpg",
+        "table_caption": [
+            "Table 4: Perceptibility Data for Figure 4 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>ResNet-50</td><td>3.36E-05</td><td>4.75E-05</td><td>6.55E-04</td><td>2.56E-03</td></tr><tr><td>ResNet-34</td><td>3.35E-05</td><td>4.84E-05</td><td>6.59E-04</td><td>2.56E-03</td></tr><tr><td>VGG-19bn</td><td>3.18E-05</td><td>3.80E-05</td><td>6.44E-04</td><td>2.56E-03</td></tr><tr><td>AlexNet</td><td>3.59E-05</td><td>7.24E-05</td><td>6.68E-04</td><td>2.56E-03</td></tr></table>",
+        "bbox": [
+            263,
+            340,
+            735,
+            439
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/7b07a9188b688b615bcb891efb10f668c9948f1b49b2ea95f2ccd7546cf68d03.jpg",
+        "table_caption": [
+            "Table 5: Perceptual Similarity Data for Figure 4 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>ResNet-50</td><td>1.48E-3</td><td>5.64E-3</td><td>0.20</td><td>0.42</td></tr><tr><td>ResNet-34</td><td>1.39E-3</td><td>5.91E-3</td><td>0.20</td><td>0.41</td></tr><tr><td>VGG-19bn</td><td>6.86E-4</td><td>2.54E-3</td><td>0.18</td><td>0.41</td></tr><tr><td>AlexNet</td><td>1.34E-2</td><td>3.50E-2</td><td>0.30</td><td>0.40</td></tr></table>",
+        "bbox": [
+            263,
+            497,
+            733,
+            597
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/d2769905aff41a3c95c88243e3c51a511a1ab2ea1e29406caaf09d8ab3699e69.jpg",
+        "table_caption": [
+            "Table 6: PieAPP Data for Figure 4 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>DeepFool</td><td>LabelFool</td><td>FGSM</td><td>SparseFool</td></tr><tr><td>ResNet-50</td><td>0.07</td><td>0.14</td><td>1.35</td><td>1.57</td></tr><tr><td>ResNet-34</td><td>0.07</td><td>0.15</td><td>1.31</td><td>1.46</td></tr><tr><td>VGG-19bn</td><td>0.05</td><td>0.08</td><td>1.27</td><td>1.59</td></tr><tr><td>AlexNet</td><td>0.07</td><td>0.21</td><td>1.28</td><td>1.21</td></tr></table>",
+        "bbox": [
+            263,
+            656,
+            735,
+            753
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "D AN EXAMPLE FOR SECTION 4.3 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            787,
+            472,
+            804
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This is an example to illustrate why our method has the highest attack rate. We only give an example of DeepFool and LabelFool. SparseFool and FGSM have similar effects with DeepFool. ",
+        "bbox": [
+            174,
+            818,
+            823,
+            847
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Figure 8 is an example where the classifier fail to give a correct classification for the input image $x$ . The ground truth of $x$ is class 2 while the predicted class is class 3. In this example, DeepFool takes class 3 as the true class. Then DeepFool finds the nearest class to class 3 in the feature space which is class 2 in this example, and moves the input image towards class 2. When the perturbed image is classified as class 2 which is different from the predicted class, DeepFool considers the attack succeed and stops the algorithm. However, it fails to attack actually because class 2 is the true class of $x$ . ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Different from DeepFool, LabelFool sample top 3 classes in this example because their probabilities are larger than 0.01 and compute the expected distance between each class in the dataset and these 3 classes (Eq.(3)). Finally, LabelFool choose class 394 as the target class because it has the minimum expected distance with top 3 classes. By moving $x$ towards class 394, LabelFool attacks successfully. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            195
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/68ba82957f5f95e3dcd977744f4aaa68e10413b4b341ab50c91d806ce43d34af.jpg",
+        "image_caption": [
+            "Figure 8: The line chart in the right part shows the top 10 elements in $\\hat { p }$ , when an image $x$ whose ground truth class is class 2 is given into the classifier VGG 19bn. The horizontal axis represents the class and the vertical axis represents the probability that $x$ belongs to this class. The predicted class $\\hat { l } _ { x }$ is class 3 and it means the classifier fail to give a correct classification. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            217,
+            779,
+            377
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "E SUPPLEMENTARY EXAMPLES FOR LABELFOOL ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            492,
+            598,
+            510
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/eccefb207082d4ff210ccb586ffbb616b97e0347e3d512eb9ef7f7ccd18ebd5c.jpg",
+        "image_caption": [
+            "Figure 9: Some other examples for illustrating what LabelFool does. The first column shows the clean image. The second column shows the ground truth label and other columns show the label after attacked. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            217,
+            559,
+            779,
+            849
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "F AN APPLICATION: FACE RECOGNITION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            531,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 10 is an application to show it is necessary to generate imperceptible adversarial examples in the label space even for image classification tasks. We take face recognition system for entrance as an example. In Figure 10, A is the person who is using the face system to go into the gate. LabelFool aims to let the system misclassify A and B who is the one looks like A, but other untargeted attacks may let the system misclassify A an C who looks totally different from A. The attack is easy to be detected by the guard if the system misclassifies A and C, but it is hard to detect if the system misclassifies A and B. Letting a fake B in will bring great potential risks to security and safety. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            231
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/549416db53bf719da6cae0d5ce3460fa6fa29dd8b3f41f41d2c957b2530bf131.jpg",
+        "image_caption": [
+            "Figure 10: A is the person who is using a face system to enter the gate. Green lines represent what LabelFool aims to do, that is to miscalssify A and B who looks like A. Red lines represent what other untargeted attacks do. They misclassify A and C who looks totally different from A and this error is easy to be detected by the guard. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            244,
+            764,
+            559
+        ],
+        "page_idx": 12
+    }
+]

parse/train/r1ltgp4FwS/r1ltgp4FwS.md

@@ -0,0 +1,351 @@
+# LEARNING TEMPORAL COHERENCE VIA SELFSUPERVISION FOR GAN-BASED VIDEO GENERATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We focus on temporal self-supervision for GAN-based video generation tasks. While adversarial training successfully yields generative models for a variety of areas, temporal relationship in the generated data is much less explored. This is crucial for sequential generation tasks , e.g. video super-resolution and unpaired video translation. For the former, state-of-the-art methods often favor simpler norm losses such as $L ^ { 2 }$ over adversarial training. However, their averaging nature easily leads to temporally smooth results with an undesirable lack of spatial detail. For unpaired video translation, existing approaches modify the generator networks to form spatio-temporal cycle consistencies. In contrast, we focus on improving the learning objectives, and propose a temporally self-supervised algorithm. For both tasks, we show that temporal adversarial learning is key to achieving temporally coherent solutions without sacrificing spatial detail. We also propose a novel Ping-Pong loss to improve the long-term temporal consistency. It effectively prevents recurrent networks from accumulating artifacts temporally without depressing detailed features. We also propose a first set of metrics to quantitatively evaluate the accuracy as well as the perceptual quality of the temporal evolution. A series of user studies confirms the rankings computed with these metrics.
+
+# 1 INTRODUCTION
+
+Generative adversarial models (GANs) have been extremely successful at learning complex distributions such as natural images (Zhu et al., 2017; Isola et al., 2017). However, for sequence generation, directly applying GANs without carefully engineered constraints typically results in strong artifacts over time due to the significant difficulties introduced by the temporal changes. In particular, conditional video generation tasks are very challenging learning problems where generators should not only learn to represent the data distribution of the target domain, but also learn to correlate the output distribution over time with conditional inputs. Their central objective is to faithfully reproduce the temporal dynamics of the target domain and not resort to trivial solutions such as features that arbitrarily appear and disappear over time.
+
+In our work, we propose a novel adversarial learning method for a recurrent training approach that supervises both spatial content as well as temporal relationships. We apply our approach to two video-related tasks that offer substantially different challenges: video super-resolution (VSR) and unpaired video translation (UVT). With no ground truth motion available, the spatio-temporal adversarial loss and the recurrent structure enable our model to generate realistic results while keeping the generated structures coherent over time. With the two learning tasks we demonstrate how spatio-temporal adversarial training can be employed in paired as well as unpaired data domains. In addition to the adversarial network which supervises the short-term temporal coherence, long-term consistency is self-supervised using a novel bi-directional loss formulation, which we refer to as “Ping-Pong” (PP) loss in the following. The PP loss effectively avoids the temporal accumulation of artifacts, which can potentially benefit a variety of recurrent architectures. The central contributions of our work are: a spatio-temporal discriminator unit together with a careful analysis of training objectives for realistic and coherent video generation tasks, a novel PP loss supervising long-term consistency, in addition to a set of metrics for quantifying temporal coherence based on motion estimation and perceptual distance. Together, our contributions lead to models that outperform previous work in terms of temporally-coherent detail, which we quantify with a wide range of metrics and user studies.
+
+![](images/34967317bd619c273a77f8cea0233ce95354866369db13e774df729e7465dd47.jpg)  
+Figure 1: When learning a mapping between Trump and Obama, the CycleGAN model gives good spatial features, but collapses to essentially static outputs of Obama. It manages to transfer facial expressions back to Trump using tiny differences encoded in its Obama outputs, instead of learning a meaningful mapping. Being able to establish the correct temporal cycle-consistency between domains, ours and RecycleGAN can generate correct blinking motions. Our model outperforms the latter in terms of coherent detail that is generated.
+
+# 2 RELATED WORK
+
+Deep learning has made great progress for image generation tasks. While regular losses such as $L ^ { 2 }$ (Kim et al., 2016; Lai et al., 2017) offer good performance for image super-resolution (SR) tasks in terms of PSNR metrics, GAN researchers found adversarial training (Goodfellow et al., 2014) to significantly improve the perceptual quality in multi-modal problems including image SR (Ledig et al., 2016), image translations (Zhu et al., 2017; Isola et al., 2017), and others. Perceptual metrics (Zhang et al., 2018; Prashnani et al., 2018) are proposed to reliably evaluate image similarity by considering semantic features instead of pixel-wise errors.
+
+Video generation tasks, on the other hand, require realistic results to change naturally over time. Recent works in VSR improve the spatial detail and temporal coherence by either using multiple low-resolution (LR) frames as inputs (Jo et al., 2018; Tao et al., 2017; Liu et al., 2017), or recurrently using previously estimated outputs (Sajjadi et al., 2018). The latter has the advantage to re-use high-frequency details over time. In general, adversarial learning is less explored for VSR and applying it in conjunction with a recurrent structure gives rise to a special form of temporal mode collapse, as we will explain below. For video translation tasks, GANs are more commonly used but discriminators typically only supervise the spatial content. E.g., Zhu et al. (2017) does not employ temporal constrains and generators can fail to learn the temporal cycle-consistency. In order to learn temporal dynamics, RecycleGAN (Bansal et al., 2018) proposes to use a prediction network in addition to a generator, while a concurrent work (Chen et al., 2019) chose to learn motion translation in addition to spatial content translation. Being orthogonal to these works, we propose a spatiotemporal adversarial training for both VSR and UVT and we show that temporal self-supervision is crucial for improving spatio-temporal correlations without sacrificing spatial detail. While $L ^ { 2 }$ temporal losses based on warping are used to enforce temporal smoothness in video style transfer tasks (Ruder et al., 2016; Chen et al., 2017), concurrent GAN-based VSR work (Perez-Pellitero ´ et al., 2018) and UVT work (Park et al., 2019), it leads to an undesirable smooth over spatial detail and temporal changes in outputs. Likewise, the $L ^ { 2 }$ temporal metric represents a sub-optimal way to quantify temporal coherence and perceptual metrics that evaluate natural temporal changes are unavailable up to now. We work on this open issue, propose two improved temporal metric and demonstrate the advantages of temporal self-supervision over direct temporal losses.
+
+Previous work, e.g. tempoGAN (Xie et al., 2018) and vid2vid (Wang et al., 2018b), have proposed adversarial temporal losses to achieve time consistency. While tempoGAN employs a second temporal discriminator with multiple aligned frames to assess the realism of temporal changes, it is not suitable for videos, as it relies on ground truth motions and employs a single-frame processing that is sub-optimal for natural images. On the other hand, vid2vid focuses on paired video translations and proposes a video discriminator based on a conditional motion input that is estimated from the ??௧ିଵ ??௧ ??௧ାଵ { }x3 ?? { paired ground-truth sequences. We focus on more difficult unpaired translation tasks instead, and ?? ௧ିଵ Conditional LR Triplet ??௔ Static Triplet ?? or Stati demonstrate the gains in quality of our approach in the evaluation section. For tracking and optical ?? flow estimation, L2-based time-cycle losses (Wang et al., 2019) were proposed to constrain motions௧ିଵ ??௧ ௧ାଵ ௧ିଵ ௧ ௧ାଵ+ ??௧ିଵ ??௧ ??௧ାଵ ??௧ିଵor and tracked correspondences using symmetric video inputs. By optimizing indirectly via motionFrame- Original Triplet ??௚ Origi compensation or tracking, this loss improves the accuracy of the results. For video generation, weRecurrent  ??௧  ??௧ିଵ  ??௧ାଵ ??௧ ??௧ିଵ ??௧ାଵ ?? ??௧ିଵ ??௧ାଵ ??௧ିଵ propose a PP loss that also makes use of symmetric sequences. However, we directly constrainGenerator Warped Triplet ?? Warped Triplet ?? ?? ?? ??or the PP loss via the generated video content, which successfully improves the long-term temporal௪௚ consistency in the video results.
+
+# 3 LEARNING TEMPORALLY COHERENT CONDITIONAL VIDEO GENERATION
+
+Generative Network Before explaining the temporal self-supervision in more detail, we outline the gener௕→௔ative model to be supervised. Our generator networks ??௧produce image sequences in a frame-recurrent manner with the help of a recurrent generator G and a flow estimator $F$ ??௔→௕. We follow previous work (Sajjadi et al., 2018), where $\mathbf { G }$ produces output $g _ { t }$ in the target doDomain main B from conditional input frame $a _ { t }$ Dfrom the in??௕→௔ ??௕→௔put domain A, and recursively uses the previous generated output $g _ { t - 1 }$ . $F$ ௕→௔is trained to estimate the motion $v _ { t }$ between $a _ { t - 1 }$ and $a _ { t }$ ௧, which is then used as a motion compensation that aligns $g _ { t - 1 }$ ??௔→௕to the current frame. This procedure, also shown in Fig. 2a), can be summarized as: $g _ { t } = \mathbf { G } ( a _ { t } , W ( g _ { t - 1 } , v _ { t } ) { \bar { ) } }$ , where $v _ { t } =$ $\Gamma ( a _ { t - 1 } , a _ { t } )$ and $W$ is the warping operation. While one generator is enough to map data from A to B for paired tasks such as VSR, unpaired generation requires a second generator to establish cycle consis??௕→௔tency. (Zhu et al., 2017). In the UVT task, we use two recurrent generators, mapping from domain A to B and back. As shown in Fig. 2b), given $g _ { t } ^ { a  b } = { \bf G } _ { \mathrm { a b } } ( a _ { t } , \tilde { W ( } g _ { t - 1 } ^ { a  b } , v _ { t } ) )$ ??௔, we can use $a _ { t }$ as the labeled data of $g _ { t } ^ { a  b  a } = \mathrm { G } _ { \mathrm { b a } } ( g _ { t } ^ { a  b } , W ( g _ { t - 1 } ^ { a  b  a } , v _ { t } ) )$ to enforce consistency. A ResNet architecture ?? ??௔→௕ ௔→௕is used for the VSR generator G and a encoder-decoder structure is applied to UVT generators and $F$ . We intentionally keep generators simple and in line with previous work, in order to demonstrate Domain BDomain Athe advantages of the temporal self-supervision that we will explain in the following paragraphs.
+
+![](images/d8939c57ff2dc4a527ca5c706e6842a6524a721dd57fcca23cfdf14fe593d6ed.jpg)  
+Figure 2: a) G. b) The UVT cycle link using recurrent G.
+
+௧Spatio-Temporal Adversarial Self-Supervision The central ??௧ିଵbuilding block of our approach is a novel spatio-temporal discriminator $D _ { s , t }$ ௧ ??that receives triplets of frames. This contrasts with typically used spatial discriminators which supervise only a single image. By concatenating multiple adjacent frames along Frame-the channel dimension, the frame triplets form an important building block for learning because they can provide networks with gradient information regarding the realism of spatial structures as well as short-term temporal information, such as first- and second-order time derivatives.
+
+![](images/052b616e4fb48fb3def3ba66b9c51aa82cf36a6ba4cb28e680a8554b41f0a7a9.jpg)  
+Figure 3: Conditional VSR $D _ { s , t }$
+
+We propose a $D _ { s , t }$ architecture, illustrated in Fig. 3 and Fig. 4, that primarily receives two types of triplets: three adjacent frames and the corresponding warped ones. We warp later frames backward and previous ones forward. While original frames contain the full spatio-temporal information, warped frames more easily yield temporal information with their aligned content. For the input variants we use the following notation: $\begin{array} { r c l } { \operatorname { I } _ { g } } & { = } & { \big \{ g _ { t - 1 } , g _ { t } , g _ { t + 1 } \big \} , \operatorname { I } _ { b } } & { = } & { \big \{ b _ { t - 1 } , b _ { t } , b _ { t + 1 } \big \} } \end{array}$ ; $\operatorname { I } _ { w g } = \{ W ( g _ { t - 1 } , v _ { t } ) , g _ { t } , W ( g _ { t + 1 } , v _ { t } ^ { \prime } ) \}$ , $\operatorname { I } _ { w b } = \{ W ( b _ { t - 1 } , v _ { t } ) , b _ { t } , W ( b _ { t + 1 } , v _ { t } ^ { \prime } ) \}$ .
+
+For VSR tasks, $D _ { s , t }$ should guide the generator to learn the correlation between LR inputs and highresolution (HR) targets. Therefore, three LR frames $\mathrm { I } _ { a } = \{ a _ { t - 1 } , a _ { t } , a _ { t + 1 } \}$ from the input domain are used as a conditional input. The input of $D _ { s , t }$ can be summarized as $\mathrm { I } _ { s , t } ^ { b } = \left\{ \mathrm { I } _ { b } , \mathrm { I } _ { w b } , \mathrm { I } _ { a } \right\}$ labelled as real and the generated inputs $\mathrm { I } _ { s , t } ^ { g } = \{ \mathrm { I } _ { g } , \mathrm { I } _ { w g } , \mathrm { I } _ { a } \}$ labelled as fake. In this way, the conditional $D _ { s , t }$ will penalize $G$ if $\mathrm { I } _ { g }$ contains less spatial details or unrealistic artifacts according to $\mathrm { I } _ { a } , \mathrm { I } _ { b }$ . At the same time, temporal relationships between the generated images $\mathrm { I } _ { w g }$ and those of the ground truth $\mathrm { I } _ { w b }$ should match. With our setup, the discriminator profits from the warped frames to classify realistic and unnatural temporal changes, and for situations where the motion estimation is less accurate, the discriminator can fall back to the original, i.e. not warped, images.
+
+For UVT tasks, we demonstrate that the temporal cycle??௧ିଵ Conditional LR Triplet ??௧consistency between different domains can be established using ௧  ??the supervision of unconditional spatio-temporal discriminators. ??௧ିଵ ??௧ ??௧ାଵ ??௧ିଵ ??௧ ??௧ାଵ+This is in contrast to previous work which focuses on the generaOriginal Triplet ??௚ Original Triplet ??௕tive networks to form spatio-temporal cycle links. Our approach Frame- ?? ?? ?? ??actually yields improved results, as we will show below, and Generator ௧ ?? ?? ௧ ??+ ??Fig. 1 shows a preview of the quality that can be achieved using Warped Triplet ??௪௚ Warped Triplet ??௪௕spatio-temporal discriminators. In practice, we found it crucial to ensure that generators first learn reasonable spatial features, and ௦,௧only then improve their temporal correlation. Therefore, different to the $D _ { s , t }$ 0/1௧ of VST that always receives 3 concatenated triplets as an input, the unconditional $D _ { s , t }$ of UVT only takes one triplet at a time. Focusing on the generated data, the input for a single batch can either be a static triplet of $\operatorname { I } _ { s g } = \{ g _ { t } , g _ { t } , g _ { t } \}$ , the warped triplet $\mathrm { I } _ { w g }$ , or the original triplet $\mathrm { I } _ { g }$ . The same holds for the reference data of the target domain, as shown in Fig. 4. With sufficient but complex information contained in these triplets, transition techniques are applied so that ??௕→௔  ??the network can consider the spatio-temporal information step by step, i.e., we initially start with $100 \%$ static triplets $\mathrm { I } _ { s g }$ ?? ??௔→௕as the input. Then, over the course of training, $2 5 \%$ ௔→௕ of them transition to $\mathrm { I } _ { w g }$ triplets with simpler temporal information, with another $2 5 \%$ ??௧ transition to $\mathrm { I } _ { g }$ ??௧afterwards, leading to a $( 5 0 \% , 2 5 \% , 2 5 \% )$ ?? ??௔→௕ ௔→௕ ௔→௕ ??௔→௕ distribution of triplets. Details of the transition calculations are given in Appendix D. Here, the warping is again performed via $F$ .
+
+![](images/1eed40303445f379f5ce1602d098baa64521a3784590cae31bfecc201319f62a.jpg)  
+Figure 4: Unconditional UVT $D _ { s , t }$
+
+??While non-adversarial training typically employs loss formulations with static goals, the GAN train??௧ିଵ   ??௧ିଵ ??௕→௔ing yields dynamic goals due to discriminative networks discovering the learning objectives over ??௧??௕→௔  ??௧??௧௕→௔the course of the training run. Therefore, their inputs have strong influence on the training process ??and the final results. Modifying the inputs in a controlled manner can lead to different results and ௔→௕ substantial improvements if done correctly, as will be shown in Sec. 4. Although the proposed concatenation of several frames seems like a simple change that has been used in a variety of projects, it is an important operation that allows discriminators to understand spatio-temporal data distributions. As will be shown below, it can effectively reduce temporal problems encountered by spatial GANs. While $L ^ { 2 }$ −based temporal losses are widely used in the field of video generation, the spatiotemporal adversarial loss is crucial for preventing the inference of blurred structures in multi-modal data-sets. Compared to GANs using multiple discriminators, the single $D _ { s , t }$ network can learn to balance the spatial and temporal aspects from the reference data and avoid inconsistent sharpness as well as overly smooth results. Additionally, by extracting shared spatio-temporal features, it allows for smaller network sizes.
+
+Self-Supervision for Long-term Temporal Consistency When relying on a previous output as input, i.e., for frame-recurrent architectures, generated structures easily accumulate frame by frame. In an adversarial training, generators learn to heavily rely on previously generated frames and can easily converge towards strongly reinforcing spatial features over longer periods of time. For videos, this especially occurs along directions of motion, and these solutions can be seen as a special form of temporal mode collapse. We have noticed this issue in a variety of recurrent architectures, examples are shown in Fig. 5 a) and the Dst in Fig. 1. While this issue could be alleviated by training with longer sequences, we generally want generators to be able to work with sequences of arbitrary length for inference. To address this inherent problem of recurrent generators, we propose a new bi-directional “Ping-Pong” loss. For natural videos, a sequence with forward order as well as its reversed counterpart offer valid information. Thus, from any input of length $n$ , we can construct a symmetric PP sequence in form of $a _ { 1 } , . . . a _ { n - 1 } , a _ { n } , a _ { n - 1 } , . . . a _ { 1 }$ as shown in Fig. 5. When inferring this in a frame-recurrent manner, the generated result should not strengthen any invalid features from frame to frame. Rather, the result should stay close to valid information and be symmetric, i.e., the forward result $g _ { t } = G ( a _ { t } , g _ { t - 1 } )$ and the one generated from the reversed part, $g _ { t } ^ { \prime } = G ( a _ { t } ,$ , $g _ { t + 1 } ^ { \prime } )$ , should be identical. Based on this observation, we train our networks with extended PP sequences and constrain the generated outputs from both “legs” to be the same using the loss: $\begin{array} { r } { \dot { \mathcal { L } _ { p p } } = \sum _ { i = 1 } ^ { n - 1 } \| g _ { t } - g _ { t } ^ { \prime } \| _ { 2 } . } \end{array}$ . Note that in contrast to the generator loss, the $L ^ { 2 }$ norm is a correct choice here: We are not faced with multi-modal data where an $L ^ { 2 }$ norm would lead to undesirable averaging, but rather aim to constrain the recurrent generator to its own, unique version over time. The PP terms provide constraints for short term consistency via $\lVert g _ { n - 1 } - { g _ { n - 1 } } ^ { \prime } \rVert _ { 2 }$ , while terms such as $\left\| g _ { 1 } - { g _ { 1 } } ^ { \prime } \right\| _ { 2 }$ prevent long-term drifts of the results. As shown in Fig. 5(b), this PP loss successfully removes drifting artifacts while appropriate high-frequency details are preserved. In addition, it effectively extends the training data set, and as such represents a useful form of data augmentation. A comparison is shown in Appendix E to disentangle the effects of the augmentation of PP sequences and the temporal constrains. The results show that the temporal constraint is the key to reliably suppressing the temporal accumulation of artifacts, achieving consistency, and allowing models to infer much longer sequences than seen during training.
+
+![](images/64bd79b06aa3430cf07173c79237823cb2de0aca7129f0963536a4257a19bc0e.jpg)  
+Figure 5: a) Result without PP loss. The VSR network is trained with a recurrent frame-length of 10. When inference on long sequences, frame 15 and latter frames of the foliage scene show the drifting artifacts. b) Result trained with PP loss. These artifacts are removed successfully for the latter. c) The ground-truth image. With our PP loss (shown on the right), the $L ^ { 2 }$ distance between $g _ { t }$ and $g _ { t } ^ { \prime }$ is minimized to remove drifting artifacts and improve temporal coherence.
+
+Perceptual Loss Terms As perceptual metrics, both pre-trained NNs (Johnson et al., 2016; Wang et al., 2018a) and in-training discriminators (Xie et al., 2018) were successfully used in previous work. Here, we use feature maps from a pre-trained VGG-19 network (Simonyan & Zisserman, 2014), as well as $D _ { s , t }$ itself. In the VSR task, we can encourage the generator to produce features similar to the ground truth ones by increasing the cosine similarity between their feature maps. In UVT tasks without paired ground truth data, we still want the generators to match the distribution of features in the target domain. Similar to a style loss in traditional style transfer (Johnson et al., 2016), we here compute the $D _ { s , t }$ feature correlations measured by the Gram matrix instead. The feature maps of $D _ { s , t }$ contain both spatial and temporal information, and hence are especially well suited for the perceptual loss.
+
+Loss and Training Summary We now explain how to integrate the spatio-temporal discriminator into the paired and unpaired tasks. We use a standard discriminator loss for the $D _ { s , t }$ of VSR and a least-square discriminator loss for the $D _ { s , t }$ of UVT. Correspondingly, a non-saturated $\mathcal { L } _ { a d v }$ is used for the $G$ and $F$ of VSR, and a least-squares one is used for the UVT generators. As summarized in Table 1, $G$ and $F$ are trained with the mean squared loss $\mathcal { L } _ { \mathrm { c o n t e n t } }$ , adversarial losses $\mathcal { L } _ { a d v }$ , perceptual losses $\mathcal { L } _ { \phi }$ , the PP loss ${ \mathcal { L } } _ { \mathrm { P P } }$ , and a warping loss ${ \mathcal { L } } _ { \mathrm { w a r p } }$ , where again $g , b$ and $\Phi$ stand for generated samples, ground truth images and feature maps of VGG-19 or $D _ { s , t }$ . We only show losses for the mapping from A to B for UVT tasks, as the backward mapping simply mirrors the terms. We refer to our full model for both tasks as TecoGAN below.1 Training parameters and details are given in Appendix G.
+
+Table 1: Summary of loss terms.   
+
+<table><tr><td rowspan=1 colspan=1>Loss for</td><td rowspan=1 colspan=1>VSR,Ds,t</td><td rowspan=1 colspan=2>UVT,D,t</td></tr><tr><td rowspan=1 colspan=1>LDs,t</td><td rowspan=1 colspan=1>-Eb~Pb(b)[log D(b,t)]-Ea~pa(a)[log(1- D(1g,t))]</td><td rowspan=1 colspan=2>Eb~p(b)[D(b,t)-1]²+Ea~p(a)[D(1g,t)]²</td></tr><tr><td rowspan=1 colspan=1>Loss for</td><td rowspan=1 colspan=1>VSR,G&amp;F</td><td rowspan=1 colspan=2>UVT, Gab</td></tr><tr><td rowspan=1 colspan=1>LG,F</td><td rowspan=1 colspan=3>XcLcontent +XaLadv+XL+XpLPp+XwLwarp</td></tr><tr><td rowspan=1 colspan=1>Lcontent</td><td rowspan=1 colspan=1>llgt-btll2</td><td rowspan=1 colspan=1>1lg→b-a-all2+</td><td rowspan=1 colspan=1>1g-a6-bll2</td></tr><tr><td rowspan=1 colspan=1>Ladv</td><td rowspan=1 colspan=1>-Ea~pa(a)[log Ds,t(1g,t)]</td><td rowspan=1 colspan=2>-Ba~pa(aDg2</td></tr><tr><td rowspan=1 colspan=1>L</td><td rowspan=1 colspan=1>1.0-(1,t）*Φ（,t）/(1g,t）*,t</td><td rowspan=1 colspan=2>GM(Φ(g,t))-GM(Φ(,t))2</td></tr><tr><td rowspan=1 colspan=1>LPP</td><td rowspan=1 colspan=3>∑-1gt-9t/12</td></tr><tr><td rowspan=1 colspan=1>Lwarp</td><td rowspan=1 colspan=1>∑llat-W(at-1,F(at-1,at))ll2</td><td rowspan=1 colspan=2>0.0,apre-trainedFis used</td></tr></table>
+
+![](images/9db868c910ed005d8a35292520959b4b52a4e1ed137a7ae3b738d5781ea4c581.jpg)  
+Figure 6: In VSR of the foliage scene, adversarial models (ENet, DsOnly, DsDt, DsDtPP, TecoGANand TecoGAN) yield better perceptual quality than methods using $L ^ { 2 }$ loss (FRVSR and DUF). In temporal profiles on the right, DsDt, DsDtPP and TecoGAN show significantly less temporal discontinuities compared to ENet and DsOnly. The temporal information of our discriminators successfully suppresses these artifacts.
+
+# 4 ANALYSIS AND EVALUATION OF LEARNING OBJECTIVES
+
+In the following, we illustrate the effects of temporal supervision using two ablation studies. In the first one, models trained with ablated loss functions show how ${ \mathcal { L } } _ { \mathrm { a d v } }$ and $\mathcal { L } _ { \mathrm { P P } }$ change the overall learning objectives. Next, full UVT models are trained with different $D _ { s , t }$ inputs. This highlights how differently the corresponding discriminators converge to different spatio-temporal equilibriums, and the general importance of providing suitable data distributions from the target domain. While we provide qualitative and quantitative evaluations in the following, we also refer the reader to our supplemental html document 2, with video clips that more clearly highlight the temporal differences.
+
+Loss Ablation Study Below we compare variants of our full TecoGAN model to EnhanceNet (ENet) (Sajjadi et al., 2017), FRVSR (Sajjadi et al., 2018), and DUF (Jo et al., 2018) for VSR, and CycleGAN (Zhu et al., 2017) and RecycleGAN (Bansal et al., 2018) for UVT. Specifically, ENet and CycleGAN represent state-of-the-art single-image adversarial models without temporal information, FRVSR and DUF are state-of-the-art VSR methods without adversarial losses, and RecycleGAN is a spatial adversarial model with a prediction network learning the temporal evolution.
+
+For VSR, we first train a DsOnly model that uses a frame-recurrent $G$ and $F$ with a VGG-19 loss and only the regular spatial discriminator. Compared to ENet, which exhibits strong incoherence due to the lack of temporal information, DsOnly improves temporal coherence thanks to the framerecurrent connection, but there are noticeable high-frequency changes between frames. The temporal profiles of DsOnly in Fig. 6 and 8, correspondingly contain sharp and broken lines. When adding a temporal discriminator in addition to the spatial one $( D s D t )$ , this version generates more coherent results, and its temporal profiles are sharp and coherent. However, DsDt often produces the drifting artifacts discussed in Sec. 3, as the generator learns to reinforce existing details from previous frames to fool $D _ { s }$ with sharpness, and satisfying $D _ { t }$ with good temporal coherence in the form of persistent detail. While this strategy works for generating short sequences during training, the strengthening effect can lead to very undesirable artifacts for long-sequence inferences. By adding the self-supervision for long-term temporal consistency $\mathcal { L } _ { p p }$ , we arrive at the $D s D t P P$ model, which effectively suppresses these drifting artifacts with an improved temporal coherence. In Fig. 6 and Fig. 8, DsDtPP results in continuous yet detailed temporal profiles without streaks from temporal drifting. Although DsDtPP generates good results, it is difficult in practice to balance the generator and the two discriminators. The results shown here were achieved only after numerous runs manually tuning the weights of the different loss terms. By using the proposed $D _ { s , t }$ discriminator instead, we get a first complete model for our method, denoted as $T e c o G A N ^ { \odot }$ . This network is trained with a discriminator that achieves an excellent quality with an effectively halved network size, as illustrated on the right of Fig. 7. The single discriminator correspondingly leads to a significant reduction in resource usage. Using two discriminators requires ca. $70 \%$ more GPU memory, and leads to a reduced training performance by ca. $20 \%$ . The TecoGANmodel yields similar perceptual and temporal quality to DsDtPP with a significantly faster and more stable training.
+
+Since the TecoGANmodel requires less training resources, we also trained a larger generator with $50 \%$ more weights. In the following we will focus on this larger single-discriminator architecture with PP loss as our full TecoGAN model for VSR. Compared to the TecoGANmodel, it can generate more details, and the training process is more stable, indicating that the larger generator and $D _ { s , t }$ are more evenly balanced. Result images and temporal profiles are shown in Fig. 6 and Fig. 8. Video results are shown in Sec. 4 of the supplemental material.
+
+We also carry out a similar ablation study for the UVT task. Again, we start from a single-image GAN-based model, a CycleGAN variant which already has two pairs of spatial generators and discriminators. Then, we train the DsOnly variant by adding flow estimation via $F$ and extending the spatial generators to frame-recurrent ones. By augmenting the two discriminators to use the triplet inputs proposed in Sec. 3, we arrive at the Dst model with spatio-temporal discriminators, which does not yet use the PP loss. Although UVT tasks are substantially different from VSR tasks, the comparisons in Fig. 1 and Sec. 4.6 of our supplemental material yield similar conclusions. In these tests, we use renderings of 3D fluid simulations of rising smoke as our unpaired training data. These simulations are generated with randomized numerical simulations using a resolution of $6 4 ^ { 3 }$ for domain A and $2 5 6 ^ { \overline { { 3 } } }$ for domain B, and both are visualized with images of size $2 5 6 ^ { 2 }$ . Therefore, video translation from domain A to B is a tough task, as the latter contains significantly more turbulent and small-scale motions. With no temporal information available, the CycleGAN variant generates HR smoke that strongly flickers. The DsOnly model offers better temporal coherence by relying on its frame-recurrent input, but it learns a solution that largely ignores the current input and fails to keep reasonable spatio-temporal cycle-consistency links between the two domains. On the contrary, our $D _ { s , t }$ enables the Dst model to learn the correlation between the spatial and temporal aspects, thus improving the cycle-consistency. However, without $\mathcal { L } _ { p p }$ , the Dst model (like the DsDt model of VSR) reinforces detail over time in an undesirable way. This manifests itself as inappropriate smoke density in empty regions. Using our full TecoGAN model which includes $\mathcal { L } _ { p p }$ , yields the best results, with detailed smoke structures and very good spatio-temporal cycle-consistency.
+
+For comparison, a DsDtPP model involving a larger number of separate networks, i.e. four discriminators, two frame-recurrent generators and the $F$ , is trained. By weighting the temporal adversarial losses from Dt with 0.3 and the spatial ones from Ds with 0.5, we arrived at a balanced training run. Although this model performs similarly to the TecoGAN model on the smoke dataset, the proposed spatio-temporal $D _ { s , t }$ architecture represents a more preferable choice in practice, as it learns a natural balance of temporal and spatial components by itself, and requires fewer resources. Continuing along this direction, it will be interesting future work to evaluate variants, such as a shared $D _ { s , t }$ for both domains, i.e. a multi-class classifier network.
+
+Besides the smoke dataset, an ablation study for the Obama and Trump dataset from Fig. 1 shows a very similar behavior, as can be seen in the supplemental material.
+
+Spatio-temporal Adversarial Equilibriums Our evaluation so far highlights that temporal adversarial learning is crucial for achieving spatial detail that is coherent over time for VSR, and for enabling the generators to learn the spatio-temporal correlation between domains in UVT. Next, we will shed light on the complex spatio-temporal adversarial learning objectives by varying the information provided to the discriminator network. The following tests $D _ { s , t }$ networks that are identical apart from changing inputs, and we focus on the smoke dataset.
+
+![](images/af93ec4a535359819f607f8cfee938e78ffa2cb742b691444dca70d812a4031a.jpg)  
+Figure 7: Visual summary of VSR models. LPIPS $\mathbf { \widetilde { x } }$ -axis) measures spatial detail and temporal coherence is measured by tLP (y-axis) and tOF (bubble size with smaller as better). The middle graph zooms in the reddashed-box region on the left, containing models in our ablation study. The right graph shows network sizes.
+
+In order to learn the spatial and temporal features of the target domain as well as their correlation, the simplest input for $D _ { s , t }$ consists of only the original, unwarped triplets, i.e. $\{ \boldsymbol { \mathrm { I } } _ { g }$ or $\mathrm { I } _ { b } \big \}$ . Using these, we train a baseline model, which yields a sub-optimal quality: it lacks sharp spatial structures, and contains coherent but dull motions. Despite containing the full information, these input triplets prevent $D _ { s , t }$ from providing the desired supervision. For paired video translation tasks, the vid2vid network achieves improved temporal coherence by using a video discriminator to supervise the output sequence conditioned with the ground-truth motion. With no ground-truth data available, we train a vid2vid variant by using the estimated motions and original triplets, i.e $\{ \mathrm { I } _ { g } + F ( g _ { t - 1 } , g _ { t } ) + F ( g _ { t + 1 } , g _ { t } )$ or $\operatorname { I } _ { b } + F ( b _ { t - 1 } , b _ { t } ) + F ( b _ { t + 1 } , b _ { t } ) \big \}$ , as the input for $D _ { s , t }$ . However, the result do not significantly improve. The motions are only partially reliable, and hence don’t help for the difficult unpaired translation task. Therefore, the discriminator still fails to fully correlate spatial and temporal features. We then train a third model, concat, using the original triplets and the warped ones, i.e. $\{ \boldsymbol { \mathrm { I } } _ { g } + \boldsymbol { \mathrm { I } } _ { w g }$ or $\operatorname { I } _ { b } + \operatorname { I } _ { w b } \}$ . In this case, the model learns to generate more spatial details with a more vivid motion. I.e., the improved temporal information from the warped triplets gives the discriminator important cues. However, the motion still does not fully resemble the target domain. We arrive at our final TecoGAN model for UVT by controlling the composition of the input data: as outlined above, we first provide only static triplets $\{ \mathrm { I } _ { s g }$ or $\mathrm { ~ \bar { I } } _ { s b } \}$ , and then apply the transitions of warped triplets $\{ \mathrm { I } _ { w g }$ or $\mathrm { I } _ { w b } \}$ , and original triplets $\{ \boldsymbol { \mathrm { I } } _ { g }$ or $\mathrm { I } _ { b } \big \}$ over the course of training. In this way, the network can first learn to extract spatial features, and build on them to establish temporal features. Finally, discriminators learn features about the correlation of spatial and temporal content by analyzing the original triplets, and provide gradients such that the generators learn to use the motion information from the input and establish a correlation between the motions in the two unpaired domains. Consequently, the discriminator, despite receiving only a single triplet at once, can guide the generator to produce detailed structures that move coherently. Video comparisons are shown in Sec 5. of the supplemental material.
+
+Results and Metric Evaluation While the visual results discussed above provide a first indicator of the quality our approach achieves, quantitative evaluations are crucial for automated evaluations across larger numbers of samples. Below we focus on the VSR task as ground-truth data is available in this case. We conduct user studies and present evaluations of the different models w.r.t. established spatial metrics. We also motivate and propose two novel temporal metrics to quantify temporal coherence. A visual summary is shown in Fig. 7.
+
+For evaluating image SR, Blau & Michaeli (2018) demonstrated that there is an inherent trade-off between the perceptual quality of the result and the distortion measured with vector norms or lowlevel structures such as PSNR and SSIM. On the other hand, metrics based on deep feature maps such as LPIPS (Zhang et al., 2018) can capture more semantic similarities. We measure the PSNR and LPIPS using the Vid4 scenes. With a PSNR decrease of less than 2dB over DUF which has twice the model size of ours, TecoGAN outperforms all methods by more than $40 \%$ on LPIPS.
+
+Table 2: Averaged VSR metric evaluations for the Vid4 data set with the following metrics, PSNR: pixel-wise accuracy. LPIPS (AlexNet): perceptual distance to the ground truth. T-diff: pixel-wise differences of warped frames. tOF: pixel-wise distance of estimated motions. tLP: perceptual distance between consecutive frames. User study: Bradley-Terry scores (Bradley & Terry, 1952). Performance is averaged over 500 images up-scaled from 320x134 to $1 2 8 0 \mathrm { x } 5 3 6$ . More details can be found in Appendix B and C.   
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>PSNR↑</td><td rowspan=1 colspan=1>LPIPS↓×10</td><td rowspan=1 colspan=1>T-diff×100</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>LP↓×100</td><td rowspan=1 colspan=1>User个Study</td><td rowspan=1 colspan=1>ModelSize(M↓</td><td rowspan=1 colspan=1>ProcessingTime(ms/rame)</td></tr><tr><td rowspan=1 colspan=1>DsOnly</td><td rowspan=1 colspan=1>24.14</td><td rowspan=1 colspan=1>1.727</td><td rowspan=1 colspan=1>6.852</td><td rowspan=1 colspan=1>2.157</td><td rowspan=1 colspan=1>2.160</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>0.8(G)+1.7(F)</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>DsDt</td><td rowspan=1 colspan=1>24.75</td><td rowspan=1 colspan=1>1.770</td><td rowspan=1 colspan=1>5.071</td><td rowspan=1 colspan=1>2.198</td><td rowspan=1 colspan=1>0.614</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>0.8(G)+1.7(F)</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>DsDtPP</td><td rowspan=1 colspan=1>25.77</td><td rowspan=1 colspan=1>1.733</td><td rowspan=1 colspan=1>4.369</td><td rowspan=1 colspan=1>2.103</td><td rowspan=1 colspan=1>0.489</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>0.8(G)+1.7(F)</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>TecoGAN</td><td rowspan=1 colspan=1>25.89</td><td rowspan=1 colspan=1>1.743</td><td rowspan=1 colspan=1>4.076</td><td rowspan=1 colspan=1>2.082</td><td rowspan=1 colspan=1>0.718</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>0.8(G)+1.7(F)</td><td rowspan=1 colspan=1>37.07</td></tr><tr><td rowspan=1 colspan=1>TecoGAN</td><td rowspan=1 colspan=1>25.57</td><td rowspan=1 colspan=1>1.623</td><td rowspan=1 colspan=1>4.961</td><td rowspan=1 colspan=1>1.897</td><td rowspan=1 colspan=1>0.668</td><td rowspan=1 colspan=1>3.258</td><td rowspan=1 colspan=1>1.3(G)+1.7(F)</td><td rowspan=1 colspan=1>41.92</td></tr><tr><td rowspan=1 colspan=1>ENet</td><td rowspan=1 colspan=1>22.31</td><td rowspan=1 colspan=1>2.458</td><td rowspan=1 colspan=1>9.281</td><td rowspan=1 colspan=1>4.009</td><td rowspan=1 colspan=1>4.848</td><td rowspan=1 colspan=1>1.616</td><td rowspan=1 colspan=1>0.8</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>FRVSR</td><td rowspan=1 colspan=1>26.91</td><td rowspan=1 colspan=1>2.506</td><td rowspan=1 colspan=1>3.648</td><td rowspan=1 colspan=1>2.090</td><td rowspan=1 colspan=1>0.957</td><td rowspan=1 colspan=1>2.600</td><td rowspan=1 colspan=1>0.8(SRNet)+1.7(F)</td><td rowspan=1 colspan=1>36.95</td></tr><tr><td rowspan=1 colspan=1>DUF</td><td rowspan=1 colspan=1>27.38</td><td rowspan=1 colspan=1>2.607</td><td rowspan=1 colspan=1>3.298</td><td rowspan=1 colspan=1>1.588</td><td rowspan=1 colspan=1>1.329</td><td rowspan=1 colspan=1>2.933</td><td rowspan=1 colspan=1>6.2</td><td rowspan=1 colspan=1>942.21</td></tr><tr><td rowspan=1 colspan=1>Bi-cubic</td><td rowspan=1 colspan=1>23.66</td><td rowspan=1 colspan=1>5.036</td><td rowspan=1 colspan=1>3.152</td><td rowspan=1 colspan=1>5.578</td><td rowspan=1 colspan=1>2.144</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+Table 3: For the Obama&Trump dataset, the averaged tLP and tOF evaluations closely correspond to our user studies. The table below summarizes user preferences as Bradley-Terry scores.
+
+<table><tr><td rowspan=1 colspan=1>UVT scenes</td><td rowspan=1 colspan=2> Trump-→Obama</td><td rowspan=1 colspan=2> Obama-→Trump</td><td rowspan=1 colspan=2>AVG</td><td rowspan=1 colspan=2>User Studies ↑, ref. to</td></tr><tr><td rowspan=1 colspan=1> metrics</td><td rowspan=1 colspan=1>tLP↓</td><td rowspan=1 colspan=1>tOF↓</td><td rowspan=1 colspan=1>tLP↓</td><td rowspan=1 colspan=1>tOF↓</td><td rowspan=1 colspan=1>tLP↓</td><td rowspan=1 colspan=1>tOF↓</td><td rowspan=1 colspan=1> original input</td><td rowspan=1 colspan=1> arbitrary target</td></tr><tr><td rowspan=1 colspan=1>CycleGAN</td><td rowspan=1 colspan=1>0.0176</td><td rowspan=1 colspan=1>0.7727</td><td rowspan=1 colspan=1>0.0277</td><td rowspan=1 colspan=1>1.1841</td><td rowspan=1 colspan=1>0.0234</td><td rowspan=1 colspan=1>0.9784</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=1>0.0</td></tr><tr><td rowspan=1 colspan=1>RecycleGAN</td><td rowspan=1 colspan=1>0.0111</td><td rowspan=1 colspan=1>0.8705</td><td rowspan=1 colspan=1>0.0248</td><td rowspan=1 colspan=1>1.1237</td><td rowspan=1 colspan=1>0.0179</td><td rowspan=1 colspan=1>0.9971</td><td rowspan=1 colspan=1>0.994</td><td rowspan=1 colspan=1>0.202</td></tr><tr><td rowspan=1 colspan=1>TecoGAN</td><td rowspan=1 colspan=1>0.0120</td><td rowspan=1 colspan=1>0.6155</td><td rowspan=1 colspan=1>0.0191</td><td rowspan=1 colspan=1>0.7670</td><td rowspan=1 colspan=1>0.0156</td><td rowspan=1 colspan=1>0.6913</td><td rowspan=1 colspan=1>1.817</td><td rowspan=1 colspan=1>0.822</td></tr></table>
+
+While traditional temporal metrics based on vector norm differences of warped frames, e.g. T-diff, can be easily deceived by very blurry results, e.g. bi-cubic interpolated ones, we propose to use a tandem of two new metrics, tOF and tLP, to measure the consistence over time. tOF measures the pixel-wise difference of motions estimated from sequences, and tLP measures perceptual changes over time using deep feature map:
+
+$$
+\begin{array} { r } { \mathsf { t O F } = \| O F ( b _ { t - 1 } , b _ { t } ) - O F ( g _ { t - 1 } , g _ { t } ) \| _ { 1 } \mathrm { a n d } \mathrm { t L P } = \| L P ( b _ { t - 1 } , b _ { t } ) - L P ( g _ { t - 1 } , g _ { t } ) \| _ { 1 } \mathrm { , } } \end{array}
+$$
+
+where $O F$ represents an optical flow estimation with LucasKanade (1981) and $\boldsymbol { L P }$ is the perceptual LPIPS metric. In tLP, the behavior of the reference is also considered, as natural videos exhibit a certain degree of changes over time. In conjunction, both pixel-wise differences and perceptual changes are crucial for quantifying realistic temporal coherence. While they could be combined into a single score, we list both measurements separately, as their relative importance could vary in different application settings. Our evaluation with these temporal metrics in Table 2 shows that all temporal adversarial models outperform spatial adversarial ones, and the full TecoGAN model performs very well: With a large amount of spatial detail, it still achieves good temporal coherence, on par with non-adversarial methods such as DUF and FRVSR. For VSR, we have confirmed these automated evaluations with several user studies. Across all of them, we find that the majority of the participants considered the TecoGAN results to be closest to the ground truth.
+
+For the UVT tasks, where no ground-truth data is available, we can still evaluate tOF and tLP metrics by comparing the motion and the perceptual changes of the output data w.r.t. the ones from the input data , i.e., $\begin{array} { r } { { \mathrm { t O F } } =  O F ( a _ { t - 1 } , a _ { t } ) - O F ( g _ { t - 1 } ^ { a  b } , g _ { t } ^ { a  b } )  _ { 1 } } \end{array}$ and $\begin{array} { r } { \mathrm { t L P } { = \left. { L P ( a _ { t - 1 } , a _ { t } ) - L P ( g _ { t - 1 } ^ { a \to b } , g _ { t } ^ { a \to b } ) } \right. } _ { 1 } . } \end{array}$ . With sharp spatial features and coherent motion, TecoGAN outperforms previous work on the Obama&Trump dataset, as shown in Table 3, although it is worth to point out that the tOF is less informative in this case, as the motion in the target domain is not necessarily pixel-wise aligned with the input. Overall, TecoGAN achieves good tLP scores thanks to its temporal coherence, on par with RecycleGAN, and its spatial detail is on par with CycleGAN. As for VSR, a perceptual evaluation by humans in the right column of Table 3 confirms our metric evaluations for the UVT task (details in Appendix C).
+
+# 5 CONCLUSIONS AND DISCUSSION
+
+In paired as well as unpaired data domains, we have demonstrated that it is possible to learn stable temporal functions with GANs thanks to the proposed discriminator architecture and PP loss. We have shown that this yields coherent and sharp details for VSR problems that go beyond what can be achieved with direct supervision. In UVT, we have shown that our architecture guides the training process to successfully establish the spatio-temporal cycle consistency between two domains. These results are reflected in the proposed metrics and user studies.
+
+While our method generates very realistic results for a wide range of natural images, our method can generate temporally coherent yet sub-optimal details in certain cases such as under-resolved faces and text in VSR, or UVT tasks with strongly different motion between two domains. For the latter case, it would be interesting to apply both our method and motion translation from concurrent work (Chen et al., 2019). This can make it easier for the generator to learn from our temporal self supervision. In our method, the interplay of the different loss terms in the non-linear training procedure does not provide a guarantee that all goals are fully reached every time. However, we found our method to be stable over a large number of training runs, and we anticipate that it will provide a very useful basis for wide range of generative models for temporal data sets.
+
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+
+# APPENDIX
+
+In the following, we first provide qualitative analysis(Appendix A) using multiple results that are mentioned but omitted in our main document due to space constraints. We then explain details of the metrics and present the quantitative analysis based on them(Appendix B). The conducted user studies are in support of our TecoGAN network and proposed temporal metrics (Appendix C). Then, we give technical details of our spatio-temporal discriminator (Sec. D), details of network architectures and training parameters (Appendix F, Appendix G). In the end, we discuss the performance of our approach in Appendix H.
+
+# A QUALITATIVE ANALYSIS
+
+For the VSR task, we test our model on a wide range of video data, including the generally used Vid4 dataset shown in Fig. 8 and 12, detailed scenes from the movie Tears of Steel (ToS, 2011) shown in Fig. 12, and others shown in Fig. 9. As mentioned in the main document, the TecoGAN model is trained with down-sampled inputs and it can similarly work with original images that were not down-sampled or filtered, such as a data-set of real-world photos (Liao et al., 2015). In Fig. 10, we compared our results to two other methods (Liao et al., 2015; Tao et al., 2017) that have used the same dataset. With the help of adversarial learning, our model is able to generate improved and realistic details in down-sampled images as well as captured images.
+
+![](images/6b759851a3116676a7d9dc498d137173bd6b475805d321ae27059b76614e0dfd.jpg)  
+Figure 8: VSR temporal profile comparisons of the calendar scene (time shown along y-axis). TecoGAN models lead to natural temporal progressions, and our final model closely matches the desired ground truth behavior over time.
+
+![](images/a8466aa454e0aad5d600d9d1470c1101413c56f45dcf2295ba4df3a7c35c3a36.jpg)  
+Figure 9: Additional VSR comparisons. The TecoGAN model generates sharp details in both scenes.
+
+![](images/4ffbf6edb04cd10c0d7b17a07f2730dea8d4a720b53748c3fe5fe906d3bdd2c3.jpg)  
+Liao et al. (2015) Ours Liao2015 Ours Tao et al. (2017) Figure 10: Comparisons for VSR of captured images.
+
+![](images/025368e02a2e75666bcb59c052b6816e8d3b435c8aeaef211c8f8402e7ae2af9.jpg)  
+Figure 11: Results of UVT tasks on different datasets.
+
+![](images/93134bc886523f5cebb6c8b1f9855df946a7a5ec86188b4a8fdc498e6e0dee2e.jpg)  
+Figure 12: Detail views of the VSR results of ToS scenes (first three columns) and Vid4 scenes (two right-most columns) with comparisons.
+
+Table 4: Metrics evaluated for the VSR Vid4 scenes.   
+
+<table><tr><td>PSNR↑</td><td>BIC</td><td>ENet</td><td>FRVSR</td><td>DUF</td><td>TecoGAN</td><td>TecoGAN</td><td>DsOnly</td><td></td><td>DsDt</td><td>DsDtPP</td></tr><tr><td>calendar</td><td>20.27</td><td>19.85</td><td>23.86</td><td>24.07 26.45</td><td>23.21</td><td>23.35 25.13</td><td></td><td>22.23</td><td>22.76</td><td>22.95</td></tr><tr><td>foliage</td><td>23.57 24.82</td><td>21.15 23.36</td><td>26.35 27.71</td><td>28.25</td><td>24.26 26.78</td><td>26.94</td><td></td><td>22.33 25.86</td><td>22.73 26.52</td><td>25.00 27.03</td></tr><tr><td>city walk</td><td>25.84</td><td>24.90</td><td>29.56</td><td>30.58</td><td>28.11</td><td>28.14</td><td></td><td>26.49</td><td>27.37</td><td>28.14</td></tr><tr><td>average</td><td>23.66</td><td>22.31</td><td>26.91</td><td>27.38</td><td>25.57</td><td>25.89</td><td></td><td>24.14</td><td>24.75</td><td>25.77</td></tr><tr><td>LPIPS↓×10 calendar</td><td>BIC 5.935</td><td>ENet 2.191</td><td>FRVSR 2.989</td><td>DUF 3.086</td><td>TecoGAN 1.511</td><td>TecoGAN 2.142</td><td></td><td>DsOnly 1.532</td><td>DsDt 2.111</td><td>DsDtPP 2.112</td></tr><tr><td>foliage</td><td>5.338</td><td>2.663</td><td>3.242</td><td>3.492</td><td>1.902</td><td></td><td>1.984</td><td>2.113</td><td>2.092</td><td>1.902</td></tr><tr><td>city</td><td>5.451</td><td>3.431</td><td>2.429</td><td>2.447</td><td>2.084</td><td>1.940</td><td></td><td>2.120</td><td>1.889</td><td>1.989</td></tr><tr><td>walk</td><td>3.655</td><td>1.794</td><td>1.374</td><td>1.380</td><td>1.106</td><td>1.011</td><td></td><td>1.215</td><td>1.057</td><td>1.051</td></tr><tr><td>average</td><td>5.036</td><td>2.458</td><td>2.506</td><td>2.607</td><td>1.623</td><td>1.743</td><td></td><td>1.727</td><td>1.770</td><td>1.733</td></tr><tr><td>tOF↓×10</td><td>BIC</td><td>ENet</td><td>FRVSR</td><td>DUF</td><td>TecoGAN</td><td>TecoGAN</td><td></td><td>DsOnly</td><td>DsDt</td><td>DsDtPP</td></tr><tr><td>calendar</td><td>4.956</td><td>3.450</td><td>1.537</td><td>1.134</td><td>1.342</td><td>1.403</td><td></td><td>1.609</td><td>1.683</td><td>1.583</td></tr><tr><td>foliage</td><td>4.922</td><td>3.775</td><td>1.489</td><td>1.356</td><td>1.238</td><td>1.444</td><td></td><td>1.543</td><td>1.562</td><td>1.373</td></tr><tr><td>city</td><td>7.967</td><td>6.225</td><td>2.992</td><td>1.724</td><td>2.612</td><td>2.905</td><td></td><td>2.920</td><td>2.936</td><td>3.062</td></tr><tr><td>walk</td><td>5.150</td><td>3.203</td><td>2.569</td><td>2.127</td><td>2.571</td><td>2.765</td><td></td><td>2.745</td><td>2.796</td><td>2.649</td></tr><tr><td>average</td><td>5.578</td><td>4.009</td><td>2.090</td><td>1.588</td><td>1.897</td><td>2.082</td><td></td><td>2.157</td><td>2.198</td><td>2.103</td></tr><tr><td>tLP↓×100</td><td>BIC</td><td>ENet</td><td>FRVSR</td><td>DUF</td><td>TecoGAN</td><td>TecoGAN</td><td></td><td>DsOnly</td><td>DsDt</td><td>DsDtPP</td></tr><tr><td>calendar</td><td>3.258</td><td>2.957</td><td>1.067</td><td>1.603</td><td>0.165</td><td>1.087</td><td></td><td>0.872</td><td>0.764</td><td>0.670</td></tr><tr><td>foliage city</td><td>2.434 2.193</td><td>6.372 7.953</td><td>1.644 0.752</td><td>2.034 1.399</td><td>0.894 0.974</td><td>0.740 0.347</td><td></td><td>3.422 2.660</td><td>0.493 0.490</td><td>0.454 0.140</td></tr><tr><td>walk</td><td>0.851</td><td>2.729</td><td>0.286</td><td>0.307</td><td>0.653</td><td>0.635</td><td></td><td>1.596</td><td>0.697</td><td>0.613</td></tr><tr><td>average</td><td>2.144</td><td>4.848</td><td>0.957</td><td>1.329</td><td>0.668</td><td>0.718</td><td></td><td>2.160</td><td>0.614</td><td>0.489</td></tr><tr><td>T-diff↓×100</td><td>BIC</td><td>ENet</td><td>FRVSR</td><td>DUF</td><td>TecoGAN</td><td>TecoGAN</td><td>DsOnly</td><td>DsDt</td><td>DsDtPP</td><td></td></tr><tr><td>calendar</td><td>2.271</td><td>9.153</td><td>3.212</td><td>2.750</td><td>4.663</td><td>3.496</td><td>6.287</td><td>4.347</td><td></td><td>GT</td></tr><tr><td>foliage</td><td>3.745</td><td>11.997</td><td>3.478</td><td>3.115</td><td>5.674</td><td>4.179</td><td></td><td></td><td>4.167</td><td>6.478</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>8.961</td><td>6.068</td><td>4.548</td><td>4.396</td></tr><tr><td>city</td><td>1.974</td><td>7.788</td><td>2.452</td><td>2.244</td><td>3.528</td><td>2.965</td><td>4.929</td><td>3.525</td><td>2.991</td><td>4.282</td></tr><tr><td>walk</td><td>4.101</td><td>7.576</td><td>5.028</td><td>4.687</td><td>5.460</td><td>5.234</td><td>6.454</td><td>5.714</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>5.305</td><td>5.525</td></tr><tr><td>average</td><td>3.152</td><td>9.281</td><td>3.648</td><td>3.298</td><td>4.961</td><td>4.076</td><td>6.852</td><td>5.071</td><td>4.369</td><td>5.184</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+For UVT tasks, we train models for Obama and Trump translations, LR- and HR- smoke simulation translations, as well as translations between smoke simulations and real-smoke captures. While smoke simulations usually contain strong numerical viscosity with details limited by the simulation resolution, the real smoke, captured using the setup from Eckert et al. (2018), contains vivid fluid motions with many vortices and high-frequency details. As shown in Fig. 11, our method can be used to narrow the gap between simulations and real-world phenomenon.
+
+# B METRICS AND QUANTITATIVE ANALYSIS
+
+Spatial Metrics We evaluate all VSR methods with PSNR together with the human-calibrated LPIPS metric (Zhang et al., 2018). While higher PSNR values indicate a better pixel-wise accuracy, lower LPIPS values represent better perceptual quality and closer semantic similarity. Mean values of the Vid4 scenes Liu & Sun (2011) are shown on the top of Table 4. Trained with direct vector norms losses, FRVSR and DUF achieve high PSNR scores. However, the undesirable smoothing induced by these losses manifests themselves in larger LPIPS distances. ENet, on the other hand, with no information from neighboring frames, yields the lowest PSNR and achieves an LPIPS score that is only slightly better than DUF and FRVSR. TecoGAN model with adversarial training achieves an excellent LPIPS score, with a PSNR decrease of less than 2dB over DUF, which is very reasonable, since PSNR and perceptual quality were shown to be anti-correlated (Blau & Michaeli, 2018), especially in regions where PSNR is very high. Based on good perceptual quality and reasonable pixel-wise accuracy, TecoGAN outperforms all other methods by more than $40 \%$ for LPIPS.
+
+Temporal Metrics For both VSR and UVT, evaluating temporal coherence without ground-truth motion is a very challenging problem. The metric $T \ – d i f f = \left. g _ { t } - W ( g _ { t - 1 } , v _ { t } ) \right. _ { 1 }$ was used by Chen et al. (2017) as a rough assessment of temporal differences. As shown on bottom of Table 4, T-diff, due to its local nature, is easily deceived by blurry method such as the bi-cubic interrelation and can not correlate well with visual assessments of coherence. By measuring the pixel-wise motion difference using tOF in together with the perceptual changes over time using tLP, we show the temporal evaluations for the VSR task in the middle of Table 4. Not surprisingly, the results of ENet show larger errors for all metrics due to their strongly flickering content. Bi-cubic up-sampling, DUF, and FRVSR achieve very low T-diff errors due to their smooth results, representing an easy, but undesirable avenue for achieving coherency. However, the overly smooth changes of the former two are identified by the tLP scores.While our DsOnly model generates sharper results at the expense of temporal coherence, it still outperforms ENet there. By adding temporal information to discriminators, our DsDt, $\mathrm { D s D t + P P }$ , TecoGANand TecoGAN improve in terms of temporal metrics. Especially the full TecoGAN model stands out here. For the UVT tasks, temporal motions are evaluated by comparing to the input sequence. With sharp spatial features and coherent motion, TecoGAN outperforms previous work on the Obama&Trump dataset, as shown in Table 3.
+
+Table 5: Metrics evaluated for the VSR of ToS.   
+
+<table><tr><td>PSNR↑</td><td>BIC</td><td>ENet</td><td>FRVSR</td><td>DUF</td><td>TecoGAN</td><td>tOF↓×10</td><td>BIC</td><td>ENet</td><td>FRVSR</td><td>DUF</td><td>TecoGAN</td></tr><tr><td>room</td><td>26.90</td><td>25.22</td><td>29.80</td><td>30.85</td><td>29.31</td><td>room</td><td>1.735</td><td>1.625</td><td>0.861</td><td>0.901</td><td>0.737</td></tr><tr><td>bridge</td><td>28.34</td><td>26.40</td><td>32.56</td><td>33.02</td><td>30.81</td><td>bridge</td><td>5.485</td><td>4.037</td><td>1.614</td><td>1.348</td><td>1.492</td></tr><tr><td>face</td><td>33.75</td><td>32.17</td><td>39.94</td><td>40.23</td><td>38.60</td><td>face</td><td>4.302</td><td>2.255</td><td>1.782</td><td>1.577</td><td>1.667</td></tr><tr><td>average</td><td>29.58</td><td>27.82</td><td>34.04</td><td>34.60</td><td>32.75</td><td>average</td><td>4.110</td><td>2.845</td><td>1.460</td><td>1.296</td><td>1.340</td></tr><tr><td>LPIPS↓×10</td><td>BIC</td><td>ENet</td><td>FRVSR</td><td>DUF</td><td>TecoGAN</td><td>tLP↓×100</td><td>BIC</td><td>ENet</td><td>FRVSR</td><td>DUF</td><td>TecoGAN</td></tr><tr><td>room</td><td>5.167</td><td>2.427</td><td>1.917</td><td>1.987</td><td>1.358</td><td>room</td><td>1.320</td><td>2.491</td><td>0.366</td><td>0.307</td><td>0.590</td></tr><tr><td>bridge</td><td>4.897</td><td>2.807</td><td>1.761</td><td>1.684</td><td>1.263</td><td>bridge</td><td>2.237</td><td>6.241</td><td>0.821</td><td>0.526</td><td>0.912</td></tr><tr><td>face</td><td>2.241</td><td>1.784</td><td>0.586</td><td>0.517</td><td>0.590</td><td>face</td><td>1.270</td><td>1.613</td><td>0.290</td><td>0.314</td><td>0.379</td></tr><tr><td>average</td><td>4.169</td><td>2.395</td><td>1.449</td><td>1.414</td><td>1.086</td><td>average</td><td>1.696</td><td>3.827</td><td>0.537</td><td>0.403</td><td>0.664</td></tr></table>
+
+Spatio-temporal Evaluations Since temporal metrics can trivially be reduced for blurry image content, we found it important to evaluate results with a combination of spatial and temporal metrics. Given that perceptual metrics are already widely used for image evaluations, we believe it is the right time to consider perceptual changes in temporal evaluations, as we did with our proposed temporal coherence metrics. Although not perfect, they are not easily deceived. Specifically, tOF is more robust than a direct pixel-wise metric as it compares motions instead of image content. In the supplemental material, we visualize the motion difference and it can well reflect the visual inconsistencies. On the other hand, we found that our calculation of tLP is a general concept that can work reliably with different perceptual metric: When repeating the tLP evaluation with the PieAPP metric (Prashnani et al., 2018) instead of $L P$ , i.e., tPieP $=$ $\lVert f ( y _ { t - 1 } , y _ { t } ) - f ( g _ { t - 1 } , g _ { t } ) \rVert _ { 1 }$ , where $\mathrm { f } ( \cdot )$ indicates the perceptual error function of PieAPP, we get close to identical results, listed in Fig. 13. The conclusions from $t P i e P$ also closely match the LPIPS-based evaluation: our network architecture can generate realistic and temporally coher
+
+![](images/e3e5f5d8e0b5c72dc8202eda113bc4ca84ec32d8b50dceeae7c28422aa235c9e.jpg)  
+Figure 13: Tables and visualization of perceptual metrics computed with PieAPP (Prashnani et al., 2018) (instead of LPIPS used in Fig. 7 previously) on ENet, FRVSR, DUF and TecoGAN for the VSR of Vid4. Bubble size indicates the tOF score.
+
+ent detail, and the metrics we propose allow for a stable, automated evaluation of the temporal perception of a generated video sequence.
+
+Besides the previously evaluated the Vid4 dataset, with graphs shown in Fig. 14, 15, we also get similar evaluation results on the Tears of Steel data-sets (room, bridge, and face, in the following referred to as $T o S$ scenes) and corresponding results are shown in Table 5 and Fig. 16. In all tests, we follow the procedures of previous work (Jo et al., 2018; Sajjadi et al., 2018) to make the outputs of all methods comparable, i.e., for all result images, we first exclude spatial borders with a distance of 8 pixels to the image sides, then further shrink borders such that the LR input image is divisible by 8 and for spatial metrics, we ignore the first two and the last two frames, while for temporal metrics, we ignore first three and last two frames, as an additional previous frame is required for inference. In the following, we conduct user studies for the Vid4 scenes. By comparing the user study results and the metric breakdowns shown in Table 4, we found our metrics to reliably capture the human temporal perception, as shown in Appendix C.
+
+![](images/5b9225aaf12d772d57724162984d8bf81e880721d167bce0b027593d690c193c.jpg)
+
+![](images/d758eeed95b126c5275a22c4af6a94d25f7f405e7627b60fa35ab559235a7aa4.jpg)  
+Figure 14: Bar graphs of temporal metrics for Vid4.
+
+![](images/02aed0cb969166db7f238959a39dd5326210b5af276a88c976dc6c2c3d1e53a3.jpg)  
+Figure 15: Spatial metrics for Vid4.   
+Figure 16: Metrics for ToS.
+
+# C USER STUDIES
+
+We conduct several user studies for the VSR task using five different methods, namely bi-cubic interpolation, ENet, FRVSR, DUF and our TecoGAN. The established 2AFC design (Fechner & Wundt, 1889; Um et al., 2017) is applied, i.e., participants have a pair-wise choice, with the groundtruth video shown as reference. One example can be seen in Fig. 17. The videos are synchronized and looped until user made the final decision. With no control to stop videos, users Participants cannot stop or influence the playback, and hence can focus more on the whole video, instead of specific spatial details. Videos positions (left/A or right/B) are randomized.
+
+After collecting 1000 votes from 50 users for every scene, i.e. twice for all possible pairs $( 5 \times 4 / 2 =$ 10 pairs), we follow common procedure and compute scores for all models with the Bradley-Terry model (1952). The outcomes for the Vid4 scenes can be seen in Fig. 18 (overall scores are listed in Table 2 of the main document).
+
+From the Bradley-Terry scores for the Vid4 scenes we can see that the TecoGAN model performs very well, and achieves the first place in three cases, as well as a second place in the walk scene. The latter is most likely caused by the overall slightly smoother images of the walk scene, in conjunction with the presence of several human faces, where our model can lead to the generation of unexpected details. However, overall the user study shows that users preferred the TecoGAN output over the other two deep-learning methods with a $6 3 . 5 \%$ probability.
+
+This result also matches with our metric evaluations. In Table 4, while TecoGAN achieves spatial (LPIPS) improvements in all scenes, DUF and FRVSR are not far behind in the walk scene. In terms of temporal metrics tOF and tLP, TecoGAN achieves similar or lower scores compared to FRVSR and DUF for calendar, foliage and city scenes. The lower performance of our model for the walk scene is likewise captured by higher tOF and tLP scores. Overall, the metrics confirm the performance of our TecoGAN approach and match the results of the user studies, which indicate that our proposed temporal metrics successfully capture important temporal aspects of human perception.
+
+For UVT tasks which have no ground-truth data, we carried out two sets of user studies: One uses an arbitrary sample from the target domain as the reference and the other uses the actual input from the source domain as the reference. On the Obama&Trump data-sets, we evaluate results from CycleGAN, RecycleGAN, and TecoGAN following the same modality, i.e. a 2AFC design with 50 users for each run. E.g., on the left of Fig. 19, users evaluate the generated Obama in reference with the input Trump on the y-axis, while an arbitrary Obama video is shown as the reference on the $\mathrm { X }$ -axis. Effectively, the y-axis is more important than the $\mathbf { X } ^ { \prime }$ -axis as it indicates whether the translated result preserves the original expression. A consistent ranking of TecoGAN $>$ RecycleGAN $>$ CycleGAN is shown on the y-axis with clear separations, i.e. standard errors don’t overlap. The $\mathbf { X }$ -axis indicates whether the inferred result matches the general spatio-temporal content of the target domain. Our TecoGAN model also receives the highest scores here, although the responses are slightly more spread out. On the right of Fig. 19, we summarize both studies in a single graph highlighting that the TecoGAN model is consistently preferred by the participants of our user studies.
+
+# D TECHNICAL DETAILS OF THE SPATIO-TEMPORAL DISCRIMINATOR
+
+Motion Compensation Used in Warped Triplet In the TecoGAN architecture, relationships $D _ { s , t }$ deteen e teand $I N _ { s , t } ^ { g }$ $\hat { I } N _ { s , t } ^ { y }$ network F. However, at the boundary of images, the output of $\mathrm { F }$ is usually less accurate due to the lack of reliable neighborhood information. There is a higher chance that objects move into the field of view, or leave suddenly, which significantly affects the images warped with the inferred motion. An example is shown in Fig. 20. This increases the difficulty for $D _ { s , t }$ , as it cannot fully rely on the images being aligned via warping. To alleviate this problem, we only use the center region of $I N _ { s , t } ^ { g }$ Ns,t and $\dot { I N } _ { s , t } ^ { y }$ as the input of the discriminator, and we reset a boundary of 16 pixels. Thus, for an input resolution of $I N _ { s , t } ^ { g }$ and $I N _ { s , t } ^ { y }$ of
+
+![](images/8fc975fc72e214ed0c51e06bfb4d57e15db34d0a9b3d77671896b1aa5d7fb5ba.jpg)  
+Figure 20: Near image boundaries, flow estimation is less accurate and warping often fails to align well. First two columns show original and warped frames and the third one shows differences after warping (ideally all black). The top row shows things move into the view with problems near lower boundaries, while the second row has objects moving out of the view.
+
+$1 2 8 \times 1 2 8$ for the VSR task, the inner part in size of $9 6 \times 9 6$ is left untouched, while the border regions are overwritten with zeros.
+
+The flow estimation network F with the loss $\mathcal { L } _ { G , F }$ should only be trained to support $\mathbf { G }$ in reaching the output quality as determined by $D _ { s , t }$ , but not the other way around. The latter could lead to $\mathrm { F }$ networks that confuse $D _ { s , t }$ with strong distortions of $I N _ { s , t } ^ { g }$ and $I N _ { s , t } ^ { y }$ . In order to avoid the this undesirable case, we stop the gradient back propagation from $I N _ { s , t } ^ { g }$ and $I N _ { s , t } ^ { y }$ to F. In this way, gradients from $D _ { s , t }$ to $\mathrm { F }$ are only back propagated through the generated samples $g _ { t - 1 } , g _ { t }$ and $g _ { t + 1 }$ into the generator network. In this way $D _ { s , t }$ can guide $\mathbf { G }$ to improve the image content, and $\mathrm { F }$ learns to warp the previous frame in accordance with the detail that G can synthesize. However, F does not adjust the motion estimation only to reduce the adversarial loss.
+
+<table><tr><td rowspan="2">Methods</td><td colspan="4">TheBradley-Terry scores (standard error)</td></tr><tr><td>calendar</td><td>foliage</td><td>city</td><td>walk</td></tr><tr><td>Bi-cubic</td><td>0.000(0.000)</td><td>0.000(0.000)</td><td>0.000(0.000)</td><td>0.000 (0.000)</td></tr><tr><td>ENet</td><td>1.834 (0.228)</td><td>1.634(0.180)</td><td>1.282 (0.205)</td><td>1.773（ (0.197)</td></tr><tr><td>FRVSR</td><td>3.043 (0.246)</td><td>2.177 (0.186)</td><td>3.173 (0.240)</td><td>2.424 (0.204)</td></tr><tr><td>DUF</td><td>3.468 (0.252)</td><td>2.243 (0.186)</td><td>3.302 (0.242)</td><td>3.175 (0.214)</td></tr><tr><td>TecoGAN</td><td>4.091 (0.262)</td><td>2.769 (0.194)</td><td>4.052 (0.255)</td><td>2.693 (0.207)</td></tr></table>
+
+![](images/443950507574f1de2fd950ce00c629f691866fc47ae345bf9b25ad248be7d3e5.jpg)  
+Figure 17: A sample setup of user study.
+
+![](images/cb829bbfd53a4c0a3a7824e4270a1a0b2f08e3d5d48c012ce0cb845fac1f8324.jpg)
+
+![](images/297b54daee522685ce45f366741c02988ceecfaabba29dec7187457f4cd877fb.jpg)  
+Figure 18: Tables and bar graphs of Bradley-Terry scores and standard errors for Vid4 VSR.   
+Figure 19: Tables and graphs of Bradley-Terry scores and standard errors for Obama&Trump UVT.
+
+Curriculum Learning for UVT Discriminators As mentioned in the main part, we train the UVT $D _ { s , t }$ with $100 \%$ spatial triplets at the very beginning. During training, $2 5 \%$ of them gradually transfer into warped triplets and another $2 5 \%$ transfer into original triplets. The transfer of the warped triplets can be represented as: $( 1 - \alpha ) \mathrm { I } _ { c g } + \alpha \mathrm { I } _ { w g }$ , with $\alpha$ growing form 0 to 1. For the original triplets, we additionally fade the “warping” operation out by using $( \bar { 1 } - \alpha ) \mathrm { I } _ { c g } + \alpha \lbrace W ( g _ { t - 1 } , \bar { v } _ { t } \ast$ $\beta ) , g _ { t } , W ( g _ { t + 1 } , v _ { t } ^ { \prime } * \beta ) \}$ , again with $\alpha$ growing form 0 to 1 and $\beta$ decreasing from 1 to 0. We found this smooth transition to be helpful for a stable training.
+
+# E DATA AUGMENTATION AND TEMPORAL CONSTRAINS IN THE PP LOSS
+
+Since training with sequences of arbitrary length is not possible with current hardware, problems such as the streaking artifacts discussed above generally arise for recurrent models. In the proposed PP loss, both the Ping-Pang data augmentation and the temporal consistency constraint contribute to solving these problems. In order to show their separated contributions, we trained another TecoGAN variant that only employs the data augmentation without the constraint (i.e., $\lambda _ { p } = 0$ in Table 1).
+
+Denoted as PP-Augment, we show its results in comparison with the DsDt and TecoGANmodels in Fig. 21. Video results are shown in the in the supplemental material.
+
+During training, the generator of DsDt receives 10 frames, and generators of PP-Augment and TecoGANsee 19 frames. While DsDt shows strong recurrent accumulation artifacts early on, the PP-Augment version slightly reduces the artifacts. In Fig. 21, it works good for frame 15, but shows artifacts from frame 32 on. Only our regular model (TecoGAN) successfully avoids temporal accumulation for all 40 frames. Hence, with the PP constraint, the model avoids recurrent accumulation of artifacts and works well for sequences that are substantially longer than the training length.
+
+![](images/90ead0e208ce193b3124cd7d7eeeb4bd6546c56a783d2afb3d5d343fc193be56.jpg)  
+Figure 21: 1st & 2nd row: Frame 15 & 40 of the Foliage scene. While DsDt leads to strong recurrent artifacts early on, PPAugment shows similar artifacts later in time (2nd row, middle). TecoGANmodel successfully removes these artifacts.
+
+Among others, we have tested our model with ToS sequences of lengths 150, 166 and 233. For all of these sequences, the TecoGAN model successfully avoids temporal accumulation or streaking artifacts.
+
+# F NETWORK ARCHITECTURE
+
+In this section, we use the following notation to specify all network architectures used: conc() represents the concatenation of two tensors along the channel dimension; $C / C T$ (input, kernel size, output channel, stride size) stands for the convolution and transposed convolution operation, respectively; $" + "$ denotes element-wise addition; BilinearUp2 up-samples input tensors by a factor of 2 using bi-linear interpolation; BicubicResize4(input) increases the resolution of the input tensor to 4 times higher via bi-cubic up-sampling; Dense(input, output size) is a densely-connected layer, which uses Xavier initialization for the kernel weights.
+
+The architecture of our VSR generator $\mathbf { G }$ is:
+
+$$
+\mathrm { c o n c } ( x _ { t } , W ( g _ { t - 1 } , v _ { t } ) ) \to l _ { i n } ; C ( l _ { i n } , 3 , 6 4 , 1 ) , \mathrm { R e L U } \to l _ { 0 }
+$$
+
+$$
+\begin{array} { r } { C T ( l _ { n } , 3 , 6 4 , 2 ) , \mathrm { R e L U }  l _ { u p 2 } ; C T ( l _ { u p 2 } , 3 , 6 4 , 2 ) , \mathrm { R e L U }  l _ { u p 4 } ; } \\ { C ( l _ { u p 4 } , 3 , 3 , 1 ) , \mathrm { R e L U }  l _ { r e s } ; \mathrm { B i c u b i c R e s i z e 4 } ( x _ { t } ) + l _ { r e s }  g _ { t } . } \end{array}
+$$
+
+In TecoGAN, there are 10 sequential residual blocks in the generator ( $l _ { n } ~ = ~ l _ { 1 0 } ~ )$ , while the TecoGAN generator has 16 residual blocks ( $l _ { n } ~ = ~ l _ { 1 6 } ~ )$ . Each ResidualBlock(li) contains the following operations: $C ( l _ { i } , 3 , 6 4 , 1 ) , \mathrm { R e L U }  r _ { i }$ ; $C ( r _ { i } , 3 , 6 4 , 1 ) + l _ { i }  l _ { i + 1 }$ .
+
+The VSR $D _ { s , t }$ ’s architecture is:
+
+$\mathrm { I N } _ { s , t } ^ { g }$ or $\mathrm { I N } _ { s , t } ^ { y } \to l _ { i n }$ ; $C ( l _ { i n } , 3 , 6 4 , 1 )$ , Leaky $\mathrm { R e L U }  l _ { 0 }$ ;   
+$C ( l _ { 0 } , 4 , 6 4 , 2 )$ , BatchNorm, Leaky ReLU $ l _ { 1 }$ ; $C ( l _ { 1 } , 4 , 6 4 , 2 )$ , BatchNorm, Leaky $\mathrm { R e L U }  l _ { 2 }$ ;   
+$C ( l _ { 2 } , 4 , 1 2 8 , 2 )$ , BatchNorm, Leaky $\mathrm { R e L U }  l _ { 3 }$ ; $C ( l _ { 3 } , 4 , 2 5 6 , 2 )$ , BatchNorm, Leaky ReLU $\to l _ { 4 }$ ; $D e n s e ( l _ { 4 } , 1 )$ , sigmoid $\to l _ { o u t }$ .
+
+VSR discriminators used in our variant models, DsDt, DsDtPP and DsOnly, have a similar architecture as $D _ { s , t }$ . They only differ in terms of their inputs.
+
+The flow estimation network F has the following architecture:
+
+conc $( x _ { t } , x _ { t - 1 } ) \to l _ { i n }$ ; $C ( l _ { i n } , 3 , 3 2 , 1 )$ , Leaky $\mathrm { R e L U }  l _ { 0 }$ ;   
+$C ( l _ { 0 } , 3 , 3 2 , 1 )$ , Leaky ReLU, MaxPooling $ l _ { 1 }$ ; $C ( l _ { 1 } , 3 , 6 4 , 1 )$ , Leaky $\mathrm { R e L U }  l _ { 2 }$ ;   
+$C ( l _ { 2 } , 3 , 6 4 , 1 )$ , Leaky ReLU, MaxPooling $ l _ { 3 }$ ; $C ( l _ { 3 } , 3 , 1 2 8 , 1 )$ , Leaky $\mathrm { R e L U }  l _ { 4 }$ ;   
+$C ( l _ { 4 } , 3 , 1 2 8 , 1 )$ , Leaky ReLU, MaxPooling $\to l _ { 5 }$ ; $C ( l _ { 5 } , 3 , 2 5 6 , 1 )$ , Leaky ReLU $\to l _ { 6 }$ ;   
+$C ( l _ { 6 } , 3 , 2 5 6 , 1 )$ , Leaky ReLU, BilinearUp2 $\to l _ { 7 }$ ; $C ( l _ { 7 } , 3 , 1 2 8 , 1 )$ , Leaky $\mathrm { R e L U } \to l _ { 8 }$ ;   
+$C ( l _ { 8 } , 3 , 1 2 8 , 1 )$ , Leaky ReLU, Bilinear $\mathrm { U p } 2 \to l _ { 9 }$ ; $C ( l _ { 9 } , 3 , 6 4 , 1 )$ , Leaky ReLU $\to l _ { 1 0 }$ ;   
+$C ( l _ { 1 0 } , 3 , 6 4 , 1 )$ , Leaky ReLU, Bilinear $\mathrm { U p } 2  l _ { 1 1 }$ ; $C ( l _ { 1 1 } , 3 , 3 2 , 1 )$ , Leaky ReLU $ l _ { 1 2 }$ ; $C ( l _ { 1 2 } , 3 , 2 , 1 )$ , tanh $ l _ { o u t }$ ; $l _ { o u t } * \mathrm { M a x V e l } \to v _ { t }$ .
+
+Here, MaxVel is a constant vector, which scales the network output to the normal velocity range.
+
+While $\mathrm { F }$ is the same for UVT tasks, UVT generators have an encoder-decoder structure:
+
+c $\operatorname { n c } ( x _ { t } , W ( g _ { t - 1 } , v _ { t } ) ) \to l _ { i n } ; C ( l _ { i n } , 7 , 3 2 , 1 )$ , InstanceNorm, $\mathrm { R e L U }  l _ { 0 }$ ; $C ( l _ { 0 } , 3 , 6 4 , 2 )$ , InstanceNorm, $\mathrm { R e L U }  l _ { 1 }$ ; $C ( l _ { 1 } , 3 , 1 2 8 , 2 )$ , InstanceNorm, ReLU $ l _ { 2 }$ ; $R e s i d u a l B l o c k ( l _ { 2 } + i )  l _ { 3 + i }$ with $i = 0 , . . . , n - 1$ ; $C T ( l _ { n + 2 } , 3 , 6 4 , 2 )$ , InstanceNorm, $\mathrm { R e L U }  l _ { n + 3 }$ ; $C T ( l _ { n + 3 } , 3 , 3 2 , 2 )$ , InstanceNorm, $\mathrm { R e L U }  l _ { n + 4 }$ ; $C T ( l _ { n + 4 } , 7 , 3 , 1 )$ , tanh $ l _ { o u t }$
+
+Residua $B l o c k ( l _ { 2 } + i )$ contains the following operations: $C ( l _ { 2 + i } , 3 , 1 2 8 , 1 )$ , InstanceNorm, ReLU $t _ { 2 + i }$ ; $C ( t _ { 2 + i } , 3 , 1 2 8 , 1 )$ , InstanceNorm $ r _ { 2 + i }$ ; $r _ { 2 + i } + l _ { 2 + i } \to l _ { 3 + i }$ . We use 10 residual blocks for all UVT generators.
+
+Since UVT generators are larger than the VSR generator, we also use a larger $D _ { s , t }$ architecture:
+
+$\mathbb { N } _ { s , t } ^ { g }$ or $\Pi _ { s , t } ^ { y }  l _ { i n } ; C ( l _ { i n } , 4 , 6 4 , 2 4 ) , \mathrm { R e L U }  l _ { 0 } \mathrm { , }$ ; $C ( l _ { 0 } , 4 , 1 2 8 , 2 )$ , InstanceNorm, Leaky $\mathrm { R e L U }  l _ { 1 }$ ; $C ( l _ { 1 } , 4 , 2 5 6 , 2 )$ , InstanceNorm, Leaky $\mathrm { R e L U } \to l _ { 2 }$ ; $C ( l _ { 2 } , 4 , 5 1 2 , 2 )$ , InstanceNorm, Leaky ReLU $ l _ { 3 }$ ; $D e n s e ( l _ { 3 } , 1 ) \to l _ { o u t }$ .
+
+Again, all ablation studies use the same architecture with different inputs.
+
+# G TRAINING DETAILS
+
+We use the non-saturated GAN for VSR and LSGAN (Mao et al., 2017) for UVT and both of them can prevent the gradient vanishing problem of a vanilla GAN (Goodfellow et al., 2014). While we train stably with a dynamic discriminator updating strategy, i.e. discriminators are not updated when there is already a large difference between $D ( \boldsymbol { \mathrm { I } } ^ { b } )$ and $D ( \mathbf { I } ^ { g } )$ , the training process could potentially be further improved with modern GAN algorithms, e.g. Wasserstein GAN (Gulrajani et al., 2017).We train $\mathbf { G }$ and $F$ together for VSR , while we simply use the pre-trained $F$ for UVT.
+
+For the VSR task, our training data-set consists of 250 short HR videos, each with 120 frames. We use sequences with a length of 10 and a batch size of 4. A black image is used as the first previous frame of each video sequence. I.e., one batch contains 40 frames and with the PP loss formulation, the NN receives gradients from 76 frames in total for every training iteration. To improve the stability of the adversarial training, we pre-train $\mathbf { G }$ and $F$ with a simple $L ^ { 2 }$ loss of $\bar { \sum \| } g _ { t } - b _ { t } \| _ { 2 } + \lambda _ { w } \bar { \mathcal { L } } _ { w a r p }$ for $5 0 0 \mathrm { k }$ batches. We use $9 0 0 \mathrm { k }$ batches for the adversarial training stage. The data-sets of the UVT tasks contain around 2400 to 3600 frames. We train the generators with a sequence length of 6 and a batch size of 1. Since temporal triplets are gradually faded in, we do not pre-train models for UVT tasks. With smaller datasets, we train UVT models with $1 0 0 \mathrm { k }$ batches.
+
+In the pre-training stage of VSR, we train the F and a generator with 10 residual blocks. An ADAM optimizer with $\beta \ : = \ : 0 . 9$ is used throughout. The learning rate starts from $1 0 ^ { - 4 }$ and decays by $50 \%$ every $5 0 \mathrm { k }$ batches until it reaches $2 . 5 * 1 0 ^ { - 5 }$ . This pre-trained model is then used for all TecoGAN variants as initial state. In the adversarial training stage of VSR, all TecoGAN variants are trained with a fixed learning rate of $5 * 1 0 ^ { - 5 }$ . The generators in DsOnly, DsDt, DsDtPP and TecoGANhave 10 residual blocks, whereas the TecoGAN model has 6 additional residual blocks in its generator. Therefore, after loading 10 residual blocks from the pre-trained model, these additional residual blocks are faded in smoothly with a factor of $2 . 5 * 1 0 ^ { - 5 }$ . We found this growing training methodology, first introduced by Growing GAN (Karras et al., 2017), to be stable and efficient in our tests. When training the VSR DsDt and DsDtPP, extra parameters are used to balance the two cooperating discriminators properly. Through experiments, we found $D _ { t }$ to be stronger. Therefore, we reduce the learning rate of $D _ { t }$ to $1 . 5 * \bar { 1 } 0 ^ { - 5 }$ in order to keep both discriminators balanced. At the same time, a factor of 0.0003 is used on the temporal adversarial loss to the generator, while the spatial adversarial loss has a factor of 0.001. During the VSR training, input LR video frames are cropped to a size of $3 2 \times 3 2$ . In all VSR models, the Leaky ReLU operation uses a tangent of 0.2 for the negative half space. Additional training parameters are listed in Table 6.
+
+Table 6: Training parameters   
+
+<table><tr><td rowspan=1 colspan=1>VSRParam</td><td rowspan=1 colspan=1>DsOnly</td><td rowspan=1 colspan=1>DsDt DsDtPP</td><td rowspan=1 colspan=1>TecoGAN</td><td rowspan=1 colspan=1>TecoGAN UVTParam</td><td rowspan=1 colspan=1>DsOnlyDst</td><td rowspan=1 colspan=1>DsDtPP</td><td rowspan=1 colspan=1>TecoGAN</td></tr><tr><td rowspan=1 colspan=1>入a</td><td rowspan=1 colspan=1>1e-3</td><td rowspan=1 colspan=1>Ds:1e-3,Dt:3e-4</td><td rowspan=1 colspan=1>1e-3</td><td rowspan=1 colspan=1>1e-3       入a</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>Ds:0.5Dt:0.3</td><td rowspan=1 colspan=1>0.5</td></tr><tr><td rowspan=1 colspan=1>入p</td><td rowspan=1 colspan=1>0.0</td><td rowspan=1 colspan=3>0.0              0.5                   Xp</td><td rowspan=1 colspan=1>0.00.0</td><td rowspan=1 colspan=2>100.0</td></tr><tr><td rowspan=1 colspan=1>入</td><td rowspan=1 colspan=4>0.02 for VGG and 1.0 forDiscriminator           入</td><td rowspan=1 colspan=3>from 10 decays to 0.0</td></tr><tr><td rowspan=1 colspan=1>入，Ac</td><td rowspan=1 colspan=4>1.0, 1.0                        入</td><td rowspan=1 colspan=3>0.0,a pre-trainedF isused for UST tasks</td></tr><tr><td rowspan=1 colspan=1>Tearning-rate</td><td rowspan=1 colspan=1>5e-5</td><td rowspan=1 colspan=1>1.5e-5 for Dt,5e-5 for others.</td><td rowspan=1 colspan=2>5e-5     5e-5       入c</td><td rowspan=1 colspan=3>10.0</td></tr></table>
+
+For all UVT tasks, we use a learning rate of $1 0 ^ { - 4 }$ to train the first $9 0 \mathrm { k }$ batches and the last $1 0 \mathrm { k }$ batches are trained with the learning rate decay from $1 0 ^ { - 4 }$ to 0. Images of the input domain are cropped into a size of $2 5 6 \times 2 5 6$ when training, while the original size is $2 8 8 \times 2 8 8$ . While the Additional training parameters are also listed in Table 6. For UVT, $\mathcal { L } _ { \mathrm { { c o n t e n t } } }$ and $\mathcal { L } _ { \phi }$ are only used to improve the convergence of the training process. We fade out the $\mathcal { L } _ { \mathrm { c o n t e n t } }$ in the first $1 0 \mathrm { k }$ batches and the $\mathcal { L } _ { \phi }$ is used for the first $8 0 \mathrm { k }$ and faded out in last $2 0 \mathrm { k }$ .
+
+# H PERFORMANCE
+
+TecoGAN is implemented in TensorFlow. While generator and discriminator are trained together, we only need the trained generator network for the inference of new outputs after training, i.e., the whole discriminator network can be discarded. We evaluate the models on a Nvidia GeForce GTX 1080Ti GPU with 11G memory, the resulting VSR performance for which is given in Table 2.
+
+The VSR TecoGANmodel and FRVSR have the same number of weights (843587 in the SRNet, i.e. generator network, and 1.7M in F), and thus show very similar performance characteristics with around $3 7 ~ \mathrm { m s }$ spent for one frame. The larger VSR TecoGAN model with 1286723 weights in the generator is slightly slower than $\mathrm { T e c o G A N ^ { \odot } }$ , spending $4 2 \mathrm { m s }$ per frame. In the UVT task, generators spend around $6 0 \mathrm { m s }$ per frame with a size of $5 1 2 \times 5 1 2$ . However, compared with the DUF model, with has more than 6 million weights in total, the TecoGAN performance significantly better thanks to its reduced size.