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+[
+    {
+        "type": "text",
+        "text": "INFERENCE SUBOPTIMALITY IN VARIATIONAL AUTOENCODERS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "There has been significant work on improving inference in variational autoencoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014) through the development of expressive approximate posteriors (Rezende & Mohamed, 2015; Kingma et al., 2016; Ranganath et al., 2016; Tomczak & Welling, 2016; 2017). These works have shown that with more expressive approximate posteriors, the model learns a better distribution over the data. ",
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+        "type": "text",
+        "text": "In this paper, we analyze inference suboptimality in VAEs: the mismatch between the true and approximate posterior. In other words, we are interested in understanding what factors cause the gap between the marginal log-likelihood and the evidence lower bound (ELBO). We refer to this as the inference gap. Moreover, we break down the inference gap into two components: the approximation gap and the amortization gap. The approximation gap comes from the inability of the approximate distribution family to exactly match the true posterior. The amortization gap refers to the difference caused by amortizing the variational parameters over the entire training set, instead of optimizing for each datapoint independently. We refer the reader to Table 1 for detailed definitions and Figure 1 for a simple illustration of the gaps. In Figure 1, ${ \\mathcal { L } } [ q ]$ refers to the ELBO using an amortized distribution $q$ , whereas $q ^ { * }$ is the optimal $q$ within its variational family. ",
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+    {
+        "type": "text",
+        "text": "Our experiments investigate how the choice of encoder, posterior approximation, decoder, and model optimization affect the approximation and amortization gaps. We train VAE models in a number of settings on the MNIST, Fashion-MNIST (Xiao et al., 2017), and CIFAR10 datasets. ",
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+        "type": "text",
+        "text": "Our contributions are: a) we investigate inference suboptimality in terms of the approximation and amortization gaps, providing insight to guide future improvements in VAE inference, b) we quantitatively demonstrate that the learned true posterior accommodates the choice of approximation, and c) we demonstrate that using parameterized functions to improve the expressiveness of the approximation plays a large role in reducing error caused by amortization. ",
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+    {
+        "type": "image",
+        "img_path": "images/7083885f712ccbaedc70c4ec67996ea743f40445ce818540472026cabebf20b2.jpg",
+        "image_caption": [
+            "Figure 1: Gaps in Inference "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/c4fddd2c40502b93ed7cf331f7e355439ee56cbabd0a92482687caf62d55ad99.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Term</td><td>Definition</td><td>VAE Formulation</td></tr><tr><td>Inference</td><td>logp(x)-L[q]</td><td>KL(q(z|x)llp(z|x))</td></tr><tr><td>Approximation</td><td>logp(x）-L[q*]</td><td>KL(q*（z|x)lp(z|x)）</td></tr><tr><td>Amortization</td><td>C-C[a]</td><td>KL(q(z|x)llp(z|x))-KL(q*(z|x)llp(z|x))</td></tr></table>",
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+        "text": "Table 1: Summary of Gap Terms. The middle column refers to the general case where our variational objective is a lower bound on the marginal log-likelihood (not necessarily the ELBO). The right most column demonstrates the specific case in VAEs. $q ^ { * } ( z | x )$ refers to the optimal approximation within a family $\\mathcal { Q }$ , i.e. $\\begin{array} { r } { q ^ { * } ( z | x ) = \\mathrm { \\bar { a r g m i n } } _ { q \\in \\mathcal { Q } } \\mathrm { K L } \\left( q ( z | \\bar { x } ) | | \\dot { p ( z | x ) } \\right) } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 INFERENCE IN VARIATIONAL AUTOENCODERS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Let $x$ be the observed variable, $z$ the latent variable, and $p ( x , z )$ be their joint distribution. Given a dataset $X = \\{ x _ { 1 } , x _ { 2 } , . . . , x _ { N } \\}$ , we would like to maximize the marginal log-likelihood: ",
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+        "type": "equation",
+        "img_path": "images/56bffca6b29e08b4eedd4e51fab95873d769405e2f6eed6cd358e18da01831f7.jpg",
+        "text": "$$\n\\log p ( X ) = \\sum _ { i = 1 } ^ { N } \\log p ( x _ { i } ) = \\sum _ { i = 1 } ^ { N } \\log \\int p ( x _ { i } , z _ { i } ) d z _ { i } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In practice, the marginal log-likelihood is computationally intractable due to the integration over the latent variable $z$ . Instead, VAEs optimize the ELBO of the marginal log-likelihood (Kingma & Welling, 2014; Rezende et al., 2014): ",
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+    {
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+        "img_path": "images/9981cd2ae42eba6533455668742d8312efea6c5a3a5acc5e076a6169be5fce28.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log p ( x ) = \\mathbb { E } _ { z \\sim q ( z \\mid x ) } \\left[ \\log \\left( \\displaystyle \\frac { p ( x , z ) } { q ( z \\mid x ) } \\right) \\right] + { \\mathrm { K L } } \\left( q ( z \\mid x ) | | p ( z | x ) \\right) } \\\\ & { \\phantom { \\exp x } \\geq \\mathbb { E } _ { z \\sim q ( z \\mid x ) } \\left[ \\log \\left( \\displaystyle \\frac { p ( x , z ) } { q ( z \\mid x ) } \\right) \\right] = \\mathcal { L } _ { \\mathrm { V A E } } [ q ] . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "From the above we can see that the lower bound is tight if $q ( z | x ) = p ( z | x )$ . The choice of $q ( z | x )$ is often a factorized Gaussian distribution for its simplicity and efficiency. VAEs perform amortized inference by utilizing a recognition network (encoder), resulting in efficient approximate inference for large datasets. The overall model is trained by stochastically optimizing the ELBO using the reparametrization trick (Kingma & Welling, 2014). ",
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+    {
+        "type": "text",
+        "text": "2.2 EXPRESSIVE APPROXIMATE POSTERIORS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "There are a number of strategies for increasing the expressiveness of approximate posteriors, going beyond the original factorized-Gaussian. We briefly summarize normalizing flows and auxiliary variables. ",
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+    {
+        "type": "text",
+        "text": "2.2.1 NORMALIZING FLOWS ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Normalizing flow (Rezende & Mohamed, 2015) is a change of variables procedure for constructing complex distributions by transforming probability densities through a series of invertible mappings. Specifically, if we transform a random variable $z _ { \\mathrm { 0 } }$ with distribution $q _ { 0 } ( z )$ , the resulting random variable $z _ { T } = T ( z _ { 0 } )$ has a distribution: ",
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+    {
+        "type": "equation",
+        "img_path": "images/4222a4b91e0898278b23c9daaae295e941330332bb6a85148f6d17aa025568e0.jpg",
+        "text": "$$\nq _ { T } ( z _ { T } ) = q _ { 0 } ( z _ { 0 } ) \\left| \\mathrm { d e t } \\frac { \\partial z _ { T } } { \\partial z _ { 0 } } \\right| ^ { - 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "By successively applying these transformations, we can build arbitrarily complex distributions. Stacking these transformations remains tractable due to the determinant being decomposable: $\\operatorname* { d e t } ( A { \\bar { B } } ) = \\operatorname* { d e t } ( A ) \\operatorname* { d e t } ( B )$ . An important property of these transformations is that we can take expectations with respect to the transformed density $q _ { T } ( z _ { T } )$ without explicitly knowing its formula known as the law of the unconscious statistician (LOTUS): ",
+        "bbox": [
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+        "img_path": "images/f7329c9a76b653fa7eb1fa5f5695895e5a2c3c7cac0699870fab1aead0669976.jpg",
+        "text": "$$\n\\mathbb { E } _ { q _ { T } } [ h ( z _ { T } ) ] = \\mathbb { E } _ { q _ { 0 } } [ h ( f _ { T } ( f _ { T - 1 } ( \\dots f _ { 1 } ( z _ { 0 } ) ) ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Using the change of variable and LOTUS, the lower bound can be written as: ",
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+        "type": "equation",
+        "img_path": "images/10550f1b7dfd6475306f929d8ec94df92fd6aa63d47a4757dacaf80d1d5c1c1f.jpg",
+        "text": "$$\n\\log p ( x ) \\geq \\mathbb { E } _ { z _ { 0 } \\sim q _ { 0 } ( z | x ) } \\left[ \\log \\left( \\frac { p ( x , z _ { T } ) } { q _ { 0 } ( z _ { 0 } | x ) \\prod _ { t = 1 } ^ { T } \\left| \\operatorname* { d e t } \\frac { \\partial z _ { t } } { \\partial z _ { t - 1 } } \\right| ^ { - 1 } } \\right) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The main constraint on these transformations is that the determinant of their Jacobian needs to be easily computable. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "2.2.2 AUXILIARY VARIABLES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Deep generative models can be extended with auxiliary variables which leave the generative model unchanged but make the variational distribution more expressive. Just as hierarchical Bayesian models induce dependencies between data, hierarchical variational models can induce dependencies between latent variables. The addition of the auxiliary variable changes the lower bound to: ",
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+        "type": "equation",
+        "img_path": "images/816d5415406b5565885747fe14695600a9c0af870de46f5a7f0d0525a26e6e14.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log p ( x ) \\geq \\mathbb { E } _ { z , v \\sim q ( z , v \\mid x ) } \\left[ \\log \\left( \\displaystyle \\frac { p ( x , z ) r ( v \\mid x , z ) } { q ( z , v \\mid x ) } \\right) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { q ( z \\mid x ) } \\left[ \\log \\left( \\displaystyle \\frac { p ( x , z ) } { q ( z \\mid x ) } \\right) - { \\mathrm { K L } \\Big ( q ( v \\mid z , x ) \\| r ( v \\mid x , z ) \\Big ) } \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "where $r ( v | x , z )$ is called the reverse model. From Eqn. 8, we see that this bound is looser than the regular ELBO, however the extra flexibility provided by the auxiliary variable can result in a higher lower bound. This idea has been employed in works such as auxiliary deep generative models (ADGM, Maaløe et al. (2016)), hierarchical variational models (HVM, Ranganath et al. (2016)) and Hamiltonian variational inference (HVI, Salimans et al. (2015)). ",
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+    {
+        "type": "text",
+        "text": "2.3 MARGINAL LOG-LIKELIHOOD ESTIMATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We use two bounds to estimate the marginal log-likelihood of a model: IWAE (Burda et al., 2016) and AIS (Neal, 2001). Here we describe the IWAE bound. See Section 6.5 in the appendix for a description of AIS. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The IWAE bound is a tighter lower bound than the VAE bound. More specifically, if we take multiple samples from the $q$ distribution, we can compute a tighter lower bound on the marginal log-likelihood: ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/b671f681db5174a6fc3eadef70e4fd2fb2f5eac2263f41c785125011cd29a498.jpg",
+        "text": "$$\n\\log p ( x ) \\geq \\mathbb { E } _ { z _ { 1 } . . . z _ { k } \\sim q ( z | x ) } \\left[ \\log \\left( \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\frac { p ( x , z _ { i } ) } { q ( z _ { i } | x ) } \\right) \\right] = \\mathcal { L } _ { \\mathrm { I W A E } } [ q ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "As the number of importance samples approaches infinity, the bound approaches the marginal loglikelihood. This importance weighted bound was introduced along with the Importance Weighted Autoencoder (Burda et al., 2016), thus we refer to it as the IWAE bound. It is often used as an evaluation metric for generative models (Burda et al., 2016; Kingma et al., 2016). As shown by Bachman & Precup (2015) and Cremer et al. (2017), the IWAE bound can be seen as using the VAE bound but with an importance weighted $q$ distribution. ",
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 APPROXIMATION AND AMORTIZATION GAPS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The inference gap $\\mathcal { G }$ is the difference between the marginal log-likelihood $\\log p ( x )$ and a lower bound ${ \\mathcal { L } } [ q ]$ . Given the distribution in the family that maximizes the bound, $q ^ { * } ( z | x ) \\ =$ arg $\\operatorname* { m a x } _ { q \\in \\mathcal { Q } } \\mathcal { L } [ q ]$ , the inference gap decomposes as the sum of approximation and amortization gaps: ",
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+        "img_path": "images/04dd23848457536dfc68dcf8374fc6ae437528541eae6bec148bdd1dbc2f49cc.jpg",
+        "text": "$$\n\\mathcal { G } = \\log p ( x ) - \\mathcal { L } [ q ] = \\underbrace { \\log p ( x ) - \\mathcal { L } [ q ^ { * } ] } _ { \\mathrm { A p p r o x i m a t i o n } } + \\underbrace { \\mathcal { L } [ q ^ { * } ] - \\mathcal { L } [ q ] } _ { \\mathrm { A m o r t i z a t i o n } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "For VAEs, we can translate the gaps to KL divergences by rearranging (2): ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4fb17592eb2f6144d7f966a688a0c03d1194b2b0f8e3d77e52dc87dbf0d07815.jpg",
+        "text": "$$\n\\mathcal { G } _ { \\mathrm { V A E } } = \\mathrm { K L } \\big ( q ^ { * } ( z | x ) | | p ( z | x ) \\big ) + \\mathrm { K L } \\big ( q ( z | x ) | | p ( z | x ) \\big ) - \\mathrm { K L } \\big ( q ^ { * } ( z | x ) | | p ( z | x ) \\big ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "3.2 FLEXIBLE APPROXIMATE POSTERIOR ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our experimentation compares two families of approximate posteriors: the fully-factorized Gaussian (FFG) and a flexible flow (Flow). Our choice of flow is a combination of the Real NVP (Dinh et al., 2017) and auxiliary variables (Ranganath et al., 2016; Maaløe et al., 2016). Our model also resembles leap-frog dynamics applied in Hamiltonian Monte Carlo (HMC, Neal et al. (2011)). ",
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+    {
+        "type": "text",
+        "text": "Let $z \\in \\mathbb { R } ^ { n }$ be the variable of interest and $v \\in \\mathbb { R } ^ { n }$ the auxiliary variable. Each flow step involves: ",
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+        "img_path": "images/d024b3b0578b2dd0f9f41e94220af3f04fa11192286a8c947dc49ac9e78e629a.jpg",
+        "text": "$$\n\\begin{array} { l } { { v ^ { \\prime } = v \\circ \\sigma _ { 1 } ( z ) + \\mu _ { 1 } ( z ) } } \\\\ { { z ^ { \\prime } = z \\circ \\sigma _ { 2 } ( v ^ { \\prime } ) + \\mu _ { 2 } ( v ^ { \\prime } ) } } \\end{array}\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\sigma _ { 1 } , \\sigma _ { 2 } , \\mu _ { 1 } , \\mu _ { 2 } : \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { n }$ are differentiable mappings parameterized by neural nets and $\\circ$ takes the Hadamard or element-wise product. The determinant of the combined transformation’s Jacobian, $| \\mathrm { d e t } ( D f ) |$ , can be easily evaluated. See section 6.2 in the Appendix for a detailed derivation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Thus, we can jointly train the generative and flow-based inference model by optimizing the bound: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bfd9253840f6460cf5590523379e8756d23bba4066c050963993a2f357155b62.jpg",
+        "text": "$$\n\\log p ( x ) \\geq \\mathbb { E } _ { z , v \\sim q ( z , v \\mid x ) } \\left[ \\log \\left( { \\frac { p ( x , z ^ { \\prime } ) r ( v ^ { \\prime } | x , z ^ { \\prime } ) } { q ( z , v | x ) \\left| \\operatorname* { d e t } ( D f ) \\right| ^ { - 1 } } } \\right) \\right] = \\mathcal { L } _ { \\mathrm { f l o w } } [ q ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            406,
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+            450
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Additionally, multiple such type of transformations can be stacked to improve expressiveness. We refer readers to section 6.1.2 in the Appendix for details of our flow configuration adopted in the experimentation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 EVALUATION BOUNDS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            512,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use several bounds to compute the inference gaps. To estimate the marginal log-likelihood, $\\log { \\hat { p } } ( x )$ , we take the maximum of our tightest lower bounds, specifically the maximum between the IWAE and AIS bounds. To compute the AIS bound, we use 100 chains, each with 500 intermediate distributions, where each transition consists of one HMC trajectory with 10 leapfrog steps. The initial distribution for AIS is the prior, so that it is encoder-independent. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For our experiments, we test two different variational distributions: the fully-factorized Gaussian $q _ { F F G }$ and the flexible approximation $q F l o w$ as described in section 3.2. When computing ${ \\mathcal { L } } _ { \\mathrm { V A E } } [ q ]$ and $\\mathcal { L } _ { \\mathrm { I W A E } } [ q ]$ , we use 5000 samples. To compute $\\mathcal { L } _ { \\mathrm { V A E } } [ q ^ { * } ]$ , we optimize the parameters of the variational distribution for every datapoint. See Section 6.4 for details of the local optimization and stopping criteria. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            704,
+            344,
+            720
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Much of the earlier work on variational inference focused on optimizing the variational parameters locally for each datapoint, e.g. the original Stochastic Variational Inference scheme (SVI, Hoffman et al. (2013)) specifies the variational parameters to be optimized locally in the inner loop. Salakhutdinov & Larochelle (2010) perform such local optimization when learning deep Boltzmann machines. More recent work has applied this idea to improve approximate inference in directed Belief networks (Hjelm et al., 2015). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Most relevant to our work is the recent work of Krishnan et al. (2017). They explicitly remark on two sources of error in variational learning with inference networks, and propose to optimize approximate inference locally from an initialization output by the inference network. They show improved training on high-dimensional, sparse data with the hybrid method, claiming that local optimization reduces the negative effects of random initialization in the inference network early on in training. Yet, their work only dwells on reducing the amortization gap and does analyze the error arising from the use of limited approximating distributions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/a1d655761a82d6b4bc4ede47b36a7fbb2611e59eeea14fc4cf9c5c21bd3472f8.jpg",
+        "image_caption": [
+            "Figure 2: True Posterior and Approximate Distributions of a VAE with 2D latent space. Columns: 4 different datapoints. FFG: Fully-factorized Gaussian. Flow: Using a flexible approximate distribution. Amortized: Using amortized parameters. Optimal: Parameters optimized for individual datapoints. The green distributions are the true posterior distributions, highlighting the mismatch with the approximation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            95,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Even though it is clear that failed inference would lead to a failed generative model, little quantitative assessment has been done showing the effect of the approximate posterior on the true posterior. Burda et al. (2016) visually demonstrate that when trained with an importance-weighted approximate posterior, the resulting true posterior is more complex than those trained with fully-factorized Gaussian approximations. We extend this observation quantitatively in the setting of flow-based approximate inference. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1 INTUITION THROUGH VISUALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To begin, we would like to gain some insight into the properties of inference in VAEs by visualizing different distributions in the latent space. To this end, we trained a VAE with a two-dimensional latent space on MNIST. We show contour plots of various distributions in the latent space in Fig. 2. The first row contains contour plots of the true posteriors $p ( z | x )$ for four different training datapoints (columns). We have selected these four examples to highlight different inference phenomena. The amortized FFG row refers to the output of the recognition net, in this case, a fully-factorized Gaussian (FFG) approximation. Optimal FFG is the FFG that best fits the posterior of the datapoint. Optimal Flow is the optimal fit of a flexible distribution to the same posterior, where the flexible distribution we use is described in Section 3.2. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Posterior A is an example of a distribution where FFG can fit well. Posterior B is an example of dependence between dimensions, demonstrating the limitation of having a factorized approximation. Posterior C highlights a shortcoming of performing amortization with a limited-capacity recognition network, where the amortized FFG shares little support with the true posterior. Posterior $\\mathbf { D }$ is a bimodal distribution which demonstrates the ability of the flexible approximation to fit to complex distributions, in contrast to the simple FFG approximation. These observations raise the following question: in more typical VAEs, is the amortization of inference the leading cause of the distribution mismatch, or is it the choice of approximation? ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d905f2c4d9a04d979f0fe40f4c3219474cd602b2da5c5d21a24b897e56a29f0.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">MNIST</td><td colspan=\"2\">Fashion-MNIST</td><td colspan=\"2\">CIFAR-10</td></tr><tr><td>qFFG</td><td>qFlow</td><td>qFFG</td><td>qFlow</td><td>qFFG</td><td>qFlow</td></tr><tr><td>log p(x)</td><td>-89.80</td><td>-88.94</td><td>-97.47</td><td>-97.41</td><td>-14913.15</td><td>-14914.45</td></tr><tr><td>LVAE[qFlow]</td><td>-90.80</td><td>-90.38</td><td>-98.92</td><td>-99.10</td><td>-14914.22</td><td>-14915.57</td></tr><tr><td>LVAE[qFFG]</td><td>-91.23</td><td>-113.54</td><td>-100.53</td><td>-132.46</td><td>-14915.40</td><td>-14919.08</td></tr><tr><td>LVAE[q]</td><td>-92.57</td><td>-91.79</td><td>-104.75</td><td>-103.76</td><td>-14976.57</td><td>-14975.12</td></tr><tr><td>Approximation</td><td>1.43</td><td>1.44</td><td>3.06</td><td>1.69</td><td>2.25</td><td>1.12</td></tr><tr><td>Amortization</td><td>1.34</td><td>1.41</td><td>4.22</td><td>4.66</td><td>61.17</td><td>59.55</td></tr><tr><td>Inference</td><td>2.77</td><td>2.85</td><td>7.28</td><td>6.35</td><td>63.42</td><td>60.67</td></tr></table>",
+        "bbox": [
+            173,
+            99,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Inference Gaps. The columns $q _ { F F G }$ and $q _ { F l o w }$ refer to the variational distribution used for training the model. All numbers are in nats. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 AMORTIZATION VS APPROXIMATION GAP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            295,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here we will compare the influence that the approximation and amortization errors have on the total inference gap. Table 2 are results from training on MNIST, Fashion-MNIST and CIFAR-10. For each dataset, we trained two different approximate posterior distributions: a fully-factorized Gaussian, $q _ { F F G }$ , and a flexible distribution, $q _ { F l o w }$ . Due to the computational cost of optimizing the local parameters for each datapoint, our evaluation is performed on a subset of 1000 datapoints for MNIST and Fashion-MNIST and a subset of 100 datapoints for CIFAR-10. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For MNIST, we see that the amortization and approximation gaps each account for nearly half of the inference gap. On Fashion-MNIST, which is a more difficult dataset to model, the amortization gap becomes larger than the approximation gap. Similarly for CIFAR-10, we see that the amortization gap is much more significant than the approximation gap. Thus, for the three datasets and model architectures that we tested, the amortization gap seems to be the prominent cause of inference suboptimality, especially when the difficulty of the dataset increases. This analysis indicates that improvements in inference will likely be a result of reducing amortization error, rather than approximation errors. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "With these results in mind, would simply increasing the capacity of the encoder improve the amortization gap? We examined this by training the MNIST and Fashion-MNIST models from above but with larger encoders. See Section 6.1.2 for implementation details. Table 3 are the results of this experiment. Comparing to Table 2, we see that for both datasets and both variational distributions, the inference gap decreases and the decrease is mainly due to a reduction in the amortization gap. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/06592f1d17c5dbedbce7c4147be55b5f350aa664be72e25fbe05bd2458d2fb30.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"7\">logp(x) LVAE[qFlow] LVAEqFFG]</td><td colspan=\"2\">MNIST</td><td colspan=\"2\">Fashion-MNIST</td></tr><tr><td>qFFG</td><td>qFlow</td><td>qFFG</td><td>qFlow</td></tr><tr><td>-89.61</td><td>-88.99</td><td>-95.99</td><td>-96.18</td></tr><tr><td>-90.65</td><td>-90.44</td><td>-97.40</td><td>-97.91</td></tr><tr><td>-91.07</td><td>-108.71</td><td>-99.64</td><td>-129.7</td></tr><tr><td>-92.18</td><td>-91.19</td><td>-102.73</td><td>-101.67</td></tr><tr><td>LVAE[q] Approximation 1.46</td><td>1.45</td><td>3.65</td><td>1.73</td></tr><tr><td>Amortization</td><td>1.11</td><td>0.75</td><td>3.09</td><td>3.76</td></tr><tr><td>Inference</td><td>2.56</td><td>2.20</td><td>6.74</td><td>5.49</td></tr></table>",
+        "bbox": [
+            272,
+            613,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Larger Encoder. The columns $q _ { F F G }$ and $q F l o w$ refer to the variational distribution used for training the model. All numbers are in nats. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2.1 INFLUENCE OF FLOWS ON AMORTIZATION GAP ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            808,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The common reasoning for increasing the expressiveness of the approximate posterior is to minimize the difference between the true and approximate, i.e. reduce the approximation gap. However, given that the expressive approximation is often accompanied by many additional parameters, we would like to know if it has an influence on the amortization error. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To investigate this, we trained a VAE in the same manner as Section 5.2. After training, we kept the generator fixed and trained new encoders to fit to the fixed posterior. Specifically, we trained a small encoder with a factorized Gaussian $q$ distribution to obtain a large amortization gap. We then trained a small encoder with a flow distribution. See Section 6.2 for the details of the experiment. The results are shown in Table 4. As expected, we observe that the small encoder has a very large amortization gap. However, when we use $q _ { F l o w }$ as the approximate distribution, we see the approximation gap decrease, but more importantly, there is a significant decrease in the amortization gap. This indicates that the parameters used for increasing the complexity of the approximation also play a large role in diminishing the amortization error. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/e48e6300e3d9c6ccda00973c572649a04270e670abd1d911f4f2f2ccd9f63306.jpg",
+        "table_caption": [
+            "Table 4: Influence of Flows on the Amortization Gap. The parameters used to increase the flexibility of the approximate distribution also reduce the amortization gap. See Section 5.2.1 for details of the experiment. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Variational Family</td><td>qFFG</td><td>qFlow</td></tr><tr><td>logp(x) LVAE[q*]</td><td>-84.70 -86.61</td><td>-84.70 -85.48</td></tr><tr><td>LVAE[q] Approximation</td><td>-129.83 1.91</td><td>-98.58 0.78</td></tr><tr><td>Amortization</td><td>43.22</td><td>13.10</td></tr><tr><td>Inference</td><td>45.13</td><td>13.88</td></tr></table>",
+        "bbox": [
+            349,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "These results are expected given that the parameterization of the Flow distribution can be interpreted as an instance of the RevNet (Gomez et al., 2017) which has demonstrated that Real-NVP like transformations (Dinh et al., 2017) can model complex functions similar to typical MLPs. Thus the flow transformations we employ should also be expected to increase the expressiveness while also increasing the capacity of the encoder. The implication of this observation is that models which improve the flexibility of their variational approximation, and attribute their improved results to the increased expressiveness, may have actually been due to the reduction in amortization error. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 INFLUENCE OF APPROXIMATE POSTERIOR ON TRUE POSTERIOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            512,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We have seen that increasing the expressiveness of the approximation improves the marginal likelihood of the trained model, but to what amount does it alter the true posterior? Will a factorized Gaussian approximation cause the true posterior to be more like a factorized Gaussian or is the true posterior mostly fixed? Just as it is hard to evaluate a generative model by visually inspecting samples from the model, its hard to say how Gaussian the true posterior is by visual inspection. We can quantitatively determine how close the posterior is to a fully factorized Gaussian (FFG) distribution by comparing the marginal log-likelihood estimate, $\\log { \\dot { \\hat { p } } } ( x )$ , and the Optimal FFG bound, $\\mathcal { L } _ { \\mathrm { V A E } } [ q _ { F F G } ^ { * } ]$ . In other words, we are estimating the KL divergence between the optimal Gaussian and the true posterior, $\\mathrm { K L } \\left( q ^ { * } ( z | x ) | | p ( z | x ) \\right)$ . ",
+        "bbox": [
+            173,
+            540,
+            825,
+            667
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Table 2 on MNIST, the Optimal Flow improves upon the Optimal FFG for the FFG trained model by 0.4 nats. In contrast, on the Flow trained model, the difference increases to 12.5 nats. This suggests that the true posterior of a FFG-trained model is closer to FFG than the true posterior of the Flow-trained model. The same observation can be made on the Fashion-MNIST dataset. This implies that the decoder can learn to have a true posterior that fits better to the approximation. Although the generative model can learn to have a posterior that fits to the approximation, it seems that not having this constraint, ie. using a flexible approximate, results in better generative models. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We can use these observations to help justify our approximation and amortization gap results of Section 5.2. Those results showed that the amortization error is often the main cause of inference suboptimality. One reason for this is that the generator accommodates to the choice of approximation, as shown above, thus reducing the approximation error. ",
+        "bbox": [
+            174,
+            776,
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+            833
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Given that we have seen that the generator could accommodate to the choice of approximation, our next question is whether a generator with more capacity can accommodate more. To this end, we trained VAEs with decoders of different sizes and measured the approximation gaps. Specifically, we trained decoders with 0, 2, and 4 hidden layers on MNIST. See Table 5 for the results. We see that as the capacity of the decoder increases, the approximation gap decreases. This result implies that the more flexible the generator, the less flexible the approximate distribution needs to be. ",
+        "bbox": [
+            174,
+            840,
+            825,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0cc491efddd1496fc48a515ff3f2cbc27273a92784e2e60ba493df7ea54403a8.jpg",
+        "table_caption": [
+            "Table 5: Increased decoder capacity reduces approximation gap. All numbers are in nats. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Generator HiddenLayers</td><td>0</td><td>2</td><td>4</td></tr><tr><td>logp(x)</td><td>-100.52</td><td>-86.61</td><td>-83.82</td></tr><tr><td>LVAE[qFFG]</td><td>-104.42</td><td>-84.78</td><td>-82.19</td></tr><tr><td>Approximation Gap</td><td>3.90</td><td>1.83</td><td>1.63</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3.1 ANNEALING THE ENTROPY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            215,
+            416,
+            229
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Typical warm-up (Bowman et al., 2015; Sønderby et al., 2016) refers to annealing $\\mathrm { K L } \\left( q ( \\boldsymbol { z } | \\boldsymbol { x } ) | | p ( \\boldsymbol { z } ) \\right)$ during training. This can also be interpreted as performing maximum likelihood estimation (MLE) early on during training. This optimization technique is known to help prevent the latent variable from degrading to the prior (Burda et al., 2016; Sønderby et al., 2016). We employ a similar annealing scheme during training. Rather than annealing the KL divergence, we anneal the entropy of the approximate distribution $q$ : ",
+        "bbox": [
+            173,
+            241,
+            825,
+            325
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7895dd88cc394d6f6df0ff187f6ff1e40185eafb0c9de2abe90374695f7c2314.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { z \\sim q ( z | x ) } \\left[ \\log p ( x , z ) - \\lambda \\log q ( z | x ) \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            337,
+            624,
+            356
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda$ is annealed from 0 to 1 over training. This can be interpreted as maximum a posteriori (MAP) in the initial phase. Due to its similarity, we will also refer to this technique as warm-up. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We find that warm-up techniques, such as annealing the entropy, are important for allowing the true posterior to be more complex. Table 6 are results from a model trained without the entropy annealing schedule. Comparing these results to Table 2, we observe that the difference between $\\mathcal { L } _ { \\mathrm { V A E } } [ q _ { F F G } ^ { * } ]$ and $\\mathcal { L } _ { \\mathrm { V A E } } [ q _ { F l o w } ^ { * } ]$ is significantly smaller without entropy annealing. This indicates that the true posterior is more Gaussian when entropy annealing is not used. This suggests that, in addition to preventing the latent variable from degrading to the prior, entropy annealing allows the true posterior to better utilize the flexibility of the expressive approximation, resulting in a better trained model. ",
+        "bbox": [
+            173,
+            401,
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+            500
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/420f77f29816604eceb64ae0c586fbbb3b495f2e362a550a087072ec8171b62c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"7\">log p(x) LVAE[qFlow LVAE[qFFG]</td><td colspan=\"2\">MNIST</td><td colspan=\"2\">Fashion-MNIST</td></tr><tr><td>qFFG</td><td>qFlow</td><td>qFFG</td><td>qFlow</td></tr><tr><td>-89.82</td><td>-89.52</td><td>-102.56</td><td>-102.88</td></tr><tr><td>-90.96</td><td>-90.45</td><td>-103.73</td><td>-104.02</td></tr><tr><td>-90.84</td><td>-92.25</td><td>-103.85</td><td>-105.80</td></tr><tr><td>-92.33</td><td>-91.75</td><td>-106.90</td><td>-107.01</td></tr><tr><td>LvAE[q] Approximation 1.02</td><td>0.93</td><td>1.29</td><td>1.14</td></tr><tr><td>Amortization</td><td>1.49</td><td>1.30</td><td>3.05</td><td>2.29</td></tr><tr><td>Inference</td><td>2.51</td><td>2.23</td><td>4.34</td><td>4.13</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Models trained without entropy annealing. The columns $q _ { F F G }$ and $q _ { F l o w }$ refer to the variational distribution used for training the model. All numbers are in nats. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            722,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we investigated how encoder capacity, approximation choice, decoder capacity, and model optimization influence inference suboptimality in terms of the approximation and amortization gaps. We found that the amortization gap is often the leading source of inference suboptimality and that the generator reduces the approximation gap by learning a true posterior that fits to the choice of approximate distribution. We showed that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation. We confirmed that increasing the capacity of the encoder reduces the amortization error. We also showed that optimization techniques, such as entropy annealing, help the generative model to better utilize the flexibility of the expressive variational distribution. Computing these gaps can be useful for guiding improvements to inference in VAEs. Future work includes evaluating other types of expressive approximations and more complex likelihood functions. ",
+        "bbox": [
+            173,
+            757,
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+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES   \nP. Bachman and D. Precup. Training Deep Generative Models: Variations on a Theme. NIPS Approximate Inference Workshop, 2015.   \nS. R. Bowman, L. Vilnis, O. Vinyals, A. M. Dai, R. Jozefowicz, and S. Bengio. Generating Sentences from a Continuous Space. ArXiv e-prints, November 2015.   \nY. Burda, R. Grosse, and R. Salakhutdinov. Importance weighted autoencoders. In ICLR, 2016.   \nDjork-Arne Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast and accurate deep network ´ learning by exponential linear units (elus). arXiv preprint arXiv:1511.07289, 2015.   \nC. Cremer, Q. Morris, and D. Duvenaud. Reinterpreting Importance-Weighted Autoencoders. ICLR Workshop, 2017.   \nL. Dinh, J. Sohl-Dickstein, and S. Bengio. Density estimation using Real NVP. ICLR, 2017.   \nXavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pp. 249–256, 2010.   \nAidan N Gomez, Mengye Ren, Raquel Urtasun, and Roger B Grosse. The reversible residual network: Backpropagation without storing activations. In Advances in Neural Information Processing Systems, pp. 2211–2221, 2017.   \nR. Grosse, Z. Ghahramani, and R. P Adams. Sandwiching the marginal likelihood using bidirectional monte carlo. arXiv preprint arXiv:1511.02543, 2015.   \nR Devon Hjelm, Kyunghyun Cho, Junyoung Chung, Russ Salakhutdinov, Vince Calhoun, and Nebojsa Jojic. Iterative refinement of approximate posterior for training directed belief networks. arXiv preprint arXiv:1511.06382, 2015.   \nMatthew D Hoffman, David M Blei, Chong Wang, and John Paisley. Stochastic variational inference. The Journal of Machine Learning Research, 14(1):1303–1347, 2013.   \nC. Jarzynski. Nonequilibrium equality for free energy differences. Physical Review Letters, 78(14): 2690, 1997.   \nDiederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   \nD.P. Kingma and M. Welling. Auto-Encoding Variational Bayes. In ICLR, 2014.   \nD.P. Kingma, T. Salimans, R. Jozefowicz, X. Chen, I. Sutskever, and M. Welling. Improving Variational Inference with Inverse Autoregressive Flow. NIPS, 2016.   \nR. G. Krishnan, D. Liang, and M. Hoffman. On the challenges of learning with inference networks on sparse, high-dimensional data. ArXiv e-prints, October 2017.   \nHugo Larochelle and Yoshua Bengio. Classification using discriminative restricted boltzmann machines. In Proceedings of the 25th international conference on Machine learning, pp. 536–543. ACM, 2008.   \nL. Maaløe, CK. Sønderby, SK. Sønderby, and O. Winther. Auxiliary Deep Generative Models. ICML, 2016.   \nRadford M Neal et al. Mcmc using hamiltonian dynamics. Handbook of Markov Chain Monte Carlo, 2(11), 2011.   \nR.M. Neal. Annealed importance sampling. Statistics and Computing, 2001.   \nR. Ranganath, D. Tran, and D. M. Blei. Hierarchical Variational Models. ICML, 2016.   \nD.J. Rezende and S Mohamed. Variational Inference with Normalizing Flows. In ICML, 2015. ",
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+    {
+        "type": "text",
+        "text": "J. M. Tomczak and M. Welling. Improving Variational Auto-Encoders using Householder Flow. ArXiv e-prints, November 2016. ",
+        "bbox": [
+            173,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "J. M. Tomczak and M. Welling. Improving Variational Auto-Encoders using convex combination linear Inverse Autoregressive Flow. ArXiv e-prints, June 2017. ",
+        "bbox": [
+            171,
+            363,
+            823,
+            393
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Y. Wu, Y. Burda, R. Salakhutdinov, and R. Grosse. On the Quantitative Analysis of Decoder-Based Generative Models. ICLR, 2017. ",
+        "bbox": [
+            173,
+            402,
+            823,
+            431
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. github.com/zalandoresearch/fashion-mnist, 2017. ",
+        "bbox": [
+            176,
+            440,
+            823,
+            470
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            498,
+            263,
+            513
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.1 MODEL ARCHITECTURES AND TRAINING HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
+            647,
+            545
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.1.1 2D VISUALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            555,
+            366,
+            570
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The VAE model of Fig. 2 uses a decoder $p ( x | z )$ with architecture: $2 - 1 0 0 - 7 8 4$ , and an encoder $q ( z | x )$ with architecture: $7 8 4 - 1 0 0 - 4$ . We use tanh activations and a batch size of 50. The model is trained for 3000 epochs with a learning rate of $1 0 ^ { - 4 }$ using the ADAM optimizer (Kingma & Ba, 2014). ",
+        "bbox": [
+            174,
+            580,
+            825,
+            636
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.1.2 MNIST & FASHION-MNIST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            651,
+            428,
+            666
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Both MNIST and Fashion-MNIST consist of a training and test set with 60k and 10k datapoints respectively, where each datapoint is a $2 8 \\mathbf { x } 2 8$ grey-scale image. We rescale the original images so that pixel values are within the range [0, 1]. For MNIST, We use the statically binarized version described by Larochelle & Bengio (2008). We also binarize Fashion-MINST statically. For both datasets, we adopt the Bernoulli likelihood for the generator. ",
+        "bbox": [
+            174,
+            676,
+            825,
+            747
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The VAE models for MNIST and Fashion-MNIST experiments have the same architecture given in table 7. The flow configuration is given in table 8. ",
+        "bbox": [
+            168,
+            753,
+            825,
+            782
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/dc294408811f93d8e09d36c639be303bc14b682ba9724782b7e0abb1f6382444.jpg",
+        "table_caption": [
+            "Table 7: Neural net architecture for MNIST/Fashion-MNIST experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Inference Network</td><td>Generator</td></tr><tr><td>Input ∈R784</td><td>Input ∈R50</td></tr><tr><td>FC.200-ELU-FC.200-ELU-FC.50+50</td><td>FC.200-ELU-FC.200-ELU-FC.784-Sigmoid</td></tr></table>",
+        "bbox": [
+            196,
+            795,
+            802,
+            840
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In the large encoder setting, we change the number of hidden units for the inference network to be 500, instead of 200. The warm-up models are trained with a linear schedule over the first 400 epochs according to Section 5.3.1. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The activation function is chosen to be the exponential linear unit (ELU, Clevert et al. (2015)), as we observe improved performance compared to tanh. We follow the same learning rate schedule and train for the same amount of epochs as described by Burda et al. (2016). All models are trained with the a batch-size of 100 with ADAM. ",
+        "bbox": [
+            174,
+            103,
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+            159
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6.1.3 CIFAR-10 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            181,
+            303,
+            196
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "CIFAR-10 consists of a training and test dataset with $5 0 \\mathrm { k }$ and $1 0 \\mathrm { k }$ datapoints respectively, where each datapoint is a $3 2 \\times 3 2$ color image. We rescale individual pixel values to be in the range [0, 1]. We follow the discretized logistic likelihood model adopted by Kingma et al. (2016), where each input channel has its own scale learned by an MLP. For the latent variable, we use a 32-dimensional factorized Gaussian for $q ( z | x )$ following Kingma et al. (2016). For all neural networks, ELU is chosen to be the activation function. The specific network architecture is shown in Table 9. ",
+        "bbox": [
+            174,
+            209,
+            825,
+            292
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We adopt a gradually decreasing learning rate with an initialize value of $1 0 ^ { - 3 }$ . Warm-up is applied with a linear schedule over the first 20 epochs. All models are trained with a batch-size of 100 with ADAM. Early-stopping is applied based on the performance on the held-out set. ",
+        "bbox": [
+            174,
+            300,
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+            342
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the model with expressive inference, we use four flow steps as opposed to only two in MNIST/Fashion-MNIST experiments. ",
+        "bbox": [
+            176,
+            349,
+            823,
+            377
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6.2 INFLUENCE OF FLOWS ON AMORTIZATION GAP EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            401,
+            642,
+            416
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The aim of this experiment is to show that the parameters used for increasing the expressiveness of the approximation also contribute to reducing the amortization error. To show this, we train a VAE on MNIST, discard the encoder, then retrain two encoders on the fixed decoder: one with a factorized Gaussian distribution and the other with a parameterized ’flow’ distribution. We use fixed decoder so that the true posterior is constant for both encoders. See 5.2.1 for the results and below for the architecture details. ",
+        "bbox": [
+            174,
+            430,
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+            513
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The architecture of the decoder is: $D _ { Z } - 2 0 0 - 2 0 0 - D _ { X }$ . The architecture of the encoder used to train the decoder is $D _ { X } - 2 0 0 - 2 0 0 - 2 D _ { Z }$ . The approximate distribution $q ( z | x )$ is a factorized Gaussian. ",
+        "bbox": [
+            174,
+            520,
+            825,
+            563
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Next, we describe the encoders which were trained on the fixed trained decoder. In order to highlight a large amortization gap, we employed a very small encoder architecture: $D _ { X } - 2 D _ { Z }$ . This encoder has no hidden layers, which greatly impoverishes its ability and results in a large amortization gap. ",
+        "bbox": [
+            174,
+            569,
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+            612
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We compare two approximate distributions $q ( z | x )$ . Firstly, we experiment with the typical fully factorized Gaussian (FFG). The second is what we call a flow distribution. Specifically, we use the transformations of Dinh et al. (2017). We also include an auxiliary variable so we don’t need to select how to divide the latent space for the transformations. The approximate distribution over the latent $z$ and auxiliary variable $v$ factorizes as: $q ( z , v | x ) = q ( z | x ) \\bar { q } ( v )$ . The $q ( v )$ distribution is simply a ${ \\bf N } ( 0 , 1 )$ distribution. Since we’re using a auxiliary variable, we also require the $r ( v | z )$ distribution which we parameterize as $r ( v | z )$ : $[ D z ] - 5 0 - 5 0 - 2 D z$ . The flow transformation is the same as in Section 3.2, which we apply twice. ",
+        "bbox": [
+            173,
+            618,
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+            731
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/ff50b3c467b9e17c1a94bf52be0b1d99c248a70921d6098e55bc9b34ef3e6fbd.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>q(uolz0)</td><td>r(Ur|zT)</td></tr><tr><td>Input ∈R50</td><td>Input ∈ R50</td></tr><tr><td>FC.100-ELU-FC.100-ELU-FC.50+50</td><td>FC.100-ELU-FC.100-ELU-FC.50+50</td></tr></table>",
+        "bbox": [
+            217,
+            758,
+            774,
+            805
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/09fb210a06ffdb5e31d4c8e2f4cde2949066fa56b3183053d7f755aaf204b46c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">q(Ut+1, Zt+1lUt, zt)</td></tr><tr><td>01（）,02（）</td><td>μ1(.)，μ2(.)</td></tr><tr><td>Input ∈ R50</td><td>Input ∈ R50</td></tr><tr><td>FC.100-ELU-FC.100-ELU-FC.50</td><td>FC.100-ELU-FC.100-ELU-FC.50</td></tr></table>",
+        "bbox": [
+            245,
+            820,
+            751,
+            881
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 8: Flow setting for MNIST/Fashion-MNIST experiments. $q ( v _ { T } , z _ { T } | v _ { 0 } , z _ { 0 } )$ consists of two normalizing flows given in the second tabular. ",
+        "bbox": [
+            173,
+            891,
+            825,
+            920
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/9d557e08e7dc3856d16358dc168e284167e37542f8eb64612d5ebecc4c3cddf5.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Inference Network</td><td>Generator</td></tr><tr><td>Input 32 × 32 color image</td><td>Input ∈ R32</td></tr><tr><td>4 × 4 conv. 64 channels.stride 2.BN 4 × 4 conv.128 channels.stride 2.BN</td><td>FC.256×2×2ELU;FC.64-ELU-FC.32-ELU-FC.3 4 × 4 deconv. 128 channels. stride 2. BN</td></tr><tr><td>4 × 4 conv. 256 channels. stride 2.BN</td><td>4 × 4 deconv.64 channels.stride 2.BN</td></tr><tr><td>FC.32 + 32.output layer for mean and log-variance</td><td>4 × 4 deconv.3 channels.stride 2. Sigmoid</td></tr></table>",
+        "bbox": [
+            174,
+            101,
+            841,
+            188
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 9: Network architecture for CIFAR-10 experiments. For the generator, one of the MLPs immediately after the input layer of the generator outputs channel-wise scales for the discretized logistic likelihood model. BN stands for batch-normalization. ",
+        "bbox": [
+            174,
+            196,
+            830,
+            239
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.3 COMPUTATION OF THE DETERMINANT FOR FLOW ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            267,
+            560,
+            282
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The overall mapping $f$ that performs $( z , v ) \\mapsto ( z ^ { \\prime } , v ^ { \\prime } )$ is the composition of two sheer mappings $f _ { 1 }$ and $f _ { 2 }$ that respectively perform $( z , v ) \\mapsto ( z , v ^ { \\prime } )$ and $( z , v ^ { \\prime } ) \\mapsto ( z ^ { \\prime } , v ^ { \\prime } )$ . Since the Jacobian of either one of the sheer mappings is diagonal, the determinant of the composed transformation’s Jacobian $D f$ can be easily computed: ",
+        "bbox": [
+            174,
+            292,
+            825,
+            349
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c320869631ff857d1c2f37a2bb721f89bd6c8f39de4c3058b4fa3d16fb728ff9.jpg",
+        "text": "$$\n\\operatorname * { d e t } ( D f ) = \\operatorname * { d e t } ( D f _ { 1 } ) \\mathrm { d e t } ( D f _ { 2 } ) = \\Bigl ( \\prod _ { i = 1 } ^ { n } \\sigma _ { 1 } ( z ) _ { i } \\Bigr ) \\Bigl ( \\prod _ { j = 1 } ^ { n } \\sigma _ { 2 } ( v ^ { \\prime } ) _ { j } \\Bigr ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            356,
+            709,
+            398
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.4 LOCAL OPTIMIZATION OF APPROXIMATE DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            414,
+            607,
+            429
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the local FFG optimization, we initialize the mean and variance as the prior, i.e. $\\mathcal { N } ( 0 , I )$ . We optimize the mean and variance using the Adam optimizer with a learning rate of $1 0 ^ { - 3 }$ . To determine convergence, after every 100 optimization steps, we compute the average of the previous 100 ELBO values and compare it to the best achieved average. If it does not improve for 10 consecutive iterations then the optimization is terminated. For the Flow model, the same process is used to optimize all of its parameters. All neural nets for the flow were initialized with a variant of the Xavier initilization (Glorot & Bengio, 2010). We use 100 Monte Carlo samples to compute the ELBO to reduce variance. ",
+        "bbox": [
+            173,
+            439,
+            826,
+            551
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.5 ANNEALED IMPORTANCE SAMPLING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            568,
+            468,
+            583
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Annealed importance sampling (AIS, Neal (2001); Jarzynski (1997)) is a means of computing a lower bound to the marginal log-likelihood. Similarly to the importance weighted bound, AIS must sample a proposal distribution $\\bar { f } _ { 1 } ( z )$ and compute the density of these samples, however, AIS then transforms the samples through a sequence of reversible transitions $\\mathcal { T } _ { t } ( z ^ { \\prime } | z )$ . The transitions anneal the proposal distribution to the desired distribution $f _ { T } ( z )$ . ",
+        "bbox": [
+            174,
+            594,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Specifically, AIS samples an initial state $z _ { 1 } \\sim f _ { 1 } ( z )$ and sets an initial weight $w _ { 1 } = 1$ . For the following annealing steps, $z _ { t }$ is sampled from $\\mathcal { T } _ { t } { \\left( z ^ { \\prime } \\right| } z )$ and the weight is updated according to: ",
+        "bbox": [
+            174,
+            671,
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+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aec817af7c8dfa871b554c2a875194e3a7922a41d5b239bc8f846b90cc5e2167.jpg",
+        "text": "$$\nw _ { t } = w _ { t - 1 } \\frac { f _ { t } ( z _ { t - 1 } ) } { f _ { t - 1 } ( z _ { t - 1 } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            707,
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+            742
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This procedure produces weight $w _ { T }$ such that $\\mathbb { E } \\left[ w _ { T } \\right] = \\mathcal { Z } _ { T } / \\mathcal { Z } _ { 1 }$ , where $Z _ { T }$ and $Z _ { 1 }$ are the normalizing constants of $f _ { T } ( z )$ and $f _ { 1 } ( z )$ respectively. This pertains to estimating the marginal likelihood when the target distribution is $p ( x , z )$ when we integrate with respect to $z$ . ",
+        "bbox": [
+            174,
+            755,
+            825,
+            799
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Typically, the intermediate distributions are simply defined to be geometric averages: $f _ { t } ( z ) ~ =$ $\\dot { f _ { 1 } } \\dot { ( z ) } ^ { 1 - \\dot { \\beta _ { t } } } f _ { T } ( z ) ^ { \\beta _ { t } }$ , where $\\beta _ { t }$ is monotonically increasing with $\\beta _ { 1 } = 0$ and $\\beta _ { T } = 1$ . When $f _ { 1 } ( z ) =$ $p ( z )$ and $f _ { T } ( z ) = p ( x , z )$ , the intermediate distributions are: $f _ { i } ( x ) = p ( z ) p ( x | z ) ^ { \\beta _ { i } }$ . ",
+        "bbox": [
+            174,
+            804,
+            825,
+            848
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Model evaluation with AIS appears early on in the setting of deep belief networks (Salakhutdinov & Murray, 2008). AIS for decoder-based models was also used by $\\mathrm { { W u } }$ et al. (2017). They validated the accuracy of the approach with Bidirectional Monte Carlo (BDMC, Grosse et al. (2015)) and demonstrated the advantage of using AIS over the IWAE bound for evaluation when the inference network overfits to the training data. ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.6 THE INFERENCE GAP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            362,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "How well is inference done in VAEs during training? Are we close to doing the optimal or is there much room for improvement? To answer this question, we quantitatively measure the inference gap: the gap between the true marginal log-likelihood and the lower bound. This amounts to measuring how well inference is being done during training. Since we cannot compute the exact marginal log-likelihood, we estimate it using the maximum of any of its lower bounds, described in 3.3. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            203
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Fig. 3a shows training curves for a FFG and Flow inference network as measured by the VAE, IWAE, and AIS bounds on the training and test set. The inference gap on the training set with the FFG model is 3.01 nats, whereas the Flow model is 2.71 nats. Accordingly, Fig. 3a shows that the training IWAE bound is slightly tighter for the Flow model compared to the FFG. Due to this lower inference gap during training, the Flow model achieves a higher AIS bound on the test set than the FFG model. ",
+        "bbox": [
+            173,
+            209,
+            825,
+            294
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate that a very small inference gap can be achieved, even with a limited approximation such as a factorized Gaussian, we train the model on a small dataset. In this experiment, our training set consists of 1000 datapoints randomly chosen from the original MNIST training set. The training curves on this small datatset are show in Fig. 3b. Even with a factorized Gaussian distribution, the inference gap is very small: the AIS and IWAE bounds are overlapping and the VAE is just slightly below. Yet, the model is overfitting as seen by the decreasing test set bounds. ",
+        "bbox": [
+            173,
+            301,
+            825,
+            385
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/27084004bf0862c2b0451b27bad5bd7bbe3e9917819b0dc3cd0ee5c7c6bd4713.jpg",
+        "image_caption": [
+            "Figure 3: Training curves for a FFG and a Flow inference model on MNIST. AIS provides the tightest lower bound and is independent of encoder overfitting. There is little difference between FFG and Flow models trained on the 1000 datapoints since inference is nearly equivalent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            424,
+            813,
+            625
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "6.6.1 ENCODER AND DECODER OVERFITTING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            722,
+            506,
+            736
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We will begin by explaining how we separate encoder from decoder overfitting. Decoder overfitting is the same as in the regular supervised learning scenario, where we compare the train and test error. To measure decoder overfitting independently from encoder overfitting, we use the AIS bound since it is encoder-independent. Thus we can observe decoder overfitting through the AIS test training curve. In contrast, the encoder can only overfit in the sense that the recognition network becomes unsuitable for computing the marginal likelihood on the test set. Thus, encoder overfitting is computed by: $\\mathcal { L } _ { \\mathrm { A I S } } \\ - \\mathcal { L } _ { \\mathrm { I W } }$ on the test set. ",
+        "bbox": [
+            173,
+            750,
+            825,
+            847
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the small dataset of Fig. 3b, it clear that there is significant encoder and decoder overfitting. A model trained in this setting would benefit from regularization. For Fig. 3a, the model is not overfit and would benefit from more training. However, there is some encoder overfitting due to the gap between the AIS and IWAE bounds on the test set. Comparing the FFG and Flow models, it appears that the Flow does not have a large effect on encoder or decoder overfitting. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "6.7 GAUSSIAN LATENTS WITH FULL COVARIANCE ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            537,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The flexiblity of the Gaussian family with arbitrary covariance lies between that of FFG and Flow. With covariance, the Gaussian distribution can model interactions between different latent dimensions. Yet, compared to Flow, its expressiveness is limited due to its inability to model higher order interactions and its unimodal nature. ",
+        "bbox": [
+            174,
+            128,
+            825,
+            185
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To apply the reparameterization trick, we perform the Cholesky decomposition on the covariance matrix: $\\overrightharpoon { \\Sigma } = L \\overrightharpoon { L } ^ { \\top }$ , where $L$ is lower triangular. A sample from $\\mathcal { N } ( \\boldsymbol { \\mu } , \\boldsymbol { \\Sigma } )$ could be obtained by first sampling from a unit Gaussian $\\epsilon \\sim \\mathcal { N } ( 0 , \\bar { I } )$ , then computing $z = \\mu + L \\epsilon$ . ",
+        "bbox": [
+            173,
+            193,
+            825,
+            234
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To analyze the capability of the Gaussian family, we train several VAEs on MNIST and FashionMNIST with the approximate posterior $q ( z | x )$ being a Gaussian with full covariance. To inspect how well inference is done, we perform the local optimizations described in Section 5.2 with FFG and Flow. ",
+        "bbox": [
+            173,
+            241,
+            825,
+            297
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/f2f3b35421bad321fcaeeb3ee3fa862b5b0861b88b7b85b84c8f3a2536d85dce.jpg",
+        "table_caption": [
+            "Table 10: Gaussian latents trained with full covariance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MNIST</td><td>Fashion-MNIST</td></tr><tr><td>logp(x)</td><td>-89.28</td><td>-96.46</td></tr><tr><td>LVAElqFlow]</td><td>-90.69</td><td>-98.19</td></tr><tr><td>LvAE[FFG]</td><td>-101.84</td><td>-107.89</td></tr><tr><td>LvAE[q]</td><td>-92.05</td><td>-102.93</td></tr></table>",
+        "bbox": [
+            348,
+            308,
+            648,
+            381
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We can see from table 10 that local optimization with FFG on a model trained with full covariance inference produces a bad lower bound. This resonates with the argument that the approximation has a significant influence on the true posterior as described in section 5.3. ",
+        "bbox": [
+            176,
+            422,
+            825,
+            465
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Comparing to numbers in table 2, we can see that the full-covariance VAE trained on MNIST is nearly on par with that trained with Flow (-89.28 vs -88.94). For Fashion-MNIST, the fullcovariance VAE even performs better by a large margin in terms of the estimated log-likelihood (-96.46 vs -97.41). ",
+        "bbox": [
+            173,
+            472,
+            825,
+            529
+        ],
+        "page_idx": 13
+    }
+]

parse/train/S1efxTVYDr/S1efxTVYDr.md

@@ -0,0 +1,440 @@
+# DATA-DEPENDENT GAUSSIAN PRIOR OBJECTIVEFOR LANGUAGE GENERATION
+
+Zuchao $\mathbf { L i } ^ { 1 , 2 , 3 }$ , Rui Wang4,∗, Kehai Chen4, Masao Utiyama4, Eiichiro Sumita4, Zhuosheng Zhang1,2,3 , Hai Zhao1,2,3,∗
+
+1Department of Computer Science and Engineering, Shanghai Jiao Tong University 2Key Laboratory of Shanghai Education Commission for Intelligent Interaction and Cognitive Engineering, Shanghai Jiao Tong University, Shanghai, China 3MoE Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University 4National Institute of Information and Communications Technology (NICT), Kyoto, Japan charlee@sjtu.edu.cn, zhangzs@sjtu.edu.cn, zhaohai@cs.sjtu.edu.cn, {wangrui, khchen, mutiyama, eiichiro.sumita}@nict.go.jp
+
+# ABSTRACT
+
+For typical sequence prediction problems such as language generation, maximum likelihood estimation (MLE) has commonly been adopted as it encourages the predicted sequence most consistent with the ground-truth sequence to have the highest probability of occurring. However, MLE focuses on once-to-all matching between the predicted sequence and gold-standard, consequently treating all incorrect predictions as being equally incorrect. We refer to this drawback as negative diversity ignorance in this paper. Treating all incorrect predictions as equal unfairly downplays the nuance of these sequences’ detailed token-wise structure. To counteract this, we augment the MLE loss by introducing an extra Kullback– Leibler divergence term derived by comparing a data-dependent Gaussian prior and the detailed training prediction. The proposed data-dependent Gaussian prior objective (D2GPo) is defined over a prior topological order of tokens and is poles apart from the data-independent Gaussian prior (L2 regularization) commonly adopted in smoothing the training of MLE. Experimental results show that the proposed method makes effective use of a more detailed prior in the data and has improved performance in typical language generation tasks, including supervised and unsupervised machine translation, text summarization, storytelling, and image captioning.
+
+# 1 INTRODUCTION
+
+Language understanding is the crown jewel of artificial intelligence. As the well-known dictum by Richard Feynman states, “what I cannot create, I do not understand.” Language generation therefore reflects the level of development of language understanding. Language generation models have seen remarkable advances in recent years, especially with the rapid development of deep neural networks (DNNs). There are several models typically used in language generation, namely sequenceto-sequence (seq2seq) models (Kalchbrenner & Blunsom, 2013; Sutskever et al., 2014; Bahdanau et al., 2015; Luong et al., 2015; Vaswani et al., 2017), generative adversarial networks (GANs) (Goodfellow et al., 2014), variational autoencoders (Kingma & Welling, 2013), and auto-regressive networks (Larochelle & Murray, 2011; Van Oord et al., 2016). Language generation is usually modeled as a sequence prediction task, which adopts maximum likelihood estimation (MLE) as the standard training criterion (i.e., objective). MLE has had much success owing to its intuitiveness and flexibility. However, sequence prediction has encountered the following series of problems due to MLE.
+
+• Exposure bias: The model is not exposed to the full range of errors during training. • Loss mismatch: During training, we maximize the log-likelihood, whereas, during inference, the model is evaluated by a different metric such as BLEU or ROUGE. Generation diversity: The generations are dull, generic (Sordoni et al., 2015; Serban et al., 2016; Li et al., 2016a), repetitive, and short-sighted (Li et al., 2016b). • Negative diversity ignorance: MLE fails to assign proper scores to different incorrect model outputs, which means that all incorrect outputs are treated equally during training.
+
+A variety of work has alleviated the above MLE training shortcomings apart from negative diversity ignorance. Negative diversity ignorance is a result of unfairly downplaying the nuance of sequences’ detailed token-wise structure. When the MLE objective compares its predicted and ground-truth sequences, it takes a once-for-all matching strategy; the predicted sequence is given a binary label, either correct or incorrect. However, these incorrect training predictions may be quite diverse and letting the model be aware of which incorrect predictions are more incorrect or less incorrect than others may more effectively guide model training. For instance, an armchair might be mistaken with a deckchair, but it should usually not be mistaken for a mushroom.
+
+To alleviate the issue of the negative diversity ignorance, we add an extra Gaussian prior objective to augment the current MLE training with an extra Kullback–Leibler divergence loss term. The extra loss is computed by comparing two probability distributions, the first of which is from the detailed model training prediction and the second of which is from a ground-truth token-wise distribution and is defined as a kind of data-dependent Gaussian prior distribution. The proposed data-dependent Gaussian prior objective (D2GPo) is then injected into the final loss through a KL divergence term. The D2GPo is poles apart from the commonly adopted data-independent Gaussian prior (L2 regularization) for the purpose of smoothing the training of MLE, which is also directly added into the MLE loss. Experimental results show that the proposed method makes effectively use of a more detailed prior in the data and improves the performance of typical language generation tasks, including supervised and unsupervised machine translation, text summarization, storytelling, and image captioning.
+
+# 2 RELATED WORK
+
+Natural language generation (NLG) has long been considered the most challenging natural language processing (NLP) task (Murty & Kabadi, 1987). NLG techniques have been widely adopted as a critical module in various tasks, including control-free sentence or poem generation (Zhang & Lapata, 2014) and input-conditioned language generation, such as machine translation, image captioning, text summarization, storytelling (Vaswani et al., 2017; Lample et al., 2018; Karpathy & Fei-Fei, 2015; Fan et al., 2018), and sentiment/tense-controlled sentence generation (Hu et al., 2017). In this work, we focus on input-conditioned language generation tasks, though our proposed method can also be applied to other language generation fields.
+
+Input-conditioned language generation tasks are challenging because there is an information imbalance between the input and output in these tasks, especially for cases with non-text input (Shapiro, 1992). Reiter & Dale (2000) discussed different ways of building complicated knowledge-based systems for NLG. In recent years, neural networks (NNs), especially DNNs, have shown promising results in many NLP tasks. Bengio et al. (2003) first proposed the NN language model (NNLM) to exploit the advantages of NNs for language generation tasks. In an NNLM, the $n$ -gram paradigm is extended by the generalization ability of NNs.
+
+Given the ground truth sequence $s = \langle w _ { 1 } , w _ { 2 } , . . . , w _ { t - 1 } \rangle$ , the NNLM adopts the equation
+
+$$
+p _ { t } \approx p ( w _ { t } | w _ { t - n } , w _ { t - n + 1 } , . . . , w _ { t - 1 } ) .
+$$
+
+Mikolov et al. (2010) developed a more general implementation for a language model (called the recurrent NN language model (RNNLM) by integrating a Markov property using a recurrent NN (RNN) to address the NNLMs’ theoretical inability to capture long-term dependencies:
+
+$$
+p _ { t } \approx p ( w _ { t } | \mathbf { R N N } ( w _ { 1 } , w _ { 2 } , . . . , w _ { t - 1 } ) ) .
+$$
+
+The RNNLM is an effective solution because it is designed to capture long-term dependencies. Because of the vanishing gradient problem in RNNs, however, the long-term dependency processing capability is limited. In contrast to an RNN, the Transformer (Vaswani et al., 2017) provides a new self-attention mechanism for handling long-term dependencies in text, resulting in robust performance across diverse tasks. Radford et al. (2018) proposed a Transformer language model called GPT, which uses a left-to-right architecture, where each token pays attention to previous tokens in the self-attention layers of the Transformer. Devlin et al. (2019) introduced a new pre-training objective: the masked language model (MLM), which enables the representation to fuse the left and right contexts and allows us to pre-train a deep bidirectional Transformer called BERT.
+
+The generators of the most current language generation model use the RNNLM or Transformer language model structure. However, as pointed out by Bengio et al. (2015), fitting the distribution of observation data does not mean that satisfactory text will be generated, because the model is not exposed to the full range of errors during training. This is called the exposure bias problem. Reinforcement learning, GANs (Goodfellow et al., 2014; Yu et al., 2017), and end-to-end re-parameterization techniques (Kusner & Hernandez-Lobato, 2016) have been proposed to solve ´ this problem. The exposure bias is no longer an issue in reinforcement learning models because the training sequences are generated by the model itself.
+
+Using MLE for the training objective leads to the problem of loss mismatch. Ranzato et al. (2015) incorporated the evaluation metric into the training of sequence-to-sequence (seq2seq) models and proposed the mixed incremental cross-entropy reinforce (MIXER) training strategy, which is similar to the idea of minimum risk training (Smith & Eisner, 2006; Li & Eisner, 2009; Ayana et al., 2016; Shen et al., 2016). MIXER uses decoder hidden states to predict the bias term and hence reduce the variance, while minimum risk training renormalizes the predicted probabilities. Zhang & Zhao (2018) introduced a new training criterion based on the Hellinger distance for the seq2seq model and empirically compared the models of two optimization categories: minimum divergence and maximum margin.
+
+For the generation diversity problem, Serban et al. (2017) applied a latent variable hierarchical encoder–decoder dialog model to introduce utterance-level variations and facilitate longer responses. Zhao et al. (2017) presented a novel framework based on conditional variational autoencoders that improves generation diversity by sampling a latent variable $z$ and optionally adding linguistic features to constrain the style further.
+
+There is an increasing interest in incorporating problem field knowledge in machine learning approaches (Taskar et al., 2004; Ganchev et al., 2010; Hu et al., 2016). One common way is to design specialized network architectures or features for specific knowledge (e.g., Liang et al. (2017; 2018)). In contrast, for structured probabilistic models, posterior regularization and related frameworks (Ganchev et al., 2010; Liang et al., 2009; Bellare et al., 2009) provide a general means to impose knowledge constraints during model estimation. Hu et al. (2018) established a mathematical correspondence between posterior regularization and reinforcement learning and, using this correspondence, expanded posterior regularization to learn knowledge constraints as the extrinsic reward in reinforcement learning. Our approach can be seen as incorporating a priori knowledge of the language field into language generation learning.
+
+Additionally, Welleck et al. (2019) proposed a new objective, unlikelihood training, which forces unlikely generations to be assigned lower probability by the model. The difference is that (Welleck et al., 2019) focuses on low-frequency words while our model focuses on negative tokens. It is claimed that the likelihood objective itself is at fault, resulting in a model that assigns too much probability to sequences containing repetition and frequent words, unlike those from the human training distribution. From this point of view, there is a point in common with our motivation, which is to make the model prediction consistent with human training distribution to some extent.
+
+# 3 BACKGROUND
+
+Consider a conditional probability model for sequence predictions ${ \pmb y } \sim p _ { \pmb \theta } ( { \pmb x } )$ with parameters $\pmb \theta$ . The target sequence $\textbf {  { y } }$ can be conditioned on any type of source $_ { \textbf { \em x } }$ (e.g., a phrase, sentence, or passage of human language or even an image), which is omitted for simplicity of notation. For the sequence $\pmb { y } = \langle y _ { 1 } , y _ { 2 } , . . . , y _ { l } \rangle$ , the probability $p _ { \theta } ( \pmb { y } | \pmb { x } )$ is
+
+$$
+p _ { \theta } ( { \pmb y } | { \pmb x } ) = p _ { \theta } ( y _ { 1 } | { \pmb x } ) p _ { \theta } ( y _ { 2 } | { \pmb x } , y _ { 1 } ) . . . p _ { \theta } ( y _ { l } | { \pmb x } , y _ { 1 : l - 1 } ) .
+$$
+
+Commonly, sequence prediction models are trained using MLE (also known as teacher forcing) (Williams & Zipser, 1989). MLE minimizes the negative log-likelihood of $p _ { \theta } ( \pmb { y } | \pmb { x } )$ :
+
+$$
+\mathcal { L } _ { \mathrm { M L E } } ( \theta ) = - \log p _ { \theta } ( \pmb { y } | \pmb { x } ) = - \sum _ { i = 1 } ^ { l } \log p _ { \theta } ( y _ { i } | \pmb { x } , \pmb { y } _ { < i } ) .
+$$
+
+Optimizing the MLE objective $\mathcal { L } _ { \mathrm { M L E } } ( \theta )$ is straightforward and meets the principle of empirical risk minimization while focusing on only minimizing losses of the correct target on the training data set.
+
+However, there may be noise in the training data, and forcibly learning the distribution of a training set does not enable the obtained model to reach good generalization. Additionally, for sequence prediction, models trained subject to MLE cursorily evaluate all predictions as either correct or incorrect and ignore the similarity between the correct and “less incorrect” predictions. Incorrect predictions might range from nearly perfect (i.e., one token is mistaken with a synonym) to completely wrong, having nothing in common with the gold sequence. However, MLE training treats all incorrect training predictions equally, which implies that MLE fails to accurately assign scores to diverse (especially negative) model predictions.
+
+# 4 D2GPO: DATA-DEPENDENT GAUSSIAN PRIOR OBJECTIVE
+
+To capture the diversity of negative training predictions, we augment the MLE objective of the model with an additional objective $\mathcal { O }$ that more accurately models such a negative diversity. Without loss of generality, supposing $\tilde { y }$ is the prediction candidate, we introduce a general evaluation function $f ( \tilde { \pmb { y } } , \pmb { y } ) \in \mathbb { R }$ independent of the model prediction, such that with a golden target token $y ^ { * }$ , a higher $f ( \tilde { y } , y ^ { * } )$ value indicates a better $p _ { \theta } ( \tilde { y } | \boldsymbol { x } )$ for a target candidate $\tilde { y } \in V$ (where $V$ is the target candidate set). Note that $f ( \tilde { \pmb y } , \pmb y )$ can also involve other factors such as latent variables and extra supervisions.
+
+There are two main methods of learning $f ( \tilde { \pmb y } , \pmb y )$ in the model. If $p _ { \theta }$ is a GAN-like implicit generative model or an explicit distribution that can be efficiently reparametrized (e.g., Gaussian) (Kingma & Welling, 2013), then one effective method is maximizing $\mathbb { E } _ { p _ { \theta } } \left[ f ( \tilde { \pmb { y } } , \pmb { y } ) \right]$ . The other method is computing the gradient $\nabla _ { \boldsymbol { \theta } } \mathbb { E } _ { p _ { \boldsymbol { \theta } } } \left[ f ( \tilde { \boldsymbol { \ y } } , \boldsymbol { y } ) \right]$ using the log-derivative trick that can suffer from high variance but is often used for the large set of non-parameterizable explicit distributions.
+
+Corresponding to the probability distribution of model predictions $p _ { \theta } ( \cdot )$ , we define a prior distribution $q ( \pmb { y } )$ (for each target $y _ { i }$ , it has its own unique distribution of $q _ { i } = q ( y _ { i } ) )$ which is extracted and derived from the ground-truth data (e.g., language text in language generation tasks). To guide the probability distribution of model predictions $p _ { \theta } ( \cdot )$ to match the prior probability distributions $q ( \cdot )$ , we adopt Kullback–Leibler divergence. Considering also the learning of the evaluation function $f ( \tilde { \boldsymbol { y } } , \boldsymbol { y } )$ , the loss for objective $\mathcal { O }$ is calculated as
+
+$$
+\mathcal { L } _ { \mathcal { O } } ( \pmb { \theta } , \boldsymbol { q } ) = K L ( \boldsymbol { q } ( \pmb { y } ) | | p _ { \theta } ( \pmb { y } | \pmb { x } ) ) - \alpha \mathbb { E } _ { \boldsymbol { q } } \left[ f ( \pmb { \tilde { y } } , \pmb { y } ) \right] ,
+$$
+
+where $\alpha$ is a weight for the evaluation function learning term. We derive the prior distribution $q ( \pmb { y } )$ from the ground-truth data (which is independent of model parameters $\theta$ ), and therefore $\mathbb { E } _ { q } \left[ f ( \tilde { \pmb { y } } , \pmb { y } ) \right] { = } 0$ . Hence, Eq. (5) becomes
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathcal { O } } ( \pmb { \theta } , q ) = K L ( q ( \pmb { y } ) \| p _ { \theta } ( \pmb { y } | \pmb { x } ) ) , } \end{array}
+$$
+
+in which KL divergence can be expanded as
+
+$$
+K L ( q \| p _ { \theta } ) = \mathbb { E } _ { p } ( \log ( \frac { q } { p } ) ) = \sum _ { i } q _ { i } * \log ( q _ { i } ) - \sum _ { i } q _ { i } * \log ( p _ { i } ) .
+$$
+
+The final objective for learning the model is written as
+
+$$
+\begin{array} { r } { \operatorname* { m i n } _ { \theta } \mathcal { L } _ { \mathrm { M L E } } ( \theta ) + \lambda \mathcal { L } _ { \mathcal { O } } ( \theta , q ) , } \end{array}
+$$
+
+where $\lambda$ is the balancing hyperparameter. Because optimizing the original model objective $\mathcal { L } _ { \mathrm { M L E } } ( \theta )$ is straightforward, in the following, we omit the discussion of $\mathcal { L } _ { \mathrm { M L E } } ( \theta )$ and focus on the proposed $\mathcal { L } _ { \mathcal { O } } ( \pmb { \theta } , \bar { \mathbf { q } } )$ .
+
+The prior probability distribution $q ( y ^ { * } )$ on $y ^ { * }$ can be obtained from the evaluation function $f ( \cdot , \cdot )$ with a softmax operation. To expose the mass of the distribution over the classes, Hinton et al. (2015) introduced a softmax temperature mechanism; therefore, the relationship between $q$ and $f ( \tilde { \pmb { y } } , \pmb { y } )$ is
+
+$$
+q ( y ^ { * } ) = \frac { e x p ( f ( \tilde { y } , y ^ { * } ) / T ) } { \sum _ { j } e x p ( f ( \tilde { y } _ { j } , y ^ { * } ) / T ) } ,
+$$
+
+where $T$ is a temperature parameter. When $T  0$ , the distribution becomes a Kronecker distribution (and is equivalent to a one-hot target vector); when $T \to + \infty$ , the distribution becomes a uniform distribution. The softmax operation always turns an evaluation function $f ( \cdot , \cdot )$ into a form of probability distribution no matter the form of the original $f ( \cdot , \cdot )$ ; thus, we only focus on $f ( \cdot , \cdot )$ .
+
+To find a good evaluation function, we have to mine token-wise diversity for all $y ^ { * }$ . Considering all token types $\tilde { y } _ { j }$ in a vocabulary, with respect to each $y ^ { * }$ , there exists a prior topological order $O R D E R ( y ^ { * } )$ among all the known tokens, in which $y ^ { * }$ is always ranked top priority. $f ( \tilde { y } _ { j } , y ^ { * } )$ can then be defined as a monotonic function over the corresponding topological order so that it has a maximal value only when the input is $y ^ { * }$ itself. Note that defining $f ( \cdot , \cdot )$ in this way leads to the resulting $q$ also being monotonic over the corresponding topological order. Considering that $q$ is $a$ priori, it is fixed throughout the learning process.
+
+The remaining questions are about how to find a meaningful evaluation function $f ( \cdot , \cdot )$ for the distribution $q$ . In language generation tasks, we may conveniently take word embedding as the token representation, and let the embedding distance determine such an order $O R D E R ( y ^ { * } )$ for each $y ^ { * }$ . In this work, we adopt the cosine similarity of pre-trained embeddings to sort the token (word/subword) order.
+
+Discussion For the evaluation function $f ( \cdot , \cdot )$ of $q$ , we adopt the Gaussian probability density function, though later we also present experimental results for other types of functions in an ablation study. As the adopted Gaussian prior used in the training objective is derived from a datadependent token-wise distribution, we call it the data-dependent Gaussian prior objective (D2GPo). This objective is a big departure from the Gaussian prior commonly adopted for smoothing in MLE training (which we the data-independent Gaussian prior). The following briefly explains why we chose the Gaussian probability density function and how our D2GPo mathematically differs from the data-independent Gaussian prior.
+
+The central limit theorem indicates that suitably standardized sums of independent random variables have an approximately normal distribution. Thus, any random variable that arises as the sum of a sufficiently large number of small random components can be modeled accurately using a normal distribution. Embedding has a linear additive property (e.g., $k i n g \textrm { - } m a n + w o m a n \approx q u e e n )$ . The additive property of embedding can be explained by inspecting the training objective (Mikolov et al., 2013). Each dimension of an embedding represents a potential feature of the token. Considering each potential feature as an independent random variable, the sum follows a Gaussian distribution centered on the correct vocabulary unit $y ^ { * }$ according to the linear additive property. We can therefore use a Gaussian distribution for the embedding-distance-determined order to effectively model the distribution $q ( y ^ { * } )$ . An overview of the concepts underlying D2GPo is illustrated in Appendix A.1.
+
+The D2GPo in this paper is different from the data-independent Gaussian prior in machine learning optimization theory. We hypothesize and experimentally verify that the embedding feature extracted from the data obeys the Gaussian distribution. The distribution from the prior knowledge of language data is used as a soft target to guide the model language generation process using knowledge distillation. The Gaussian prior in the machine learning optimization theory assumes that each component in the parameter $\theta$ is subject to a zero-mean Gaussian prior distribution, which is equivalent to L2 regularization. In general, our Gaussian prior objective is to act on the guiding target probability, while the Gaussian prior in machine learning is applied to the selection of model parameters.
+
+# 5 EXPERIMENTS AND RESULTS
+
+This section describes the experimental evaluation of D2GPo on a variety of typical language generation tasks: neural machine translation (NMT), text summarization, storytelling, and image captioning. The hyperparameters in D2GPo and effect analysis are given in Appendix A.7.
+
+# 5.1 EMBEDDING PRE-TRAINING
+
+Our proposed D2GPo approach for experimental tasks requires either word embeddings or bytepair-encoding (BPE) (Sennrich et al., 2016b) subword embeddings. We generated the pretrained embeddings using fastText (Bojanowski et al., 2017) with an embedding dimension of 512, a context window of size 5, and 10 negative samples. For NMT, fastText was applied to the concatenation of source and target language monolingual corpora, resulting in cross-lingual BPE subword embedding. For text summarization, we generated BPE subword embedding only on the English monolingual corpora, while for the storytelling and image captioning, we obtained the word embedding also on the English monolingual corpora.
+
+# 5.2 SUPERVISED NMT
+
+We evaluated the model on several widely used translation tasks: WMT14 English-to-German (EN– DE), English-to-French (EN–FR), and WMT16 English-to-Romanian (EN–RO)1 tasks, which all have standard large-scale corpora for NMT evaluation. Owing to space limits, the data details are provided in Appendix A.3. The sentences were encoded using sub-word types based on BPE, which has a shared vocabulary of 40,000 sub-word units for all three tasks. We chose the Transformer NMT (Vaswani et al., 2017) model as our baseline. For the hyperparameters of the Transformer (base/big) models, we followed the settings used by Vaswani et al. (2017). The BLEU (Papineni et al., 2002) score with multi-bleu.pl was calculated during the evaluation.
+
+<table><tr><td>System</td><td>EN-DE</td><td>EN-FR</td><td>EN-RO</td><td>EN-RO + STD</td></tr><tr><td>Vaswani et al. (2017) (base)</td><td>27.30</td><td>38.10</td><td>-</td><td></td></tr><tr><td>Vaswani et al. (2017) (big)</td><td>28.40</td><td>41.00</td><td>1</td><td>-</td></tr><tr><td>Transformer (base)</td><td>27.35</td><td>38.44</td><td>33.22</td><td>36.68</td></tr><tr><td>+ D2GPo</td><td>27.93++</td><td>39.23++</td><td>34.00+</td><td>37.11+</td></tr><tr><td>Transformer (big)</td><td>28.51</td><td>41.05</td><td>33.45</td><td>37.55</td></tr><tr><td>+ D2GP0</td><td>29.10+</td><td>41.77++</td><td>34.13+</td><td>37.92+</td></tr></table>
+
+Table 1: Comparison with baseline and existing systems on supervised translation tasks. Here, $" + + / + "$ after the BLEU score indicates that the proposed method was significantly better than the corresponding baseline Transformer (base or big) at significance levels $p < 0 . 0 1 / 0 . 0 5$ . “STD” represents synthetic training data from (Sennrich et al., 2016b).
+
+In Table 1, we report the performance of our full model, the baseline, and existing systems. Our baseline model obtains results similar to those of Vaswani et al. (2017), the existing strong model used for these tasks. The results indicate that our method performed better than the strong baselines for all language pairs. Our model is not only an improvement on the translation model of large-scale training sets but also performs better for small-scale training sets. Refer to Appendix A.8 and A.9 for analysis of the low-resource scenario and generation diversity.
+
+# 5.3 UNSUPERVISED NMT
+
+For unsupervised machine translation, we also used the three language pairs EN–DE, EN–FR, and EN–RO as our evaluation targets. Note that the evaluation performed on EN–DE uses newstest2016 instead of newstest2014 to ensure the results are comparable with the results of other works; this is unlike supervised machine translation. We used the masked sequence to sequence the pre-training (MASS) model (Song et al., 2019) as our baseline. Following the practice of Song et al. (2019), we pretrained our model with a masked sequence-to-sequence pre-training (MASS) objective (without D2GPo) on EN, FR, DE, and RO monolingual data samples from WMT 2007–2018 News Crawl datasets that respectively cover 190M, 60M, 270M, and 10M sentences. We then fine-tuned the models on the same monolingual data using the back-translation cross-entropy loss (Lample et al., 2018) and our D2GPo loss. For the training dataset, we filtered out sentences longer than 175 words in length and jointly learned 60K BPE sub-word units for each language pair.
+
+<table><tr><td>Method</td><td>EN-FR</td><td>FR-EN</td><td>EN-DE</td><td>DE-EN</td><td>EN-RO</td><td>RO-EN</td></tr><tr><td>Artetxe et al. (2017)</td><td>15.13</td><td>15.56</td><td>6.89</td><td>10.16</td><td></td><td>=</td></tr><tr><td>Lample et al. (2017)</td><td>15.05</td><td>14.31</td><td>9.75</td><td>13.33</td><td>=</td><td>1</td></tr><tr><td>Yang et al. (2018)</td><td>16.97</td><td>15.58</td><td>10.86</td><td>14.62</td><td>=</td><td>=</td></tr><tr><td>Lample et al. (2018)</td><td>25.14</td><td>24.18</td><td>17.16</td><td>21.00</td><td>21.18</td><td>19.44</td></tr><tr><td>XLM (Lample &amp; Conneau, 2019)</td><td>33.40</td><td>33.30</td><td>27.00</td><td>34.30</td><td>33.30</td><td>31.80</td></tr><tr><td>MASS (Song et al.,2019)</td><td>37.50</td><td>34.90</td><td>28.30</td><td>35.20</td><td>35.20</td><td>33.10</td></tr><tr><td>MASS +D2GPo</td><td>37.92</td><td>34.94</td><td>28.42</td><td>35.62</td><td>36.31</td><td>33.41</td></tr></table>
+
+Table 2: BLEU score comparisons between MASS and previous methods of unsupervised NMT.
+
+As shown in Table 2, D2GPo consistently outperformed MASS (the state-of-the-art baseline) on all unsupervised translation pairs. Meanwhile, the MASS and XLMsystems leverage large-scale monolingual pre-training, and the decoder (generator, language model) can still be improved by our D2GPo loss in the fine-tuning phase. This demonstrates the efficiency of the proposed method.
+
+# 5.4 TEXT SUMMARIZATION
+
+Text summarization is a typical language generation task that creates a short and fluent summary of the given long-text document. Song et al. (2019) fine-tuned the MASS pretrained model on the text summarization task and achieved state-of-the-art results. We chose this model as our baseline, maintained consistent pre-training, and used D2GPo loss for enhancements in the fine-tuning phase. We used the Annotated Gigaword corpus as the benchmark, as detailed in Appendix A.4. In the evaluation, ROUGE-1, ROUGE-2, and ROUGE-L (Lin, 2004) are reported.
+
+Table 3: Performance on the text summarization task   
+
+<table><tr><td colspan="2">Model</td><td>ROUGE-1</td><td>ROUGE-2</td><td>ROUGE-L</td></tr><tr><td>Supervised</td><td>RNN-based seq2seq Nallapati et al. (2016)</td><td>35.50 34.97</td><td>15.54 17.17</td><td>32.45 32.70</td></tr><tr><td>Semi-supervised</td><td>MLM pre-training ( g (Song et al.,2019) DAE pre-training (Song et al., 2019) MASS pre-training (Song et al., 2019) MASS + D2GP0</td><td>37.75 35.97 38.73 39.23</td><td>18.45 17.17 19.71 20.11</td><td>34.85 33.14 35.96 36.48</td></tr></table>
+
+Our results for text summarization are listed in Table 3. We compared our $+ { \bf D } 2 \mathrm { { G P o } }$ with our baseline MASS, which is the current state-of-the-art model; $+ { \bf D } 2 \mathrm { { G P o } }$ consistently outperformed the baseline on all evaluation metrics. The models with a semi-supervised setting yielded a large-margin improvement relative to the model without any pre-training, which demonstrates that the supervised pre-training is effective in the text summarization task.
+
+# 5.5 STORYTELLING
+
+Storytelling is at the frontier of current language generation technologies; i.e., stories must maintain a consistent theme throughout and require long-distance dependency modeling. Additionally, stories require creativity and a high-level plot with planning ahead rather than word-by-word generation (Wiseman et al., 2017).
+
+We used the hierarchical story generation model (Fan et al., 2018) (which is introduced in Appendix A.5) as our baseline to test the improvements of D2GPo for the storytelling task. To guarantee the single-variable principle, we added only the D2GPo loss to the story generation model. The prompt generation model is consistent with Fan et al. (2018).
+
+For automatic evaluation, we measured the model perplexity on validation and test sets. Table 4 shows results obtained using D2GPo. It is seen that with the addition of D2GPo, the Conv seq2seq $^ +$ self-attention model substantially improved the likelihood of human-generated stories and even outperformed the ensemble or fusion models without increasing the number of parameters. Perplexity was further reduced with the addition of the fusion mechanism. These results suggest that
+
+Table 4: Perplexity on WRITINGPROMPTS.   
+
+<table><tr><td>Model</td><td>Params</td><td>Valid Perplexity</td><td>Test Perplexity</td></tr><tr><td>GCNN LM</td><td>123.4 M</td><td>54.50</td><td>54.79</td></tr><tr><td>GCNN + self-attention LM</td><td>126.4 M</td><td>51.84</td><td>51.18</td></tr><tr><td>LSTM seq2seq</td><td>110.3M</td><td>46.83</td><td>46.79</td></tr><tr><td>Conv seq2seq</td><td>113.0 M</td><td>45.27</td><td>45.54</td></tr><tr><td>Conv seq2seq + self-attention</td><td>134.7M</td><td>37.37</td><td>37.94</td></tr><tr><td>Ensemble: Conv seq2seq + self-attention</td><td>270.3M</td><td>36.63</td><td>36.93</td></tr><tr><td>Fusion: Conv seq2seq + self-attention</td><td>255.4 M</td><td>36.08</td><td>36.56</td></tr><tr><td>Conv seq2seq + self-attention + D2GPo</td><td>134.7M</td><td>35.56</td><td>35.74</td></tr><tr><td>Fusion: Conv seq2seq + self-attention +D2GPo</td><td>255.4 M</td><td>33.82</td><td>33.90</td></tr></table>
+
+D2GPo improves the quality of language generation greatly, especially in settings where there are fewer restrictions on story generation tasks.
+
+# 5.6 IMAGE CAPTIONING
+
+Image captioning is a task that combines image understanding and language generation. It continues to inspire considerable research at the boundary of computer vision and natural language processing. We elected to experiment with image captioning to verify the performance of D2GPo on a language generation model having diverse types of input.
+
+In our experiments, we evaluated our model on an ablated baseline (top-down, as detailed in Appendix A.6) (Anderson et al., 2018) against prior work on the MSCOCO 2014 caption dataset (Lin et al., 2014), which has became the standard benchmark for image captioning. For validation of model hyperparameters and offline testing, we used Karpathy splits (Karpathy & Fei-Fei, 2015), which have been used extensively in prior work. SPICE (Anderson et al., 2016), CIDEr (Vedantam et al., 2015), METEOR (Denkowski & Lavie, 2014), ROUGE-L, and BLEU were used to evaluate the caption quality.
+
+Table 5: Image caption performance on the MSCOCO Karpathy test split.   
+
+<table><tr><td></td><td>BLEU-1</td><td>BLEU-4</td><td>METEOR</td><td>ROUGE-L</td><td>CIDEr</td><td>SPICE</td></tr><tr><td>Att2in (Rennie et al., 2017)</td><td>=</td><td>31.3</td><td>26.0</td><td>54.3</td><td>101.3</td><td>1</td></tr><tr><td>Att2all (Rennie et al., 2017)</td><td>-</td><td>30.0</td><td>25.9</td><td>53.4</td><td>99.4</td><td>-</td></tr><tr><td>Baseline: Top-down Baseline + D2GPo</td><td>74.5</td><td>33.4</td><td>26.1</td><td>54.4</td><td>105.4</td><td>19.2</td></tr><tr><td></td><td>75.2</td><td>33.6</td><td>26.3</td><td>55.1</td><td>106.6</td><td>19.7</td></tr><tr><td>Baseline + SCST</td><td>77.8</td><td>34.4</td><td>26.6</td><td>56.1</td><td>114.3</td><td>19.9</td></tr><tr><td>Baseline + SCST + D2GPo</td><td>78.0</td><td>34.7</td><td>26.8</td><td>56.3</td><td>116.8</td><td>20.2</td></tr></table>
+
+Table 5 summarizes the performance of our full model and the ResNet Top-down baseline in comparison with the existing strong Self-critical Sequence Training (SCST) (Rennie et al., 2017) approach on the test portion of the Karpathy splits. To ensure a fair comparison, results are only reported for models trained with standard cross-entropy loss (i.e., MLE). All results are reported for a single model with no fine tuning of the input ResNet model. Our ResNet baseline performs slightly better than the SCST models. After incorporating our proposed D2GPo loss, our model improves further across all metrics.
+
+# 6 EVALUATION FUNCTION
+
+According to the analysis in Section 4, for the embedding, we used the Gaussian probability density function as our evaluation function $f ( \cdot )$ ; however, to evaluate the effectiveness of different evaluation functions, we changed the function and tested the performance changes on the supervised NMT
+
+EN-DE task. We used the same experiment settings as described in Section 5.2 and compared the BLEU score changes on the test set, as listed in Table 6.
+
+<table><tr><td>Evaluation Function</td><td>BLEU</td><td>△</td></tr><tr><td>Baseline</td><td>27.35</td><td></td></tr><tr><td>Gaussian</td><td>27.93</td><td>0.58个</td></tr><tr><td>Random</td><td>26.34</td><td>1.01↓</td></tr><tr><td>Linear</td><td>27.45</td><td>0.10个</td></tr><tr><td>Cosine</td><td>27.62</td><td>0.27个</td></tr></table>
+
+Table 6: Ablation study on our proposed D2GPo with different evaluation functions on the supervised NMT WMT14 EN-DE task, with the Transformer-base model.
+
+The table shows that the performance of Gaussian density, linear, and cosine functions increased while the performance of the random function decreased. This shows that the distance information obtained from embedding can effectively guide the generation process. Among these functions, the Gaussian density function had the greatest improvement, which agrees with our analysis of the embedding features obeying the Gaussian distribution. We postulate that because the linear and cosine functions are rough approximations of the Gaussian density function, they perform similarly to the Gaussian density function.
+
+# 7 CONCLUSION
+
+This work proposed a data-dependent Gaussian prior objective (D2GPo) for language generation tasks with the hope of alleviating the difficulty of negative diversity ignorance. D2GPo imposes the prior from (linguistic) data over the sequence prediction models. D2GPo outperformed strong baselines in experiments on classic language generation tasks (i.e., neural machine translation, text summarization, storytelling, and image captioning tasks).
+
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+
+# A APPENDIX
+
+# A.1 CONCEPTS UNDERLYING D2GPO
+
+![](images/f1fc5b6248dc4220592c08e2456a64213eea95997bf88d3423ec9fc6b5cf30c0.jpg)  
+Figure 1: Overview of the concepts underlying D2GPo taking the example of the sentence The little boy sits on the armchair.
+
+# A.2 TOPOLOGICAL ORDER
+
+Specifically, for target $y ^ { * }$ , we calculate the embedding cosine similarity as the distance $d i s t ( \tilde { y } _ { j } , y ^ { * } )$ of $y ^ { * }$ and all other token types in the vocabulary $\tilde { y } _ { j }$ , which are used to give the distance:
+
+$$
+d i s t ( \tilde { y } _ { j } , y ^ { * } ) = c o s i n e \_ s i m i l a r i t y ( e m b ( \tilde { y } _ { j } ) , e m b ( y ^ { * } ) ) .
+$$
+
+Sorting by distance from small to large to obtain the topological order of token types yields
+
+$$
+O R D E R ( y ^ { * } ) = I N D E X ( s o r t ( [ d i s t ( \tilde { y } _ { 1 } , y ^ { * } ) , d i s t ( \tilde { y } _ { 2 } , y ^ { * } ) , . . . , d i s t ( \tilde { y } _ { N } , y ^ { * } ) ] ) ) .
+$$
+
+where $N$ is the vocabulary size and $I N D E X ( \cdot )$ is used to obtain the new sequential index according to the distance sort. We define $O R D E R ( y ^ { * } ) _ { j }$ as the $j$ -th value in $O R D \bar { E } R ( y ^ { * } )$ . Therefore, the evaluation function $f ( \tilde { y } _ { j } , y ^ { * } )$ is converted to a function defined on $O R D E R ( y ^ { * } )$ :
+
+$$
+f ( { \tilde { y } } _ { j } , y ^ { * } ) = f ( O R D E R ( y ^ { * } ) _ { j } ) .
+$$
+
+For example, suppose the vocabulary has 5 tokens (i.e. ${ \mathrm { N } } { = } 5$ ), for the golden target $y ^ { * }$ to be predicted, there is a relationship: $d i s t ( \tilde { y } _ { 2 } , y ^ { * } ) < d i s t ( \tilde { y } _ { 3 } , y ^ { * } ) < d i s t ( \tilde { y } _ { 1 } , y ^ { * } ) < d i s t ( \tilde { y } _ { 5 } , y ^ { * } ) < d i s t ( \tilde { y } _ { 4 } , y ^ { * } )$ , the $O R D E R ( y ^ { * } )$ is $[ 3 , 1 , 2 , 5 , 4 ]$ .
+
+# A.3 SUPERVISED NMT DATA
+
+For the EN–DE translation task, 4.43M bilingual sentence pairs from the WMT’14 dataset, which includes the Common Crawl, News Commentary, and Europarl v7 datasets, were used as training data. The newstest2013 and newstest2014 datasets were used as the dev set and test set, respectively.
+
+For the EN–FR translation task, 36M bilingual sentence pairs from the WMT’14 dataset were used as training data. The newstest2012 and newstest2013 datasets were combined for validation and newstest2014 was used as the test set, following the configuration of Gehring et al. (2017).
+
+For the EN–RO task, we tested two settings; i.e., Europarl v7, which uses only the officially provided parallel corpus, and SETIMES2, which yields 600,000 sentence pairs for a low-resource supervised machine translation study. Alternatively, following the work of Sennrich et al. (2016a), we used synthetic training data (STD) of Sennrich et al. (2016a), which provides $2 . 8 \mathbf { M }$ sentence pairs for training. We used newsdev2016 as the dev set and newstest2016 as the test set. Our reported results on EN-RO are evaluated on a reference for which diacritics are removed from letters.
+
+# A.4 TEXT SUMMARIZATION DATA
+
+The Annotated Gigaword corpus (Napoles et al., 2012) was used as a benchmark (Rush et al., 2015). This data set is derived from news articles and comprises pairs of main sentences in the article (longer) and headline (shorter). The article and headline were respectively used as the source input sentence and reference. The data include approximately $3 . 8 \mathbf { M }$ training samples, 400,000 validation samples, and 2000 test samples.
+
+# A.5 HIERARCHICAL STORY GENERATION MODEL
+
+The hierarchical story generation model (Fan et al., 2018) was proposed for the situation in which a sentence called a prompt that describes the topic of the upcoming story generation is first generated, and then conditions on the prompt are applied when generating the story. Specifically, Fan et al. (2018) used a self-attention gated convolutional language model (GCNN) (Dauphin et al., 2017) as the sequence-to-sequence prompt generation model with top- $k$ random sampling. For prompt-tostory generation, they collected a dataset from Reddit’s WRITINGPROMPTS forum in which each prompt has multiple story responses. With the dataset, they trained a story generation model that benefitted from a novel form of model fusion that improved the relevance of the story to the prompt and added a new gated multi-scale self-attention mechanism to model the long-range context.
+
+# A.6 TOP-DOWN IMAGE CAPTION MODEL
+
+The top-down image captioning model uses a ResNet (He et al., 2016) convolutional neural net pretrained on ImageNet (Deng et al., 2009) to encode each image. Similar to previous work (Rennie et al., 2017), the cited study encoded the full-sized input image with the final convolutional layer of Resnet-101 and used bilinear interpolation to resize the output to a fixed-size spatial representation of $1 0 \times 1 0$ . This is equivalent to the maximum number of spatial regions used in our full model.
+
+# A.7 HYPERPARAMETERS IN D2GPO
+
+During training with our D2GPo, the value of the standard deviation of the KL diversity item $\lambda$ was set to 0.1, and the softmax temperature was $T = 2 . 0$ in all experiments.
+
+To study the effects of hyperparameters (i.e., the standard deviation $\lambda$ and softmax temperature $T$ in D2GPo) on the experimental results, we carried out experiments on WMT14 EN-DE with the Transformer-base model as the baseline 2 and set $\lambda$ as $[ 0 , 0 . \dot { 1 } , 0 . 2 , 0 . 5 , 1 . 0 ]$ , $T$ as [1.0, 2.0, 5.0, 10.0].
+
+![](images/825d1d96a12dfd1681ce96c3d89d2e69a6460828a4ce411118bf06a8c4a3866a.jpg)  
+Figure 3: Performances on WMT14 EN-DE with different $\lambda$ values.
+
+![](images/7b9709370e22bbca43f79b0a064c049145f53adeee68285fa3123042fec658fa.jpg)  
+Figure 4: Performances on WMT14 EN-DE with different $T$ values.
+
+The experimental results reveal that $\lambda$ affects the model training process. We believe that the reason is that a small value of $\lambda$ results in the model being unable to make full use of the prior knowledge (distribution), while a larger value of $\lambda$ will make the model more uncertain because of the higher probability of there being incorrect or even opposite words whose fastText embeddings are similar.
+
+In addition, experimental results show that a small value of $T$ can improve the model to some extent, whereas a large value of $T$ will seriously decrease the performance of the model. Theoretically, when $T$ approaches infinity, the distribution $q$ becomes uniform, and there is no prior knowledge with which to guide the model. A loss penalty is applied to any model prediction, and an excessively high value of $T$ is thus harmful to training.
+
+# A.8 D2GPO UNDER A LOW-RESOURCE SETTING
+
+Priors are generally more helpful in low-data regimes. We sampled 10,000, 100,000, and 600,000 paired sentences from the bilingual training data of WMT16 EN-RO to explore the performance of D2GPo in different low-resource scenarios. We used the same BPE code and learned fastText embeddings in all WMT16 EN-RO training data.
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>10K</td><td rowspan=1 colspan=1>100K</td><td rowspan=1 colspan=1>600K</td></tr><tr><td rowspan=2 colspan=1>Baseline+ D2GPo</td><td rowspan=1 colspan=1>1.01</td><td rowspan=1 colspan=1>17.80</td><td rowspan=2 colspan=1>33.2234.00</td></tr><tr><td rowspan=1 colspan=1>4.33</td><td rowspan=1 colspan=1>20.48</td></tr></table>
+
+Table 7: Comparison of our baseline and our D2GPo method under different training data scales in terms of BLEU on the WMT16 EN-RO test set.
+
+As shown in Table 7, D2GPo outperforms the baseline model, demonstrating the effectiveness of our method in low-resource scenarios. At the same time, the results show that the performance improvement provided by D2GPo increases with fewer training data. This shows that prior knowledge can substantially improve the performance of the model when training data are scarce. A possible reason is that the training data are insufficient to train a robust model. In this case, the injection of prior knowledge can help train the parameters of the model and substantially improve the translation performance. However, with an increase in the number of training data, the model itself can be optimized well, and the improvement gained by introducing prior knowledge is not as substantial as before.
+
+# A.9 EXAMPLES OF IMAGE CAPTIONING
+
+![](images/5be2cb0a62bed3ae756d3a97d55ac465950b55986c5cc7a45df24e40cf6d5a3e.jpg)  
+Table 8: Captions generated for the left image by the various models described in the paper. The models trained with SCST return a more accurate and more detailed summary of the image. The models trained with D2GPo return a more grammatically complete sentence.
+
+# A.10 ANALYSIS ON GENERATION DIVERSITY
+
+Compared with traditional MLE training, D2GPo encourages negative diversity. To examine differences between D2GPo and MLE models, we counted high- and low-frequency words in the training set and compared the frequencies of low-frequency words predicted by the two models and the golden reference on the test set.
+
+<table><tr><td></td><td>#GOLD</td><td>Baseline</td><td>+D2GPo</td></tr><tr><td>#LF</td><td>4915</td><td>3900</td><td>3998</td></tr><tr><td>#SUM</td><td>63086</td><td>55234</td><td>56129</td></tr><tr><td>#RATIO</td><td>7.79%</td><td>7.06%</td><td>7.12%</td></tr></table>
+
+Table 9: The statics of low frequency words in the reference and generations.
+
+The experiment was carried out on WMT14 EN-DE, the baseline model was Transformer-base, and the statistics were calculated at the word level. We chose words with a frequency less than or equal to 100 in the training set as low-frequency words. We used the golden reference (#GOLD), baseline model prediction output, and $+ { \bf D } 2 \mathrm { G P o }$ model prediction output to count the total number of tokens (#SUM) and the number of low-frequency words (#LF). Results are given in Table 8.
+
+The results show that compared with the baseline, the D2GPo optimized model generates more lowfrequency words and has a higher ratio of low-frequency words. However, the number is still far less than the golden reference. It is thus demonstrated that D2GPo increases the diversity of model output.
+
+A.11 EXAMPLES OF STORYTELLING   
+Table 10: Example stories generated by the baselines and our full models.   
+
+<table><tr><td></td><td rowspan=1 colspan=1>Example Prompt: The scientists have discovered something terrible .</td></tr><tr><td></td><td rowspan=1 colspan=1>Generation with Conv seq2seq + Self-Attention (PPL 37.37):</td></tr><tr><td></td><td rowspan=1 colspan=1>The discovery of the discovery was a Iong and boring process. It was a small,(unk),and a simple andelegant solution .The scientists,scientists in charge,and the scientific community,was able to decipherthe meaning of the universe .The theory that the universe was not a very real one ，was that the universe itself was not a planet ,but avery,very real planet.The discovery of a particular object in the universe that the universe was not .We found something thatresembled something that could not be described,but something that could not be described .Wefound a way to make the object a lot more specific .We found that it was a very ，very large thing ，asmall,very small and very large .It was the first thing we noticed that we found that it was a masive,large ,enormous,enormous ,giant ,giant .We found a way to make the whole thing.We found that the entire</td></tr><tr><td></td><td rowspan=1 colspan=1>Generation with_Conv seq2seq + Self-Attention_+ D2GPo (PPL_35.74):</td></tr><tr><td></td><td rowspan=1 colspan=1>Idon&#x27;tunderstand.Whydid you tell us aboutit？&quot;The scientist&#x27;saistant,Dr.harry,looked up fromhis tablet and back at the scientist “ It &#x27;s... you see.” He looked back up at Dr. harry.”What ’s it mean ? ”“ It means that the experiment is on the cusp of time .It’s not the same as it looks .There &#x27;s no explanation It’&#x27;s just that...“ How is it possible that the experiment is going to fail ?”The scientist looked ut the window and then backat Dr. harry,who was still looking out the window ,at the vast blackness of the room .“ You think they &#x27;ll do it ？”“ Well , no, we can ’t be sure,Dr. harry . They &#x27;ll do it.”,Dr. harry turned to the scientist in front of him .”We can &#x27;t risk losing our best friend in the world</td></tr><tr><td></td><td rowspan=1 colspan=1>Generation with_ Fusion of Conv_seq2seq_+_Self-Attention (PPL 36.08):We thought that we should be able to see our entire species . The scientists were n&#x27;t sure what we wouldfind . But they had to be prepared for it . They could n’t be ready .The scientists had never seen anything like this,and yet they hadn&#x27;t seen anything like this {unk)in months. The scientists were just about to be in their own business , when they saw the results .They were n&#x27;t readyfor anything:We thought that we should take a deep breath before we go . But it was n&#x27;t . There was n&#x27;t any sign of any other intelligent life in the universe . It was nothing .The scientists had no idea what was going on . They just watched,waiting for a signal .Theyhad no ideawhat it would mean .But they could n&#x27;t see us .The scientists just knew . They were allafraid of what they would see .They could n&#x27;t see the future .They could n’t see the future .</td></tr><tr><td rowspan=1 colspan=2>Generation with_Fusion of Conv_seq2seq_+_Self-Attention + D2GPo (PPL 33.82):&quot;Im afraid Iwo n&#x27;t be able to find out why my experiment is working .“Well , we ’ve been working on the project for about a month now .“ It&#x27;s been a month and a half since Ilast saw it .&quot;“ We &#x27;re all looking at the results . ”“ You&#x27;ve already been working on it for months now . You think we&#x27;ve found that ? ”“ I do n&#x27;t know, but we do have a lot of research to do .&quot;“ But it ’s not like it was working ,is it ? ”“ We do n&#x27;t know . We &#x27;re not looking for a breakthrough , it &#x27;s just an experiment .”“ It&#x27;s just an experiment ? People will die and the world may be destroyed .The disaster is about to happen,we have to act.&quot;“ What do you mean,it’ll not . It &#x27;s just an experiment . ”“ No ,no ,no ,it is something terrible we cannot ignore &quot;</td></tr></table>

parse/train/Syx72jC9tm/Syx72jC9tm.md

@@ -0,0 +1,345 @@
+# INVARIANT AND EQUIVARIANT GRAPH NETWORKS
+
+Haggai Maron, Heli Ben-Hamu, Nadav Shamir & Yaron Lipman
+
+Department of Computer Science and Applied Mathematics   
+Weizmann Institute of Science   
+Rehovot, Israel
+
+# ABSTRACT
+
+Invariant and equivariant networks have been successfully used for learning images, sets, point clouds, and graphs. A basic challenge in developing such networks is finding the maximal collection of invariant and equivariant linear layers. Although this question is answered for the first three examples (for popular transformations, at-least), a full characterization of invariant and equivariant linear layers for graphs is not known.
+
+In this paper we provide a characterization of all permutation invariant and equivariant linear layers for (hyper-)graph data, and show that their dimension, in case of edge-value graph data, is 2 and 15, respectively. More generally, for graph data defined on $k$ -tuples of nodes, the dimension is the $k$ -th and $2 k$ -th Bell numbers. Orthogonal bases for the layers are computed, including generalization to multigraph data. The constant number of basis elements and their characteristics allow successfully applying the networks to different size graphs. From the theoretical point of view, our results generalize and unify recent advancement in equivariant deep learning. In particular, we show that our model is capable of approximating any message passing neural network.
+
+Applying these new linear layers in a simple deep neural network framework is shown to achieve comparable results to state-of-the-art and to have better expressivity than previous invariant and equivariant bases.
+
+# 1 INTRODUCTION
+
+We consider the problem of graph learning, namely finding a functional relation between input graphs (more generally, hyper-graphs) $\mathcal { G } ^ { \ell }$ and corresponding targets $T ^ { \ell }$ , e.g., labels. As graphs are common data representations, this task received quite a bit of recent attention in the machine learning community Bruna et al. (2013); Henaff et al. (2015); Monti et al. (2017); Ying et al. (2018).
+
+More specifically, a (hyper-)graph data point $\mathcal { G } = ( \mathbb { V } , \pmb { \mathsf { A } } )$ consists of a set of $n$ nodes $\mathbb { V }$ , and values $\pmb { \mathsf { A } }$ attached to its hyper-edges1. These values are encoded in a tensor $\pmb { \mathsf { A } }$ . The order of the tensor $\pmb { \mathsf { A } }$ , or equivalently, the number of indices used to represent its elements, indicates the type of data it represents, as follows: First order tensor represents node-values where $\pmb { \mathsf { A } } _ { i }$ is the value of the $i$ -th node; Second order tensor represents edge-values, where $\pmb { \mathsf { A } } _ { i j }$ is the value attached to the $( i , j )$ edge; in general, $k$ -th order tensor encodes hyper-edge-values, where $\mathsf { \pmb { A } } _ { i _ { 1 } , \dots , i _ { k } }$ represents the value of the hyper-edge represented by $( i _ { 1 } , \dots , i _ { k } )$ . For example, it is customary to represent a graph using a binary adjacency matrix $\pmb { \mathsf { A } }$ , where $\mathsf { \pmb { A } } _ { i j }$ equals one if vertex $i$ is connected to vertex $j$ and zero otherwise. We denote the set of order- $k$ tensors by $\mathbb { R } ^ { n ^ { k } }$ .
+
+The task at hand is constructing a functional relation $f ( \mathbf { A } ^ { \ell } ) \approx T ^ { \ell }$ , where $f$ is a neural network. If $T ^ { \ell } = t ^ { \ell }$ is a single output response then it is natural to ask that $f$ is order invariant, namely it should produce the same output regardless of the node numbering used to encode $\pmb { \mathsf { A } }$ . For example, if we represent a graph using an adjacency matrix $\pmb { \mathsf { A } } = \pmb { \cal A } \in \mathbb { R } ^ { n \times n }$ , then for an arbitrary permutation matrix $_ { r }$ and an arbitrary adjacency matrix $\pmb { A }$ , the function $f$ is order invariant if it satisfies $f ( P ^ { T } A P ) = f ( A )$ . If the targets $T ^ { \ell }$ specify output response in a form of a tensor, $T ^ { \ell } = \boldsymbol { \mathsf { T } } ^ { \ell }$ , then it is natural to ask that $f$ is order equivariant, that is, $f$ commutes with the renumbering of nodes operator acting on tensors. Using the above adjacency matrix example, for every adjacency matrix $\pmb { A }$ and every permutation matrix $_ { P }$ , the function $f$ is equivariant if it satisfies $f ( P ^ { T } A P ) = P ^ { T } f ( A ) P$ . To define invariance and equivariance for functions acting on general tensors $\pmb { \mathsf { A } } \in \mathbb { R } ^ { n ^ { k } }$ we use the reordering operator: $\pmb { P } \star \pmb { \mathsf { A } }$ is defined to be the tensor that results from renumbering the nodes $\mathbb { V }$ according to the permutation defined by $_ { r }$ . Invariance now reads as $f ( P \star \mathsf { \pmb { A } } ) = f ( \mathsf { \pmb { A } } )$ ; while equivariance means $f ( P \star \mathsf { \pmb { A } } ) = P \star f ( \mathsf { \pmb { A } } )$ . Note that the latter equivariance definition also holds for functions between different order tensors, $f : \mathbb { R } ^ { n ^ { k } }  \mathbb { R } ^ { n ^ { l } }$ .
+
+![](images/bda79cfcda8f8f068304197cd21d3aa2f231a9483fe2d24dc5bf83b22825e92e.jpg)  
+Figure 1: The full basis for equivariant linear layers for edge-value data $\ b { \sf A } \in \mathbb { R } ^ { n \times n }$ , for $n = 5$ . The purely linear 15 basis elements, $\mathtt { B } ^ { \mu }$ , are represented by matrices $n ^ { 2 } \times n ^ { 2 }$ , and the 2 bias basis elements (right), ${ \pmb { \complement } } ^ { \lambda }$ , by matrices $n \times n$ , see equation 9.
+
+Following the standard paradigm of neural-networks where a network $f$ is defined by alternating compositions of linear layers and non-linear activations, we set as a goal to characterize all linear invariant and equivariant layers. The case of node-value input $\pmb { \mathsf { A } } = \pmb { a } \in \mathbb { R } ^ { n }$ was treated in the pioneering works of Zaheer et al. (2017); Qi et al. (2017). These works characterize all linear permutation invariant and equivariant operators acting on node-value (i.e., first order) tensors, $\mathbb { R } ^ { n }$ . In particular it it shown that the linear space of invariant linear operators $L : \mathbb { R } ^ { n } \to \mathbb { R }$ is of dimension one, containing essentially only the sum operator, $\mathbf { \Psi } _ { L ( \mathbf { a } ) } = \alpha \mathbf { \dot { 1 } } ^ { T } \mathbf { a }$ . The space of equivariant linear operators $L : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ is of dimension two, $L ( \mathbf { \boldsymbol { a } } ) = \left[ \alpha \mathbf { \boldsymbol { I } } + \beta ( \mathbf { 1 1 } ^ { T } - I ) \right] \mathbf { \boldsymbol { a } } .$ .
+
+The general equivariant tensor case was partially treated in Kondor et al. (2018) where the authors make the observation that the set of standard tensor operators: product, element-wise product, summation, and contraction are all equivariant, and due to linearity the same applies to their linear combinations. However, these do not exhaust nor provide a full and complete basis for all possible tensor equivariant linear layers.
+
+In this paper we provide a full characterization of permutation invariant and equivariant linear layers for general tensor input and output data. We show that the space of invariant linear layers $L : \mathbb { R } ^ { k } \to \mathbb { R }$ is of dimension $\mathrm { b } ( k )$ , where $\mathrm { b } ( k )$ is the $k$ -th Bell number. The $k$ -th Bell number is the number of possible partitions of a set of size $k$ ; see inset for the case $k = 3$ . Furthermore, the space of equivariant linear layers
+
+![](images/96b69817489233f9c7a052ff2c2fb95ad5f2cbad66d6cfaa1e750f649cbcbf6b.jpg)
+
+$L : \mathbb { R } ^ { n ^ { k } } \to \mathbb { R } ^ { n ^ { l } }$ is of dimension $\mathrm { b } ( k + l )$ . Remarkably, this dimension is independent of the size $n$ of the node set $\mathbb { V }$ . This allows applying the same network on graphs of different sizes. For both types of layers we provide a general formula for an orthogonal basis that can be readily used to build linear invariant or equivariant layers with maximal expressive power. Going back to the example of a graph represented by an adjacency matrix $\pmb { A } \in \mathbb { R } ^ { \bar { n } \times n }$ we have $k = 2$ and the linear invariant layers $L : \mathbb { R } ^ { n \times n }  \mathbb { R }$ have dimension ${ \mathrm { b } } ( 2 ) = 2$ , while linear equivariant layers $L : \mathbb { R } ^ { n \times n }  \mathbb { R } ^ { n \times n }$ have dimension $\mathrm { b } ( 4 ) = 1 5$ . Figure 1 shows visualization of the basis to the linear equivariant layers acting on edge-value data such as adjacency matrices.
+
+In Hartford et al. (2018) the authors provide an impressive generalization of the case of node-value data to several node sets, $\mathbb { V } _ { 1 } , \mathbb { V } _ { 2 } , \ldots , \mathbb { V } _ { m }$ of sizes $n _ { 1 } , n _ { 2 } , \ldots , n _ { m }$ . Their goal is to learn interactions across sets. That is, an input data point is a tensor $\mathbf { A } \in \mathbb { R } ^ { n _ { 1 } \times n _ { 2 } \times \cdots \times n _ { m } }$ that assigns a value to each element in the cartesian product $\mathbb { V } _ { 1 } \times \mathbb { V } _ { 2 } \times \dots \times \mathbb { V } _ { m }$ . Renumbering the nodes in each node set using permutation matrices $P _ { 1 } , \ldots , P _ { m }$ (resp.) results in a new tensor we denote by $P _ { 1 : m } \star \pmb { \mathsf { A } }$ . Order invariance means $f ( P _ { 1 : m } \star \mathbf { A } ) = f ( \mathbf { A } )$ and order equivariance is $f ( P _ { 1 : m } \star \pmb { \mathsf { A } } ) = \dot { P } _ { 1 : m } \star f ( \pmb { \mathsf { A } } )$ . Hartford et al. (2018) introduce bases for linear invariant and equivariant layers. Although the layers in Hartford et al. (2018) satisfy the order invariance and equivariance, they do not exhaust all possible such layers in case some node sets coincide. For example, if $\mathbb { V } _ { 1 } = \mathbb { V } _ { 2 }$ they have 4 independent learnable parameters where our model has the maximal number of 15 parameters.
+
+Our analysis allows generalizing the multi-node set case to arbitrary tensor data over $\mathbb { V } _ { 1 } \times \mathbb { V } _ { 2 } \times$ $\cdots \times \mathbb { V } _ { m }$ . Namely, for data points in the form of a tensor $\pmb { \ A } \in \mathbb { R } ^ { k _ { 1 } } \times n _ { 2 } ^ { k _ { 2 } } \times \cdots \times n _ { m } ^ { k _ { m } }$ . The tensor $\pmb { \mathsf { A } }$ attaches a value to every element of the Cartesian product $\mathbb { V } _ { 1 } ^ { k _ { 1 } } \times \dots \times \mathbb { V } _ { 2 } ^ { k _ { 2 } }$ , that is, $k _ { 1 }$ -tuple from $\mathbb { V } _ { 1 }$ , $k _ { 2 }$ -tuple from $\mathbb { V } _ { 2 }$ and so forth. We show that the linear space of invariant linear layers $L : \mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times n _ { 2 } ^ { k _ { 2 } } \times \dots \times n _ { m } ^ { k _ { m } } }  \mathbb { R }$ is of dimension $\textstyle \prod _ { i = 1 } ^ { m } { \mathrm { b } ( k _ { i } ) }$ , while the equivariant linear layers $L :$
+
+$\mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times n _ { 2 } ^ { k _ { 2 } } \times \dots \times n _ { m } ^ { k _ { m } } }  \mathbb { R } ^ { n _ { 1 } ^ { l _ { 1 } } \times n _ { 2 } ^ { l _ { 2 } } \times \dots \times n _ { m } ^ { l _ { m } } }$ has dimension $\textstyle \prod _ { i = 1 } ^ { m } \ b ( k _ { i } + l _ { i } )$ . We also provide orthogonal bases for these spaces. Note that, for clarity, the discussion above disregards biases and features; we detail these in the paper.
+
+In appendix C we show that our model is capable of approximating any message-passing neural network as defined in Gilmer et al. (2017) which encapsulate several popular graph learning models. One immediate corollary is that the universal approximation power of our model is not lower than message passing neural nets.
+
+In the experimental part of the paper we concentrated on possibly the most popular instantiation of graph learning, namely that of a single node set and edge-value data, e.g., with adjacency matrices. We created simple networks by composing our invariant or equivariant linear layers in standard ways and tested the networks in learning invariant and equivariant graph functions: (i) We compared identical networks with our basis and the basis of Hartford et al. (2018) and showed we can learn graph functions like trace, diagonal, and maximal singular vector. The basis in Hartford et al. (2018), tailored to the multi-set setting, cannot learn these functions demonstrating it is not maximal in the graph-learning (i.e., multi-set with repetitions) scenario. We also demonstrate our representation allows extrapolation: learning on one size graphs and testing on another size; (ii) We also tested our networks on a collection of graph learning datasets, achieving results that are comparable to the state-of-the-art in 3 social network datasets.
+
+# 2 PREVIOUS WORK
+
+Our work builds on two main sub-fields of deep learning: group invariant or equivariant networks, and deep learning on graphs. Here we briefly review the relevant works.
+
+Invariance and equivariance in deep learning. In many learning tasks the functions that we want to learn are invariant or equivariant to certain symmetries of the input object description. Maybe the first example is the celebrated translation invariance of Convolutional Neural Networks (CNNs) (LeCun et al., 1989; Krizhevsky et al., 2012); in this case, the image label is invariant to a translation of the input image. In recent years this idea was generalized to other types of symmetries such as rotational symmetries (Cohen & Welling, 2016a;b; Weiler et al., 2018; Cohen et al., 2018). Cohen & Welling (2016a) introduced Group Equivariant Neural Networks that use a generalization of the convolution operator to groups of rotations and reflections; Weiler et al. (2018); Cohen et al. (2018) also considered rotational symmetries but in the case of 3D shapes and spherical functions. Ravanbakhsh et al. (2017) showed that any equivariant layer is equivalent to a certain parameter sharing scheme. If we adopt this point of view, our work reveals the structure of the parameter sharing in the case of graphs and hyper-graphs. In another work, Kondor & Trivedi (2018) show that a neural network layer is equivariant to the action of some compact group iff it implements a generalized form of the convolution operator. Yarotsky (2018) suggested certain group invariant/equivariant models and proved their universality. To the best of our knowledge these models were not implemented.
+
+Learning of graphs. Learning of graphs is of huge interest in machine learning and we restrict our attention to recent advancements in deep learning on graphs. Gori et al. (2005); Scarselli et al. (2009) introduced Graph Neural Networks (GNN): GNNs hold a state (a real valued vector) for each node in the graph, and propagate these states according to the graph structure and learned parametric functions. This idea was further developed in Li et al. (2015) that use gated recurrent units. Following the success of CNNs, numerous works suggested ways to define convolution operator on graphs. One promising approach is to define convolution by imitating its spectral properties using the Laplacian operator to define generalized Fourier basis on graphs (Bruna et al., 2013). Multiple follow-up works (Henaff et al., 2015; Defferrard et al., 2016; Kipf & Welling, 2016; Levie et al., 2017) suggest more efficient and spatially localized filters. The main drawback of spectral approaches is that the generalized Fourier basis is graph-dependent and applying the same network to different graphs can be challenging. Another popular way to generalize the convolution operator to graphs is learning stationary functions that operate on neighbors of each node and update its current state (Atwood & Towsley, 2016; Duvenaud et al., 2015; Hamilton et al., 2017; Niepert et al., 2016; Velickovi ˇ c´ et al., 2017; Monti et al., 2017; Simonovsky & Komodakis, 2017). This idea generalizes the locality and weight sharing properties of the standard convolution operators on regular grids. As shown in the important work of Gilmer et al. (2017), most of the the above mentioned methods (including the spectral methods) can be seen as instances of the general class of Message Passing Neural Networks.
+
+In this section we characterize the collection of linear invariant and equivariant layers. We start with the case of a single node set $\mathbb { V }$ of size $n$ and edge-value data, that is order 2 tensors $\pmb { \mathsf { A } } = \pmb { \cal A } \in \mathbb { R } ^ { n \times n }$ . As a typical example imagine, as above, an adjacency matrix of a graph. We set a bit of notation. Given a matrix $\dot { \boldsymbol { X } _ { \mathrm { ~ \scriptsize ~ \in ~ } \mathbb { R } ^ { a \times b } } }$ we denote $\mathrm { v e c } ( \bar { \boldsymbol X } ) \in \dot { \mathbb R } ^ { a b \times 1 }$ its column stack, and by brackets the inverse action of reshaping to a square matrix, namely $[ \mathrm { v e c } ( X ) ] = X$ . Let $p$ denote an arbitrary permutation and $_ { r }$ its corresponding permutation matrix.
+
+Let $\pmb { L } \in \mathbb { R } ^ { 1 \times n ^ { 2 } }$ denote the matrix representing a general linear operator $L : \mathbb { R } ^ { n \times n }  \mathbb { R }$ in the standard basis, then $L$ is order invariant iff $\underline { { L } } \mathrm { v e c } ( \bar { P ^ { T } } A P ) = \underline { { L } } \mathrm { v e c } ( A )$ . Using the property of the Kronecker product that $\operatorname { v e c } ( X A Y ) = Y ^ { T } \otimes X \operatorname { v e c } ( A )$ , we get the equivalent equality $\dot { L } P ^ { T } \otimes$ $P ^ { T } \mathrm { v e c } ( A ) \stackrel { - } { = } L \mathrm { v e c } ( A )$ . Since the latter equality should hold for every $\pmb { A }$ we get (after transposing both sides of the equation) that order invariant $\pmb { L }$ is equivalent to the equation
+
+for every permutation matrix $_ { r }$ . Note that we used $\pmb { L } ^ { T } = \mathrm { v e c } ( \pmb { L } )$ .
+
+For equivariant layers we consider a general linear operator $L : \mathbb { R } ^ { n \times n }  \mathbb { R } ^ { n \times n }$ and its corresponding matrix $\boldsymbol { L } \in \mathbb { R } ^ { n ^ { 2 } \times n ^ { 2 } }$ . Equivariance of $L$ is now equivalent to $\left[ L \mathrm { v e c } ( P ^ { T } A P ) \right] =$ $P ^ { \acute { T } } [ L \mathrm { v e c } \acute { ( } A ) ] P$ . Using the above property of the Kronecker product again we get $ { \mathbf { } } ^ { L P ^ { T } \otimes }$ $P ^ { T } \mathrm { { v e c } } ( A ) = P ^ { T } \otimes \bar { P ^ { T } } L \mathrm { v e c } ( A )$ . Noting that $P ^ { T } \otimes P ^ { T }$ is an $n ^ { 2 } \times n ^ { \bar { 2 } }$ permutation matrix and its inverse is $P \otimes P$ we get to the equivalent equality $P \otimes P L P ^ { T } \otimes P ^ { T } \mathrm { v \bar { e c } } ( A ) = L \mathrm { v e c } ( A )$ . As before, since this holds for every $\pmb { A }$ and using the properties of the Kronecker product we get that $\pmb { L }$ is order equivariant iff for all permutation matrices $_ { r }$
+
+$$
+P \otimes P \otimes P \otimes P \operatorname { v e c } ( L ) = \operatorname { v e c } ( L ) .
+$$
+
+From equations 1 and 2 we see that finding invariant and equivariant linear layers for the order-2 tensor data over one node set requires finding fixed points of the permutation matrix group represented by Kronecker powers $P \otimes P \otimes \cdots \otimes P$ of permutation matrices $_ { P }$ . As we show next, this is also the general case for order- $k$ tensor data $\pmb { \mathsf { A } } \in \bar { \mathbb { R } } ^ { k }$ over one node set, $\mathbb { V }$ . That is,
+
+$$
+\begin{array} { r l } { \operatorname { i n v a r i a n t } L : \quad } & { P ^ { \otimes k } \mathrm { v e c } ( L ) = \mathrm { v e c } ( L ) } \\ { \mathrm { e q u i v a r i a n t } L : \quad } & { P ^ { \otimes 2 k } \mathrm { v e c } ( L ) = \mathrm { v e c } ( L ) } \end{array}
+$$
+
+k for every permutation matrix $_ { r }$ , where $P ^ { \otimes k } = \overbrace { P \otimes \cdots \otimes P } ^ { \substack { \longleftrightarrow } }$ . In equation 3, $\pmb { L } \in \mathbb { R } ^ { 1 \times n ^ { k } }$ is the matrix of an invariant operator; and in equation 4, L ∈ Rnk×nk i s the matrix of an equivariant operator. We call equations 3,4 the fixed-point equations.
+
+To see this, let us add a bit of notation first. Let $p$ denote the permutation corresponding to the permutation matrix $_ { r }$ . We let $\pmb { P } \star \pmb { \mathsf { A } }$ denote the tensor that results from expressing the tensor $\pmb { \mathsf { A } }$ after renumbering the nodes in $\mathbb { V }$ according to permutation $_ { r }$ . Explicitly, the $( p ( i _ { 1 } ) , p ( i _ { 2 } ) , \dots , p ( i _ { k } ) )$ -th entry of $\pmb { P } \star \pmb { \mathsf { A } }$ equals the $( i _ { 1 } , i _ { 2 } , \ldots , i _ { k } )$ -th entry of $\pmb { \mathsf { A } }$ . The matrix that corresponds to the operator $P \star$ in the standard tensor basis $e ^ { ( i _ { 1 } ) } \otimes \cdots \otimes e ^ { ( i _ { k } ) }$ is the Kronecker power $P ^ { T \otimes k } = ( P ^ { T } ) ^ { \otimes k }$ . Note that $\operatorname { v e c } ( \pmb { \mathsf { A } } )$ is exactly the coordinate vector of the tensor $\pmb { \mathsf { A } }$ in this standard basis and therefore we have $\mathrm { v e c } ( \pmb { P } \star \mathbf { A } ) = \mathbf { \dot { P } } ^ { T \otimes k } \mathrm { v e c } ( \mathbf { A } )$ . We now show:
+
+Proposition 1. A linear layer is invariant (equivariant) if and only if its coefficient matrix satisfies the fixed-point equations, namely equation $^ 3$ (equation 4).
+
+Proof. Similarly to the argument from the order-2 case, let $\pmb { L } \in \mathbb { R } ^ { 1 \times n ^ { k } }$ denote the matrix corresponding to a general linear operator $L : \mathbb { R } ^ { n ^ { k } } \to \mathbb { R }$ . Order invariance means
+
+$$
+L \mathrm { v e c } ( P \star \mathsf { \pmb { A } } ) = L \mathrm { v e c } ( \mathsf { \pmb { A } } ) .
+$$
+
+Using the matrix $P ^ { T \otimes k }$ we have equivalently $L P ^ { T \otimes k } \mathrm { v e c } ( \pmb { \mathsf { A } } ) = L \mathrm { v e c } ( \pmb { \mathsf { A } } )$ which is in turn equivalent to $P ^ { \otimes k } \mathrm { v e c } ( L ) \ = \ \mathrm { v e c } ( L )$ for all permutation matrices $_ { r }$ . For order equivariance, let $\pmb { L } \in \mathbb { R } ^ { n ^ { k } \times n ^ { k } }$ denote the matrix of a general linear operator $L : \mathbb { R } ^ { n ^ { k } } \to \mathbb { R } ^ { n ^ { k } }$ . Now equivariance of $L$ is equivalent to
+
+$$
+[ L \mathrm { v e c } ( P \star \mathsf { \pmb A } ) ] = P \star [ L \mathrm { v e c } ( \mathsf { \pmb A } ) ] .
+$$
+
+Similarly to above this is equivalent to $L P ^ { T \otimes k } \mathrm { v e c } ( \mathbf { A } ) = P ^ { T \otimes k } L \mathrm { v e c } ( \mathbf { A } ) .$ which in turn leads to $P ^ { \otimes k } L \dot { P } ^ { T \otimes k } = L$ , and using the Kronecker product properties we get $P ^ { \otimes 2 k } \mathrm { v e c } ( L ) = \mathrm { v e c } ( L )$ .
+
+We have reduced the problem of finding all invariant and equivariant linear operators $L$ to finding all solutions $\pmb { L }$ of equations 3 and 4. Although the fixed point equations consist of an exponential number of equations with only a polynomial number of unknowns they actually possess a solution space of constant dimension (i.e., independent of $n$ ).
+
+To find the solution of $P ^ { \otimes \ell } \mathrm { v e c } ( { \pmb X } ) = \mathrm { v e c } ( { \pmb X } )$ , where $\pmb { \mathsf { X } } \in \mathbb { R } ^ { n ^ { \ell } }$ , note that $P ^ { \otimes \ell } \mathrm { v e c } ( { \pmb X } ) = \mathrm { v e c } ( { \pmb Q } \star { \pmb X } )$ , where $Q = P ^ { T }$ . As above, the tensor $Q \star \mathsf { X }$ is the tensor resulted from renumbering the nodes in $\mathbb { V }$ using permutation $Q$ . Equivalently, the fixed-point equations we need to solve can be formulated as
+
+$$
+Q \star { \mathsf { X } } = { \mathsf { X } } , \quad \forall Q \ \mathrm { p e r m u t a t i o n \ m a t r i c e s }
+$$
+
+The permutation group is acting on tensors $\mathbf { X } \in \mathbb { R } ^ { n ^ { \ell } }$ with the action ${ \pmb x } \mapsto { \pmb Q } \star { \pmb X }$ . We are looking for fixed points under this action. To that end, let us define an equivalence relation in the index space of tensors $\mathbb { R } ^ { n ^ { \ell } }$ , namely in $[ n ] ^ { \ell }$ , where with a slight abuse of notation (we use light brackets) we set $[ n ] = \{ 1 , 2 , \dots , n \}$ . For multi-indices $a , b \in [ n ] ^ { \ell }$ we set $\mathbf { \mu } _ { a \sim b }$ iff $\mathbf { \delta } _ { a , b }$ have the same equality pattern, that is $\pmb { a } _ { i } = \pmb { a } _ { j } \Leftrightarrow b _ { i } = b _ { j }$ for all $i , j \in [ \ell ]$ .
+
+The equality pattern equivalence relation partitions the index set $[ n ] ^ { \ell }$ into equivalence classes, the collection of which is denoted $[ n ] ^ { \ell } / _ { \sim }$ . Each equivalence class can be represented by a unique partition of the set $[ \ell ]$ where each set in the partition indicates maximal set of identical values. Let us exemplify. For $\ell = 2$ we have two equivalence classes $\gamma _ { 1 } = \{ \{ 1 \} , \{ 2 \} \}$ and $\gamma _ { 2 } = \{ \{ 1 , 2 \} \}$ ; $\gamma _ { 1 }$ represents all multi-indices $( i , j )$ where $i \neq j$ , while $\gamma _ { 2 }$ represents all multi-indices $( i , j )$ where $i = j$ . For $\ell = 4$ , there are 15 equivalence classes $\gamma _ { 1 } = \{ \{ 1 \} , \{ 2 \} , \{ 3 \} , \{ 4 \} \}$ , $\gamma _ { 2 } = \{ \{ 1 \} , \{ 2 \} , \{ 3 , 4 \} \}$ , $\gamma _ { 3 } = \{ \{ 1 , 2 \} , \{ 3 \} , \{ 4 \} \} , . .$ . ; $\gamma _ { 3 }$ represents multi-indices $( i _ { 1 } , i _ { 2 } , i _ { 3 } , i _ { 4 } )$ so that $i _ { 1 } = i _ { 2 }$ , $i _ { 2 } \neq i _ { 3 }$ , $i _ { 3 } \neq i _ { 4 }$ , $i _ { 2 } \neq i _ { 4 }$ .
+
+For each equivalence class $\gamma \in [ n ] ^ { \ell } / _ { \sim }$ we define an order- $\ell$ tensor $\mathbf { B } ^ { \gamma } \in \mathbb { R } ^ { n ^ { \ell } }$ by setting
+
+$$
+\mathbf { { \vec { 8 } } } _ { a } ^ { \gamma } = \left\{ { \begin{array} { l l } { 1 } & { \mathbf { { \vec { a } } } \in \gamma } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} } \right.
+$$
+
+Since we have a tensor $\pmb { \mathsf { B } } ^ { \gamma }$ for every equivalence class $\gamma$ , and the equivalence classes are in oneto-one correspondence with partitions of the set $[ \ell ]$ we have ${ \bf b } ( \ell )$ tensors $\pmb { \mathsf { B } } ^ { \gamma }$ . (Remember that ${ \mathrm { b } } ( \ell )$ denotes the $\ell$ -th Bell number.) We next prove:
+
+Proposition 2. The tensors $\pmb { \mathsf { B } } ^ { \gamma }$ in equation 8 form an orthogonal basis (in the standard innerproduct) to the solution set of equations 7. The dimension of the solution set is therefore ${ \bf b } ( \ell )$ .
+
+Proof. Let us first show that: $\pmb { \chi }$ is a solution to equation 7 iff $\pmb { \chi }$ is constant on equivalence classes of the equality pattern relation, $\sim$ . Since permutation $q : [ n ]  [ n ]$ is a bijection the equality patterns of $\pmb { a } = ( i _ { 1 } , i _ { 2 } , \dotsc , i _ { \ell } ) \in [ n ] ^ { \ell }$ and $q ( \pmb { a } ) = ( q ( i _ { 1 } ) , q ( i _ { 2 } ) , \dots , \bar { q } ( i _ { \ell } ) ) \in [ n ] ^ { \ell }$ are identical, i.e., $a \sim q ( a )$ . Taking the $a \in [ \dot { n } ] ^ { \ell }$ entry of both sides of equation 7 gives $\pmb { \mathsf { X } } _ { q ( \pmb { a } ) } = \pmb { \mathsf { X } } _ { \pmb { a } }$ . Now, if $\pmb { \chi }$ is constant on equivalence classes then in particular it will have the same value at $\textbf { \em a }$ and $q ( a )$ for all ${ \pmb a } \in [ n ] ^ { \ell }$ and permutations $q$ . Therefore $\pmb { \chi }$ is a solution to equation 7. For the only if part, consider a tensor $\pmb { \chi }$ for which there exist multi-indices $\mathbf { \mu } _ { a \sim b }$ (with identical equality patterns) and ${ \pmb X } _ { a } \ne { \pmb X } _ { b }$ then $\pmb { \chi }$ is not a solution to equation 7. Indeed, since $\mathbf { \mu } _ { a \sim b }$ one can find a permutation $q$ so that $\pmb { b } = \pmb { q } ( \pmb { a } )$ and using the equation above, $\pmb { \mathsf { X } } _ { b } = \pmb { \mathsf { X } } _ { q ( \pmb { a } ) } = \pmb { \mathsf { X } } _ { a }$ which leads to a contradiction.
+
+To finish the proof note that any tensor $\pmb { \chi }$ , constant on equivalence classes, can be written as a linear combination of $\pmb { \mathsf { B } } ^ { \gamma }$ , which are merely indicators of the equivalence class. Furthermore, the collection $\pmb { \mathsf { B } } ^ { \gamma }$ have pairwise disjoint supports and therefore are an orthogonal basis. □
+
+Combining propositions 1 and 2 we get the characterization of invariant and equivariant linear layers acting on general $k$ -order tensor data over a single node set $\mathbb { V }$ :
+
+Theorem 1. The space of invariant (equivariant) linear layers $\mathbb { R } ^ { n ^ { k } }  \mathbb { R }$ $( \mathbb { R } ^ { n ^ { k } } \to \mathbb { R } ^ { n ^ { k } } ,$ ) is of dimension $\mathrm { b } ( k )$ $( \mathrm { b } ( 2 k ) )$ with basis elements $\pmb { \mathsf { B } } ^ { \gamma }$ defined in equation 8, where $\gamma$ are equivalence classes in $[ n ] ^ { k } / _ { \sim } ( [ n ] ^ { 2 k } / _ { \sim } )$ .
+
+Biases Theorem 1 deals with purely linear layers, that is without bias, i.e., without constant part. Nevertheless extending the previous analysis to constant layers is straight-forward. First, any constant layer Rnk $\mathbb { R } ^ { n ^ { k } }  \mathbb { R }$ is also invariant so all constant invariant layers are represented by constants $c \in \mathbb { R }$ . For equivariant layers $L : \mathbb { R } ^ { k } \to \mathbb { R } ^ { n ^ { k } }$ we note that equivariance means ${ \mathsf { \pmb { \mathsf { C } } } } = { \cal L } ( { \pmb { P } } \star { \pmb { \mathsf { A } } } ) = { \pmb { P } } \star { \cal L } ( { \pmb { \mathsf { A } } } ) = { \pmb { P } } \star { \pmb { \mathsf { C } } }$ . Representing this equation in matrix form we get $P ^ { T \otimes k } \mathrm { v e c } ( { \bf { C } } ) = \mathrm { v e c } ( { \bf { C } } )$ . This shows that constant equivariant layers on one node set acting on general $k$ -order tensors are also characterized by the fixed-point equations, and in fact have the same form and dimensionality as invariant layers on $k$ -order tensors, see equation 3. Specifically, their basis is $\boldsymbol { \mathsf { B } } ^ { \lambda }$ , $\lambda \in [ n ] ^ { k } / _ { \sim }$ . For example, for $k = 2$ , the biases are shown on the right in figure 1.
+
+Features. It is pretty common that input tensors have vector values (i.e., features) attached to each hyper-edge ( $k$ -tuple of nodes) in $\mathbb { V }$ , that is $\pmb { \mathsf { A } } \in \mathbb { R } ^ { n ^ { k } \times d }$ . Now linear invariant $\mathbb { R } ^ { n ^ { k } \times d } \ \to \ \mathbb { R } ^ { 1 \times d ^ { \prime } }$ or equivariant $\bar { \mathbb { R } ^ { n ^ { k } \times d } } \to \mathbb { R } ^ { n ^ { k } \times d ^ { \prime } }$ layers can be formulated using a slight generalization of the previous analysis. The operator $\pmb { P } \star \pmb { \mathsf { A } }$ is defined to act only on the nodal indices, i.e., $i _ { 1 } , \dots , i _ { k }$ (the first $k$ indices). Explicitly, the $( p ( i _ { 1 } ) , p ( i _ { 2 } ) , \dots , p ( i _ { k } ) , i _ { k + 1 } )$ -th entry of $\pmb { P } \star \pmb { \mathsf { A } }$ equals the $( i _ { 1 } , i _ { 2 } , \ldots , i _ { k } , i _ { k + 1 } ) $ -th entry of $\pmb { \mathsf { A } }$ .
+
+Invariance is now formulated exactly as before, equation 5, namely ${ \cal L } \mathrm { v e c } ( P \star { \bf A } ) = { \cal L } \mathrm { v e c } ( { \bf A } )$ . The matrix that corresponds to $\mathbfcal { P } \star$ acting on $\mathbb { R } ^ { n ^ { k } \times d }$ in the standard basis is $P ^ { T \otimes k } \otimes I _ { d }$ and therefore $L ( P ^ { T \otimes k } \otimes I _ { d } ) \mathrm { v e c } ( \mathbf { A } ) = L \mathrm { v e c } ( \mathbf { A } )$ . Since this is true for all $\pmb { \mathsf { A } }$ we have $\begin{array} { r } { ( P ^ { \otimes k } \otimes I _ { d } \otimes I _ { d ^ { \prime } } ) \operatorname { v e c } ( L ) = } \end{array}$ $\mathrm { v e c } ( L )$ , using the properties of the Kronecker product. Equivariance is written as in equation 6, $[ L \mathrm { v e c } ( P \star \mathsf { \pmb { A } } ) ] = P \star [ L \mathrm { v e c } ( \mathsf { \pmb { A } } ) ]$ . In matrix form, the equivariance equation becomes ${ \pmb { L } } ( { \pmb { P } } ^ { T \otimes k } \otimes$ ${ \bar { { \cal I } } } _ { d } ) \mathrm { v e c } ( { \bf A } ) = \bar { ( } P ^ { T \otimes k } \stackrel { \cdot } { \otimes } { \cal I } _ { d ^ { \prime } } ) { \cal L } \mathrm { v e c } ( { \bf A } )$ , since this is true for all $\pmb { \mathsf { A } }$ and using the properties of the Kronecker product again we get to $\mathring { P ^ { \otimes k } } \otimes I _ { d } \otimes P ^ { \otimes k } \otimes I _ { d ^ { \prime } } \ : \mathrm { v e c } ( L ) = \mathrm { v e c } ( L ) .$ . The basis (with biases) to the solution space of these fixed-point equations is defined as follows. We use $a , b \in [ n ] ^ { k }$ , $i , j \in [ d ] , \ i ^ { \prime } , j ^ { \prime } \in [ d ^ { \prime } ] , \ \hat { \lambda } \in [ n ] ^ { k } / \sim , \mu \in [ n ] ^ { 2 k } / \sim$ .
+
+$$
+\begin{array} { r l } & { \mathsf { B } _ { a , i , i ^ { \prime } } ^ { \lambda , j , j ^ { \prime } } = \left\{ \begin{array} { l l } { 1 } & { a \in \lambda , \ i = j , \ i ^ { \prime } = j ^ { \prime } } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. ; \quad \mathsf { C } _ { i ^ { \prime } } ^ { j ^ { \prime } } = \left\{ \begin{array} { l l } { 1 } & { i ^ { \prime } = j ^ { \prime } } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \\ & { \mathsf { B } _ { a , i , b , i ^ { \prime } } ^ { \mu , j , j ^ { \prime } } = \left\{ \begin{array} { l l } { 1 } & { ( a , b ) \in \mu , \ i = j , \ i ^ { \prime } = j ^ { \prime } } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. ; \quad \mathsf { C } _ { b , i ^ { \prime } } ^ { \lambda , j ^ { \prime } } = \left\{ \begin{array} { l l } { 1 } & { b \in \lambda , \ i ^ { \prime } = j ^ { \prime } } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array}
+$$
+
+Note that these basis elements are similar to the ones in equation 8 with the difference that we have different basis tensor for each pair of input $j$ and output $j ^ { \prime }$ feature channels.
+
+An invariant (equation 10a)/ equivariant (equation 10b) linear layer $L$ including the biases can be written as follows for input $\pmb { \mathsf { A } } \in \mathbb { R } ^ { n ^ { k } \times d }$ :
+
+$$
+\begin{array} { r l } & { L ( \pmb { \mathsf { A } } ) _ { i ^ { \prime } } = \displaystyle \sum _ { a , i } \mathbf { \mathsf { T } } _ { a , i , i ^ { \prime } } \mathbf { A } _ { a , i } + \mathbf { Y } _ { i ^ { \prime } } ; \quad \mathsf { T } = \displaystyle \sum _ { \lambda , j , j ^ { \prime } } w _ { \lambda , j , j ^ { \prime } } \pmb { \mathsf { B } } ^ { \lambda , j , j ^ { \prime } } ; \pmb { \mathsf { Y } } = \displaystyle \sum _ { j ^ { \prime } } b _ { j ^ { \prime } } \pmb { \mathsf { C } } ^ { j ^ { \prime } } } \\ & { L ( \pmb { \mathsf { A } } ) _ { b , i ^ { \prime } } = \displaystyle \sum _ { a , i } \mathbf { \mathsf { T } } _ { a , i , b , i ^ { \prime } } \mathbf { A } _ { a , i } + \mathbf { Y } _ { b , i ^ { \prime } } ; \quad \mathsf { T } = \displaystyle \sum _ { \mu , j , j ^ { \prime } } w _ { \mu , j , j ^ { \prime } } \mathbf { B } ^ { \mu , j , j ^ { \prime } } ; \pmb { \mathsf { Y } } = \displaystyle \sum _ { \lambda , j ^ { \prime } } b _ { \lambda , j ^ { \prime } } \mathbf { C } ^ { \lambda , j ^ { \prime } } } \end{array}
+$$
+
+where the learnable parameters are $w \in \mathbb { R } ^ { \mathrm { b } ( k ) \times d \times d ^ { \prime } }$ and $b \in \mathbb { R } ^ { d ^ { \prime } }$ for a single linear invariant layer $\mathbb { R } ^ { n ^ { k } \times d } \to \mathbb { R } ^ { d ^ { \prime } }$ ; and it is $w \in \mathbb { R } ^ { \mathrm { b } ( 2 k ) \times d \times d ^ { \prime } }$ and $b \in \mathbb { R } ^ { \mathrm { b } ( k ) \times d ^ { \prime } }$ for a single linear equivariant layer $\mathbb { R } ^ { n ^ { k } \times d } \to \mathbb { R } ^ { n ^ { k } \times d ^ { \prime } }$ . The natural generalization of theorem 1 to include bias and features is therefore:
+
+Theorem 2. The space of invariant (equivariant) linear layers $\mathbb { R } ^ { n ^ { k } , d } \to \mathbb { R } ^ { d ^ { \prime } }$ $\mathbb { R } ^ { n ^ { k } \times d }  \mathbb { R } ^ { n ^ { k } \times d ^ { \prime } } )$ is of dimension $d d ^ { \prime } \mathrm { b } ( k ) + d ^ { \prime }$ (for equivariant: $d d ^ { \prime } \mathrm { b } ( 2 k ) + \bar { d ^ { \prime } } \mathrm { b } ( k ) )$ with basis elements defined in equation $^ { g }$ ; equation 10a (10b) show the general form of such layers.
+
+Since, by similar arguments to proposition 2, the purely linear parts $\mathbf { B }$ and biases $\circ$ in equation 9 are independent solutions to the relevant fixed-point equations, theorem 2 will be proved if their number equals the dimension of the solution space of these fixed-point equations, namely $d d ^ { \prime } \mathrm { b } ( k )$ for purely linear part and $d ^ { \prime }$ for bias in the invariant case, and $d d ^ { \prime } \mathrm { b } ( \mathrm { 2 } k )$ for purely linear and $d ^ { \prime } \mathrm { b } ( k )$ for bias in the equivariant case. This can be shown by repeating the arguments of the proof of proposition 2 slightly adapted to this case, or by a combinatorial identity we show in Appendix $\mathbf { B }$ .
+
+For example, figure 1 depicts the 15 basis elements for linear equivariant layers $\mathbb { R } ^ { n \times n } \to \mathbb { R } ^ { n \times n }$ taking as input edge-value (order-2) tensor data $\ b { \sf A } \in \mathbb { R } ^ { n \times n }$ and outputting the same dimension tensor. The basis for the purely linear part are shown as $n ^ { 2 } \times n ^ { 2 }$ matrices while the bias part as $n \times n$ matrices (far right); the size of the node set is $| \mathbb { V } | = n = 5$ .
+
+Mixed order equivariant layers. Another useful generalization of order equivariant linear layers is to linear layers between different order tensor layers, that is, $L : \mathbb { R } ^ { n ^ { k } } \to \bar { \mathbb { R } } ^ { n ^ { l } }$ , where $l \neq k$ . For example, one can think of a layer mapping an adjacency matrix to per-node features. For simplicity we will discuss the purely linear scalar-valued case, however generalization to include bias and/or general feature vectors can be done as discussed above. Consider the matrix $\pmb { L } \in \mathbb { R } ^ { n ^ { l } \times n ^ { k } }$ representing the linear layer $L$ , using the renumbering operator, $P \star$ , order equivariance is equivalent to $[ L \mathrm { v e c } ( P \star \mathsf { \pmb A } ) ] = P \star [ L \mathrm { v e c } ( \mathsf { \pmb A } ) ]$ . Note that while this equation looks identical to equation 6 it is nevertheless different in the sense that the $P \star$ operator in the l.h.s. of this equation acts on $k$ -order tensors while the one on the r.h.s. acts on $l$ -order tensor. Still, we can transform this equation to a matrix equation as before by remembering that $P ^ { T \otimes k }$ is the matrix representation of the renumbering operator $P \star$ acting on $k$ -tensors in the standard basis. Therefore, repeating the arguments in proof of proposition 1, equivariance is equivalent to $P ^ { \otimes ( k + l ) } \mathrm { v e c } ( L ) = \mathrm { v e c } ( L )$ , for all permutation matrices $_ { r }$ . This equation is solved as in section 3.1. The corresponding bases to such equivariant layers are computed as in equation 9b, with the only difference that now $\mathbf { a } \in [ n ] ^ { k }$ , $b \in { \overline { { [ n ] } } } ^ { l }$ , an d $\mu \doteq [ n ] ^ { k + l } / \sim$ .
+
+# 4 EXPERIMENTS
+
+Implementation details. We implemented our method in Tensorflow (Abadi et al., 2016). The equivariant linear basis was implemented efficiently using basic row/column/diagonal summation operators, see appendix A for details. The networks we used are composition of $1 - 4$ equivariant linear layers with ReLU activation between them for the equivariant function setting. For invariant function setting we further added a max over the invariant basis and $1 - 3$ fully-connected layers with ReLU activations.
+
+Table 1: Comparison to baseline methods on synthetic experiments.   
+
+<table><tr><td></td><td colspan="3">Symmetric projection</td><td colspan="3">Diagonal extraction</td><td colspan="4">Max singular vector</td><td colspan="3">Trace</td></tr><tr><td>#Layers</td><td>1</td><td>2</td><td>3</td><td>1</td><td>2</td><td>3</td><td>1</td><td>2</td><td>3</td><td>4</td><td>1</td><td>2</td><td>3</td></tr><tr><td>Trivial predictor Hartford et al.</td><td>4.17</td><td>4.17</td><td>4.17</td><td>0.21</td><td>0.21</td><td>0.21</td><td>0.025</td><td>0.025</td><td>0.025</td><td>0.025</td><td>333.33</td><td>333.33</td><td>333.33</td></tr><tr><td></td><td>2.09</td><td>2.09</td><td>2.09</td><td>0.81</td><td>0.81</td><td>0.81</td><td>0.043</td><td>0.044</td><td>0.043</td><td>0.043</td><td>316.22</td><td>311.55</td><td>307.97</td></tr><tr><td>Ours</td><td>1E-05</td><td>7E-06</td><td>2E-05</td><td>8E-06</td><td>7E-06</td><td>1E-04</td><td>0.015</td><td>0.0084</td><td>0.0054</td><td>0.0016</td><td>0.005</td><td>0.001</td><td>0.003</td></tr></table>
+
+Synthetic datasets. We tested our method on several synthetic equivariant and invariant graph functions that highlight the differences in expressivity between our linear basis and the basis of Hartford et al. (2018). Given an input matrix data $A \in \mathbb { R } ^ { n \times n }$ we considered: (i) projection onto the symmetric matrices $\scriptstyle { \frac { 1 } { 2 } } ( A + A ^ { T } )$ ; (ii) diagonal extraction $\operatorname { l i a g } ( \operatorname { d i a g } ( A ) )$ (keeps only the diagonal and plugs zeros elsewhere); (iii) computing the maximal right singular vector arg $\operatorname* { m a x } _ { \| \pmb { v } \| _ { 2 } = 1 } \| \pmb { A } \pmb { v } \| _ { 2 }$ ; and (iv) computing the trace $\operatorname { t r } ( A )$ . Tasks (i)-(iii) are equivariant while task (iv) is invariant. We created accordingly 4 datasets with $1 0 K$ train and $1 K$ test examples of $4 0 \times 4 0$ matrices; for tasks (i), (ii), (iv) we used i.i.d. random matrices with uniform distribution in $[ 0 , 1 0 ]$ ; we used mean-squared error (MSE) as loss; for task (iii) we random matrices with uniform distribution of singular values in $[ 0 , 0 . 5 ]$ and spectral $\mathrm { g a p } \geq 0 . 5$ ; due to sign ambiguity in this task we used cosine loss of the form $l ( \pmb { x } , \pmb { y } ) = 1 - \left. \pmb { x } / \left\| \pmb { x } \right\| , \pmb { y } / \left\| \pmb { y } \right\| \right. ^ { 2 }$ .
+
+We trained networks with 1, 2, and 3 hidden layers with 8 feature channels each and a single fullyconnected layer. Both our models as well as Hartford et al. (2018) use the same architecture but with different bases for the linear layers. Table 1 logs the best mean-square error of each method over a set of hyper-parameters. We add the MSE for the trivial mean predictor.
+
+Table 2: Generalization.   
+
+<table><tr><td></td><td>30</td><td>40</td><td>50</td></tr><tr><td>sym</td><td>0.0053</td><td>3.8E-05</td><td>0.0013</td></tr><tr><td>svd</td><td>0.0108</td><td>0.0084</td><td>0.0096</td></tr><tr><td>diag</td><td>0.0150</td><td>1.5E-05</td><td>0.0055</td></tr></table>
+
+This experiment emphasizes simple cases in which the additional parameters in our model, with respect to Hartford et al. (2018), are needed. We note that Hartford et al. (2018) target a different scenario where the permutations acting on the rows and columns of the input matrix are not necessarily the same. The assumption taken in this paper, namely, that the same permutation acts on both rows and columns, gives rise to additional parameters that are associated with the diagonal and with the transpose of the matrix (for a complete list of layers for the $k = 2$ case see appendix A). In case of an input matrix that represents graphs, these parameters can be understood as parameters that control self-edges or node features, and incoming/outgoing edges in a different way. Table 2 shows the result of applying the learned equivariant networks from the above experiment to graphs (matrices) of unseen sizes of $n = 3 0$ and $n = 5 0$ . Note, that although the network was trained on a fixed size, the network provides plausible generalization to different size graphs. We note that the generalization of the invariant task of computing the trace did not generalize well to unseen sizes and probably requires training on different sizes as was done in the datasets below.
+
+Table 3: Graph Classification Results.   
+
+<table><tr><td>dataset</td><td>MUTAG</td><td>PTC</td><td>PROTEINS</td><td>NCI1</td><td>NCI109</td><td>COLLAB</td><td>IMDB-B</td><td>IMDB-M</td></tr><tr><td>size</td><td>188</td><td>344</td><td>1113</td><td>4110</td><td>4127</td><td>5000</td><td>1000</td><td>1500</td></tr><tr><td>classes</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>3</td><td>2</td><td>3</td></tr><tr><td>avg node #</td><td>17.9</td><td>25.5</td><td>39.1</td><td>29.8</td><td>29.6</td><td>74.4</td><td>19.7</td><td>13</td></tr><tr><td colspan="9">Results</td></tr><tr><td>DGCNN</td><td>85.83±1.7</td><td>58.59±2.5</td><td>75.54±0.9</td><td>74.44±0.5</td><td>NA</td><td>73.76±0.5</td><td>70.03±0.9</td><td>47.83±0.9</td></tr><tr><td>PSCN (k=10)</td><td>88.95±4.4</td><td>62.29±5.7</td><td>75±2.5</td><td>76.34±1.7</td><td>NA</td><td>72.6±2.2</td><td>71±2.3</td><td>45.23±2.8</td></tr><tr><td>DCNN</td><td>NA</td><td>NA</td><td>61.29±1.6</td><td>56.61± 1.0</td><td>NA</td><td>52.11±0.7</td><td>49.06±1.4</td><td>33.49±1.4</td></tr><tr><td>ECC</td><td>76.11</td><td>NA</td><td>NA</td><td>76.82</td><td>75.03</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>DGK</td><td>87.44±2.7</td><td>60.08±2.6</td><td>75.68±0.5</td><td>80.31±0.5</td><td>80.32±0.3</td><td>73.09±0.3</td><td>66.96±0.6</td><td>44.55±0.5</td></tr><tr><td>DiffPool</td><td>NA</td><td>NA</td><td>78.1</td><td>NA</td><td>NA</td><td>75.5</td><td>NA</td><td>NA</td></tr><tr><td>CCN</td><td>91.64±7.2</td><td>70.62±7.0</td><td>NA</td><td>76.27±4.1</td><td>75.54±3.4</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>GK</td><td>81.39±1.7</td><td>55.65±0.5</td><td>71.39±0.3</td><td>62.49±0.3</td><td>62.35±0.3</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>RW</td><td>79.17±2.1</td><td>55.91±0.3</td><td>59.57±0.1</td><td>&gt;3days</td><td>NA</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>PK</td><td>76±2.7</td><td>59.5±2.4</td><td>73.68±0.7</td><td>82.54±0.5</td><td>NA</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>WL</td><td>84.11±1.9</td><td>57.97±2.5</td><td>74.68±0.5</td><td>84.46±0.5</td><td>85.12±0.3</td><td>NA</td><td>NA</td><td>NA</td></tr><tr><td>FGSD</td><td>92.12</td><td>62.80</td><td>73.42</td><td>79.80</td><td>78.84</td><td>80.02</td><td>73.62</td><td>52.41</td></tr><tr><td>AWE-DD</td><td>NA</td><td>NA</td><td>NA</td><td>NA</td><td>NA</td><td>73.93±1.9</td><td>74.45 ±5.8</td><td>51.54 ± 3.6</td></tr><tr><td>AWE-FB</td><td>87.87±9.7</td><td>NA</td><td>NA</td><td>NA</td><td>NA</td><td>70.99 ± 1.4</td><td>73.13 ±3.2</td><td>51.58 ± 4.6</td></tr><tr><td>ours</td><td>84.61±10</td><td>59.47±7.3</td><td>75.19±4.3</td><td>73.71±2.6</td><td>72.48±2.5</td><td>77.92±1.7</td><td>71.27±4.5</td><td>48.55±3.9</td></tr></table>
+
+Graph classification. We tested our method on standard benchmarks of graph classification. We use 8 different real world datasets from the benchmark of Yanardag & Vishwanathan (2015): five of these datasets originate from bioinformatics while the other three come from social networks. In all datasets the adjacency matrix of each graph is used as input and a categorial label is assigned as output. In the bioinformatics datasets node labels are also provided as inputs. These node labels can be used in our framework by placing their 1-hot representations on the diagonal of the input.
+
+Table 3 specifies the results for our method compared to state-of-the-art deep and non-deep graph learning methods. We follow the evaluation protocol including the 10-fold splits of Zhang et al. (2018). For each dataset we selected learning and decay rates on one random fold. In all experiments we used a fixed simple architecture of 3 layers with (16, 32, 256) features accordingly. The last equivariant layer is followed by an invariant max layer according to the invariant basis. We then add two fully-connected hidden layers with (512, 256) features.
+
+We compared our results to seven deep learning methods: DGCNN (Zhang et al., 2018), PSCN (Niepert et al., 2016), DCNN (Atwood & Towsley, 2016), ECC (Simonovsky & Komodakis, 2017), DGK (Yanardag & Vishwanathan, 2015), DiffPool (Ying et al., 2018) and CCN (Kondor et al., 2018). We also compare our results to four popular graph kernel methods: Graphlet Kernel (GK) (Shervashidze et al., 2009),Random Walk Kernel (RW) (Vishwanathan et al., 2010), Propagation Kernel (PK) (Neumann et al., 2016), and Weisfeiler-lehman kernels (WL) (Shervashidze et al., 2011) and two recent feature-based methods: Family of Graph Spectral Distance (FGSD) (Verma & Zhang, 2017) and Anonymous Walk Embeddings (AWE) (Ivanov & Burnaev, 2018). Our method achieved results comparable to the state-of-the-art on the three social networks datasets, and slightly worse results than state-of-the-art on the biological datasets.
+
+# 5 GENERALIZATIONS TO MULTI-NODE SETS
+
+Lastly, we provide a generalization of our framework to data that is given on tuples of nodes from a collection of node sets $\mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times n _ { 2 } ^ { k _ { 2 } } \times \dots \times n _ { m } ^ { k _ { m } } }$ . We characterize invariant linear layers $\mathbb { V } _ { 1 } , \mathbb { V } _ { 2 } , \ldots , \mathbb { V } _ { m }$ of sizes $L : \mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times \dots \times n _ { m } ^ { k _ { m } } }  \mathbb { R }$ $n _ { 1 } , n _ { 2 } , \ldots , n _ { m }$ (resp.), namely and equivariant ${ \textbf { \textsf { A } } } \in$ linear layer $L : \mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times \dots \times n _ { m } ^ { k _ { m } } }  \mathbb { R } ^ { n _ { 1 } ^ { l _ { 1 } } \times \dots \times n _ { m } ^ { l _ { m } } }$ , where for simplicity we do not discuss features that can be readily added as discussed in section 3. Note that the case of $k _ { i } = l _ { i } = 1$ for all $i = 1 , \ldots , m$ is treated in Hartford et al. (2018). The reordering operator now is built out of permutation matrices $P _ { i } \in \mathbb { R } ^ { n _ { i } \times n _ { i } }$ ${ p } _ { i }$ denotes the permutation), $i = 1 , \ldots , m$ , denoted $P _ { 1 : m } \star$ , and defined as follows: the $( p _ { 1 } ( { \pmb a } _ { 1 } ) , p _ { 2 } ( { \pmb a } _ { 2 } ) , \dots , p _ { m } ( { \pmb a } _ { m } ) )$ -th entry of the tensor $P _ { 1 : m } \star \pmb { \mathsf { A } }$ , where $\mathbf { a } _ { i } \in [ n _ { i } ] ^ { k _ { i } }$ is defined to be the $( \pmb { a } _ { 1 } , \pmb { a } _ { 2 } , \dots , \pmb { a } _ { m } )$ -th entry of the tensor $\pmb { \mathsf { A } }$ . Rewriting the invariant and equivariant equations, i.e., equation 5, 6, in matrix format, similarly to before, we get the fixed-point equations: $M \mathrm { v e c } ( { \cal L } ) = \mathrm { v e c } ( { \cal L } )$ for invariant, and $M \otimes M \mathrm { v e c } ( L ) = \mathrm { v e c } ( L )$ for equivariant, where $M = P _ { 1 } ^ { \otimes k _ { 1 } } \otimes \cdot \cdot \cdot \otimes P _ { m } ^ { \otimes k _ { m } }$ . The solution of these equations would be linear combinations of basis tensor similar to equation 9 of the form
+
+$$
+\begin{array} { r } { \mathsf { t } : \mathsf { B } _ { a _ { 1 } , \ldots , a _ { m } } ^ { \lambda _ { 1 } , \ldots , \lambda _ { m } } = \left\{ \begin{array} { l l } { 1 } & { a _ { i } \in \lambda _ { i } , \forall i } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. ; \quad \mathrm { e q u i v a r i a n t : } \ \mathsf { B } _ { a _ { 1 } , \ldots , a _ { m } , b _ { 1 } , \ldots , b _ { m } } ^ { \mu _ { 1 } , \ldots , \mu _ { m } } = \left\{ \begin{array} { l l } { 1 } & { ( a _ { i } , b _ { i } ) \in \mu _ { i } , \forall i } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array}
+$$
+
+where $\lambda _ { i } \in [ n _ { i } ] ^ { k _ { i } } , \mu _ { i } \in [ n _ { i } ] ^ { k _ { i } + l _ { i } } , { \pmb a } \in [ n _ { i } ] ^ { k _ { i } } , { \pmb b } _ { i } \in [ n _ { i } ] ^ { l _ { i } }$ . The number of these tensors is $\textstyle \prod _ { i = 1 } ^ { m } \mathrm { b } ( i )$ for invariant layers and $\textstyle \prod _ { i = 1 } ^ { m } \mathrm { b } ( k _ { i } + l _ { i } )$ i=1  for equivariant layers. Since these are all linear independent (pairwise disjoint support of non-zero entries) we need to show that their number equal the dimension of the solution of the relevant fixed-point equations above. This can be done again by similar arguments to the proof of proposition 2 or as shown in appendix B. To summarize:
+
+Theorem 3. The linear space of invariant linear layers L : Rnk11 × $L : \mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times n _ { 2 } ^ { k _ { 2 } } \times \dots \times n _ { m } ^ { k _ { m } } }  \mathbb { R }$ is of dimension $\textstyle \prod _ { i = 1 } ^ { m } { \mathrm { b } ( k _ { i } ) }$ . The equivariant linear layers $L : \mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times n _ { 2 } ^ { k _ { 2 } } \times \dots \times n _ { m } ^ { k _ { m } } }  \mathbb { R } ^ { n _ { 1 } ^ { l _ { 1 } } \times n _ { 2 } ^ { l _ { 2 } } \times \dots \times n _ { m } ^ { l _ { m } } }$ has dimension $\textstyle \prod _ { i = 1 } ^ { m } \ b ( k _ { i } + l _ { i } )$ . Orthogonal bases for these layers are listed in equation $1 l$ .
+
+# ACKNOWLEDGMENTS
+
+This research was supported in part by the European Research Council (ERC Consolidator Grant, ”LiftMatch” 771136) and the Israel Science Foundation (Grant No. 1830/17).
+
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+
+# APPENDIX A EFFICIENT IMPLEMENTATION OF LAYERS
+
+For fast execution of order-2 layers we implemented the following 15 operations which can be easily shown to span the basis discussed in the paper. We denote by $\mathbf { 1 } \in \mathbb { R } ^ { n }$ the vector of all ones.
+
+1. The identity and transpose operations: $L ( A ) = A$ , $L ( A ) = A ^ { T }$ .   
+2. The diag operation: $L ( A ) = \mathrm { d i a g } ( \mathrm { d i a g } ( A ) )$ .   
+3. Sum of rows replicated on rows/ columns/ diagonal: $L ( A ) = A \mathbf { 1 1 } ^ { T } , L ( A ) = \mathbf { 1 } ( A \mathbf { 1 } ) ^ { T }$ , $L ( A ) = \mathrm { d i a g } ( { \bar { A } } \mathbf { 1 } )$ .   
+4. Sum of columns replicated on rows/ columns/ diagonal: $L ( A ) \ = \ A ^ { T } \mathbf { 1 } \mathbf { 1 } ^ { T }$ , $L ( A ) =$ $\mathbf { 1 } ( A ^ { T } \mathbf { 1 } ) ^ { T } , L ( A ) = { \mathrm { { d i a g } } } ( A ^ { T } \mathbf { 1 } )$ .   
+5. Sum of all elements replicated on all matrix/ diagonal: $L ( A ) = ( \mathbf { 1 } ^ { T } A \mathbf { 1 } ) \cdot \mathbf { 1 } \mathbf { 1 } ^ { T } ,$ $L ( A ) =$ $( \mathbf { 1 } ^ { T } A \mathbf { 1 } ) \cdot \mathrm { { d i a g } ( \mathbf { 1 } ) }$ .   
+6. Sum of diagonal elements replicated on all matrix/diagonal: $L ( A ) = ( \mathbf { 1 } ^ { T } \mathrm { d i a g } ( A ) ) \cdot \mathbf { 1 } \mathbf { 1 } ^ { T }$ , $L ( A ) = ( \bar { \mathbf { 1 } ^ { T } } \mathrm { d i a g } ( A ) ) \cdot \mathrm { d i a g } \bar { ( } \mathbf { 1 } )$ .   
+7. Replicate diagonal elements on rows/columns: $L ( A ) = \mathrm { d i a g } ( A ) \mathbf { 1 } ^ { T }$ , $L ( A ) = \mathbf { 1 } \mathrm { d i a g } ( A ) ^ { T }$ .
+
+We normalize each operation to have unit max operator norm. We note that in case the input matrix is symmetric, our basis reduces to 11 elements in the first layer. If we further assume the matrix has zero diagonal we get a 6 element basis in the first layer. In both cases our model is more expressive than the 4 element basis of Hartford et al. (2018) and as the output of the first layer (or other inner states) need not be symmetric nor have zero diagonal the deeper layers can potentially make good use of the full 15 element basis.
+
+# APPENDIX B INVARIANT AND EQUIVARIANT SUBSPACE DIMENSIONS
+
+We prove a useful combinatorial fact as a corollary of proposition 2. This fact will be used later to easily compute the dimensions of more general spaces of invariant and equivariant linear layers. We use the fact that if $V$ is a representation of a finite group $G$ then
+
+$$
+\phi = { \frac { 1 } { | G | } } \sum _ { g \in G } g \in { \mathrm { E n d } } ( V )
+$$
+
+is a projection onto $V ^ { G } = \{ \pmb { v } \in V | g \pmb { v } = \pmb { v } , \forall g \in G \}$ , the subspace of fixed points in $V$ under the action of $G$ , and consequently that $\operatorname { t r } ( \phi ) = \dim ( V ^ { G } )$ (see Fulton & Harris (2013) for simple proofs).
+
+Proposition 3. The following formula holds:
+
+$$
+{ \frac { 1 } { n ! } } \sum _ { P \in \Pi _ { n } } t r ( P ) ^ { k } = \mathrm { b } ( k ) ,
+$$
+
+where $\Pi _ { n }$ is the matrix permutation group of dimensions $n \times n$ .
+
+Proof. In our case, the vector space is the space of order- $k$ tensors and the group acting on it is the matrix group $G = \left\{ P ^ { \otimes k } \mid P \in \overline { { \Pi } } _ { m } \right\}$ .
+
+$$
+\dim ( V ^ { G } ) = \operatorname { t r } ( \phi ) = { \frac { 1 } { | G | } } \sum _ { g \in G } \operatorname { t r } ( g ) = { \frac { 1 } { n ! } } \sum _ { P \in \Pi _ { n } } \operatorname { t r } ( P ^ { \otimes k } ) = { \frac { 1 } { n ! } } \sum _ { P \in \Pi _ { n } } \operatorname { t r } ( P ) ^ { k } ,
+$$
+
+where we used the multiplicative law of the trace with respect to Kronecker product. Now we use proposition 2 noting that in this case $V ^ { G }$ is the solution space of the fixed-point equations. Therefore, $\mathrm { d i m } ( V ^ { G } ) = \mathrm { b } ( k )$ and the proof is finished. □
+
+Recall that for a permutation matrix $_ { r }$ , $\operatorname { t r } ( P ) = | \{ i \in [ n ]$ s.t. $_ { r }$ fixes $e _ { i } \} |$ . Using this, we can interpret the equation in proposition 3 as the $k$ -th moment of a random variable counting the number of fixed points of a permutation, with uniform distribution over the permutation group. Proposition 3 proves that the $k$ -th moment of this random variable is the $k$ -th Bell number.
+
+We can now use proposition 3 to calculate the dimensions of two linear layer spaces: (i) Equivariant layers acting on order- $k$ tensors with features (as in 3); and (ii) multi-node sets (as in section 5).
+
+Theorem 2. The space of invariant (equivariant) linear layers $\mathbb { R } ^ { n ^ { k } , d } \to \mathbb { R } ^ { d ^ { \prime } }$ $\mathbb { R } ^ { n ^ { k } \times d }  \mathbb { R } ^ { n ^ { k } \times d ^ { \prime } } )$ is of dimension $d d ^ { \prime } \mathrm { b } ( k ) + d ^ { \prime }$ (for equivariant: $d d ^ { \prime } \mathrm { b } ( 2 k ) + d ^ { \prime } \mathrm { b } ( k ) )$ with basis elements defined in equation $^ { 9 }$ ; equations 10a (10b) show the general form of such layers.
+
+Proof. We prove the dimension formulas for the invariant case. The equivariant case is proved similarly. The solution space for the fixed point equations is the set $V ^ { G }$ for the matrix group $G =$ $\left\{ P ^ { \otimes k } { \overset { \cdot } { \otimes } } I _ { d } \otimes I _ { d ^ { \prime } } | P \in { \overset { \cdot } { \Pi } } _ { n } \right\}$ . Using the projection formula 12 we get that the dimension of the solution subspace, which is the space of invariant linear layers, can be computed as follows:
+
+$$
+\dim ( V ^ { G } ) = { \frac { 1 } { n ! } } \sum _ { P \in \Pi _ { n } } \operatorname { t r } ( P ) ^ { k } \operatorname { t r } ( I _ { d } ) \operatorname { t r } ( I _ { d ^ { \prime } } ) = \left( { \frac { 1 } { n ! } } \sum _ { P \in \Pi _ { n } } \operatorname { t r } ( P ) ^ { k } \right) \operatorname { t r } ( I _ { d } ) \operatorname { t r } ( I _ { d ^ { \prime } } ) = d \cdot d ^ { \prime } \cdot \operatorname { b } ( k ) .
+$$
+
+Theorem 3. The linear space of invariant linear layers $L : \mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times n _ { 2 } ^ { k _ { 2 } } \times \dots \times n _ { m } ^ { k _ { m } } }  \mathbb { R }$ is of dimension $\textstyle \prod _ { i = 1 } ^ { m } { \mathrm { b } ( k _ { i } ) }$ . The equivariant linear layers $L : \mathbb { R } ^ { n _ { 1 } ^ { k _ { 1 } } \times n _ { 2 } ^ { k _ { 2 } } \times \dots \times n _ { m } ^ { k _ { m } } }  \mathbb { R } ^ { n _ { 1 } ^ { l _ { 1 } } \times n _ { 2 } ^ { l _ { 2 } } \times \dots \times n _ { m } ^ { l _ { m } } }$ has dimension $\textstyle \prod _ { i = 1 } ^ { m } \ b ( k _ { i } + l _ { i } )$ . Orthogonal bases for these layers are listed in equation $1 l$ .
+
+Proof. In this case we get the fixed-point equations: $M \mathrm { v e c } ( { \cal L } ) = \mathrm { v e c } ( { \cal L } )$ for invariant, and $M \otimes$ $M \mathrm { v e c } ( { \cal L } ) = \mathrm { v e c } ( { \cal L } )$ for equivariant, where $M = P _ { 1 } ^ { \otimes k _ { 1 } } \otimes \cdot \cdot \cdot \otimes P _ { m } ^ { \otimes k _ { m } }$ . Similarly to the previous theorem, plugging $M$ into equation 12, using the trace multiplication rule and proposition 3 we get the above formulas. □
+
+# APPENDIX C IMPLEMENTING MESSAGE PASSING WITH OUR MODEL
+
+In this appendix we show that our model can approximate message passing layers as defined in Gilmer et al. (2017) to an arbitrary precision, and consequently that our model is able to approximate any network consisting of several such layers. The key idea is to mimic multiplication of features by the adjacency matrix, which allows summing over local neighborhoods. This can be implemented using our basis.
+
+Theorem 4. Our model can represent message passing layers to an arbitrary precision on compact sets.
+
+Proof. Consider input vertex data $\pmb { H } = ( h _ { u } ) \in \mathbb { R } ^ { n \times d }$ ( $n$ is the number of vertices in the graph, and $d$ is the input feature depth), adjacency matrix $\pmb { A } = ( a _ { u v } ) \in \mathbb { R } ^ { n \times n }$ of the graph, and additional edge features $\pmb { \dot { \mathsf { E } } } = ( e _ { u v } ) \in \dot { \mathbb { R } } ^ { n \times n \times l }$ . Recall that a message passing layer of Gilmer et al. (2017) is of the form:
+
+$$
+\begin{array} { l } { { m _ { u } ^ { t + 1 } = \displaystyle \sum _ { v \in N ( u ) } { \cal M } _ { t } ( h _ { u } ^ { t } , h _ { v } ^ { t } , e _ { u v } ) } } \\ { { \displaystyle h _ { u } ^ { t + 1 } = { \cal U } _ { t } ( h _ { u } ^ { t } , m _ { u } ^ { t + 1 } ) } } \end{array}
+$$
+
+where $u , v$ are nodes in the graph, $h _ { u } ^ { t }$ is the feature vector associated with $u$ in layer $t$ , and $e _ { u v }$ are additional edge features. We denote the number of output features of $M _ { t }$ by $d ^ { \prime }$ .
+
+In our setting we represent this data using a tensor $\pmb { \gamma } \in \mathbb { R } ^ { n \times n \times ( 1 + l + d ) }$ where the first channel is the adjacency matrix $\pmb { A }$ , the next $l$ channels are edge features, and the last $d$ channels are diagonal matrices that hold $\boldsymbol { X }$ .
+
+Let us construct a message passing layer using our model:
+
+1. Our first step is constructing an $n \times n \times ( 1 + l + 2 d )$ tensor. In the first channels we put the adjacency matrix $\pmb { A }$ and the edge features $\mathsf E$ . In the next $d$ channels we replicate the features on the rows, and in the last $d$ channels we replicate features on the columns. The output tensor ${ \pmb Z } ^ { 1 }$ has the form ${ \bf { Z } } _ { u , v } ^ { 1 } = [ a _ { u v } , e _ { u v } , h _ { u } ^ { t } , h _ { v } ^ { t } ]$ .   
+2. Next, we copy the feature channels $[ a _ { u v } , e _ { u v } , h _ { u } ^ { t } ]$ to the output tensor ${ \pmb Z } ^ { 2 }$ . We then apply a multilayer perceptron (MLP) on the last $l + 2 d$ feature dimensions of ${ \pmb Z } ^ { 1 }$ that approximates $M _ { t }$ (Hornik, 1991). The output tensor of this stage is $\begin{array} { r l } { \pmb { { Z } } _ { u , v } ^ { 2 } } & { { } = } \end{array}$ $\left[ a _ { u v } , e _ { u v } , h _ { u } ^ { t } , M _ { t } ( h _ { u } ^ { t } , h _ { v } ^ { t } , e _ { u v } ) + \epsilon _ { 1 } \right]$ .   
+3. Next, we would like to perform poi ise multiplication $\mathbf { Z } _ { u , v , 1 } ^ { 2 } \odot \mathbf { Z } _ { u , v , ( l + d + 2 ) : e n d } ^ { 2 }$ . This $M _ { t }$ for non-adjacent nodes $u , v$ . As this point-wise multiplication is not a part of our framework we can use an MLP on the feature dimension to approximate it and get $\pmb { \mathrm { Z } } _ { u , v } ^ { 3 } = [ a _ { u v } , e _ { u v } , h _ { u } ^ { t } , a _ { u v } M _ { t } ( h _ { u } ^ { t } , h _ { v } ^ { t } ) + \epsilon _ { 2 } ]$ .   
+4. As before we copy the feature channels $[ a _ { u v } , e _ { u v } , h _ { u } ^ { t } ]$ . We now apply a sum over the rows ( $v$ dimension) on the $M _ { t }$ output channels. We put the output of this sum on the diagonal of ${ \pmb Z } ^ { 4 }$ in separate channels. We get $\begin{array} { r } { \pmb { \mathsf { Z } } _ { u , v } ^ { 4 } = [ a _ { u v } , e _ { u v } , h _ { u } ^ { t } , \delta _ { u v } \sum _ { w \in N ( u ) } M _ { t } ( h _ { u } ^ { t } , h _ { w } ^ { t } ) + \epsilon _ { 3 } ] . } \end{array}$ , where $\delta _ { u v }$ is the Kronecker delta. We get a tensor $\pmb { \mathrm { Z } } ^ { 4 } \in \mathbb { R } ^ { n \times n \times ( 1 + l + d + d ^ { \prime } ) }$ .
+
+5. The last step is to apply an MLP to the last $d + d ^ { \prime }$ feature channels of the diagonal of ${ \pmb Z } ^ { 4 }$ . After this last step we have $\pmb { \mathrm { Z } } _ { u , v } ^ { 5 } = [ a _ { u v } , e _ { u v } , \delta _ { u v } U _ { t } ( h _ { u } ^ { t } , m _ { u } ^ { t + 1 } ) + \epsilon _ { 4 } ]$ .
+
+The errors $\epsilon _ { i }$ depend on the approximation error of the MLP to the relevant function, the previous errors $\epsilon _ { i - 1 }$ (for $i > 1 \AA$ ), and uniform bounds as-well as uniform continuity of the approximated functions. □
+
+Corollary 1. Our model can represent any message passing network to an arbitrary precision on compact sets. In other words, in terms of universality our model is at-least as powerful as any message passing neural network (MPNN) that falls into the framework of Gilmer et al. (2017).

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+# Deep Equals Shallow for ReLU Networks in Kernel Regimes
+
+Alberto Bietti∗ NYU† alberto.bietti@nyu.edu
+
+Francis Bach   
+Inria‡   
+francis.bach@inria.fr
+
+# Abstract
+
+Deep networks are often considered to be more expressive than shallow ones in terms of approximation. Indeed, certain functions can be approximated by deep networks provably more efficiently than by shallow ones, however, no tractable algorithms are known for learning such deep models. Separately, a recent line of work has shown that deep networks trained with gradient descent may behave like (tractable) kernel methods in a certain over-parameterized regime, where the kernel is determined by the architecture and initialization, and this paper focuses on approximation for such kernels. We show that for ReLU activations, the kernels derived from deep fully-connected networks have essentially the same approximation properties as their “shallow” two-layer counterpart, namely the same eigenvalue decay for the corresponding integral operator. This highlights the limitations of the kernel framework for understanding the benefits of such deep architectures. Our main theoretical result relies on characterizing such eigenvalue decays through differentiability properties of the kernel function, which also easily applies to the study of other kernels defined on the sphere.
+
+# 1 Introduction
+
+The question of which functions can be well approximated by neural networks is crucial for understanding when these models are successful, and has always been at the heart of the theoretical study of neural networks (e.g., Hornik et al., 1989; Pinkus, 1999). While early works have mostly focused on shallow networks with only two layers, more recent works have shown benefits of deep networks for approximating certain classes of functions (Eldan & Shamir, 2016; Mhaskar & Poggio, 2016; Telgarsky, 2016; Daniely, 2017; Yarotsky, 2017; Schmidt-Hieber et al., 2020). Unfortunately, many of these approaches rely on constructions that are not currently known to be learnable using efficient algorithms.
+
+A separate line of work has considered over-parameterized networks with random neurons (Neal, 1996), which also display universal approximation properties while additionally providing efficient algorithms based on kernel methods or their approximations such as random features (Rahimi & Recht, 2007; Bach, 2017b). Many recent results on gradient-based optimization of certain over-parameterized networks have been shown to be equivalent to kernel methods with an architecture-specific kernel called the neural tangent kernel (NTK) and thus also fall in this category (e.g., Jacot et al., 2018; Li & Liang, 2018; Allen-Zhu et al., 2019b; Du et al., 2019a;b; Zou et al., 2019). This regime has been coined lazy (Chizat et al., 2019), as it does not capture the common phenomenon where weights move significantly away from random initialization and thus may not provide a satisfying model for learning adaptive representations, in contrast to other settings such as the mean field or active regime, which captures complex training dynamics where weights may move in a non-trivial manner and adapt to the data (e.g., Chizat & Bach, 2018; Mei et al., 2018). Nevertheless, one benefit compared to the mean field regime is that the kernel approach easily extends to deep architectures, leading to compositional kernels similar to the ones of Cho & Saul (2009); Daniely et al. (2016). Our goal in this paper is to study the role of depth in determining approximation properties for such kernels, with a focus on fully-connected deep ReLU networks.
+
+Our approximation results rely on the study of eigenvalue decays of integral operators associated to the obtained dot-product kernels on the sphere, which are diagonalized in the basis of spherical harmonics. This provides a characterization of the functions in the corresponding reproducing kernel Hilbert space (RKHS) in terms of their smoothness, and leads to convergence rates for non-parametric regression when the data are uniformly distributed on the sphere. We show that for ReLU networks, the eigenvalue decays for the corresponding deep kernels remain the same regardless of the depth of the network. Our key result is that the decay for a certain class of kernels is characterized by a property related to differentiability of the kernel function around the point where the two inputs are aligned. In particular, the property is preserved when adding layers with ReLU activations, showing that depth plays essentially no role for such networks in kernel regimes. This highlights the limitations of the kernel regime for understanding the power of depth in fully-connected networks, and calls for new models of deep networks beyond kernels (see, e.g., Allen-Zhu & Li, 2020; Chen et al., 2020, for recent works in this direction). We also provide applications of our result to other kernels and architectures, and illustrate our results with numerical experiments on synthetic and real datasets.
+
+Related work. Kernels for deep learning were originally derived by Neal (1996) for shallow networks, and later for deep networks (Cho & Saul, 2009; Daniely et al., 2016; Lee et al., 2018; Matthews et al., 2018). Smola et al. (2001); Minh et al. (2006) study regularization properties of dot-product kernels on the sphere using spherical harmonics, and Bach (2017a) derives eigenvalue decays for such dot-product kernels arising from shallow networks with positively homogeneous activations including the ReLU. Extensions to shallow NTK or Laplace kernels are studied by Basri et al. (2019); Bietti & Mairal (2019b); Geifman et al. (2020). The observation that depth does not change the decay of the NTK was previously made by Basri et al. (2020) empirically, and Geifman et al. (2020) provide a lower bound on the eigenvalues for deep networks; our work makes this observation rigorous by providing tight asymptotic decays. Spectral properties of wide neural networks were also considered in (Cao et al., 2019; Fan & Wang, 2020; Ghorbani et al., 2019; Xie et al., 2017; Yang & Salman, 2019). Azevedo & Menegatto (2014); Scetbon & Harchaoui (2020) also study eigenvalue decays for dot-product kernels but focus on kernels with geometric decays, while our main focus is on polynomial decays. Additional works on over-parameterized or infinite-width networks in lazy regimes include (Allen-Zhu et al., 2019a;b; Arora et al., 2019a;b; Brand et al., 2020; Lee et al., 2020; Song & Yang, 2019).
+
+Concurrently to our work, Chen & Xu (2021) also studied the RKHS of the NTK for deep ReLU networks, showing that it is the same as for the Laplace kernel on the sphere. They achieve this by studying asymptotic decays of Taylor coefficients of the kernel function at zero using complex-analytic extensions of the kernel functions, and leveraging this to obtain both inclusions between the two RKHSs. In contrast, we obtain precise descriptions of the RKHS and regularization properties in the basis of spherical harmonics for various dot-product kernels through spectral decompositions of integral operators, using (real) asymptotic expansions of the kernel function around endpoints. The equality between the RKHS of the deep NTK and Laplace kernel then easily follows from our results by the fact that the two kernels have the same spectral decay.
+
+# 2 Review of Approximation with Dot-Product Kernels
+
+In this section, we provide a brief review of the kernels that arise from neural networks and their approximation properties.
+
+# 2.1 Kernels for wide neural networks
+
+Wide neural networks with random weights or weights close to random initialization naturally lead to certain dot-product kernels that depend on the architecture and activation function, which we now present, with a focus on fully-connected architectures.
+
+Random feature kernels. We first consider a two-layer (shallow) network of the form $\begin{array} { r } { f ( \boldsymbol { x } ) = \frac { 1 } { \sqrt { m } } \sum _ { j = 1 } ^ { m } v _ { j } \sigma ( w _ { j } ^ { \top } \boldsymbol { x } ) } \end{array}$ , for some activation function $\sigma$ . When $w _ { j } \sim \mathcal { N } ( 0 , I ) \in \mathbb { R } ^ { d }$ are fixed and only $v _ { j } \in \mathbb { R }$ are trained with $\ell _ { 2 }$ regularization, this corresponds to using a random feature approximation Rahimi $\&$ Recht (2007) of the kernel
+
+$$
+\begin{array} { r } { k ( x , x ^ { \prime } ) = \mathbb { E } _ { w \sim \mathcal { N } ( 0 , I ) } [ \sigma ( w ^ { \top } x ) \sigma ( w ^ { \top } x ^ { \prime } ) ] . } \end{array}
+$$
+
+If $x , x ^ { \prime }$ are on the sphere, then by spherical symmetry of the Gaussian distribution, one may show that $k$ is invariant to unitary transformations and takes the form $k ( x , x ^ { \prime } ) = \kappa ( x ^ { \prime } x ^ { \prime } )$ for a certain function $\kappa$ . More precisely, if $\textstyle { \boldsymbol { \sigma } } ( u ) = \sum _ { i \geq 0 } a _ { i } h _ { i } ( u )$ is the decomposition of $\sigma$ in the basis of Hermite polynomials $h _ { i }$ , which are orthogonal w.r.t. the Gaussian measure, then we have (Daniely et al., 2016):
+
+$$
+\kappa ( u ) = \sum _ { i \geq 0 } a _ { i } ^ { 2 } u ^ { i } .
+$$
+
+Conversely, given a kernel function of the form above with $\begin{array} { r } { \kappa ( u ) = \sum _ { i \geq 0 } b _ { i } u ^ { i } } \end{array}$ with $b _ { i } \geq 0$ , one may construct corresponding activations using Hermite polynomials by taking
+
+$$
+\sigma ( u ) = \sum _ { i } a _ { i } h _ { i } ( u ) , \quad a _ { i } \in \{ \pm \sqrt { b _ { i } } \} .
+$$
+
+In the case where $o$ is $s$ -positively homogeneous, such as the ReLU $\sigma ( u ) = \mathrm { m a x } ( u , 0 )$ (with $s \ = \ 1$ ), or more generally $\sigma _ { s } ( u ) ~ = ~ \operatorname* { m a x } ( u , 0 ) ^ { s }$ , then the kernel (1) takes the form $\begin{array} { r } { k ( x , x ^ { \prime } ) = \| x \| ^ { s } \| x ^ { \prime } \| ^ { s } \kappa \big ( \frac { x ^ { \intercal } x ^ { \prime } } { \| x \| \| x ^ { \prime } \| } \big ) } \end{array}$ ( x>x0kxkkx0k ) for any x, x0. This leads to RKHS functions of the form $f ( x ) = \| x \| ^ { s } g ( { \frac { x } { \| x \| } } )$ , with $g$ in the RKHS of the kernel restricted to the sphere (Bietti $\&$ Mairal, 2019b, Prop. 8). In particular, for the step and ReLU activations $\sigma _ { 0 }$ and $\sigma _ { 1 }$ , the functions $\kappa$ are given by the following arc-cosine kernels (Cho & Saul, 2009):1
+
+$$
+\kappa _ { 0 } ( u ) = \frac { 1 } { \pi } \left( \pi - \operatorname { a r c c o s } ( u ) \right) , \qquad \kappa _ { 1 } ( u ) = \frac { 1 } { \pi } \left( u \cdot ( \pi - \operatorname { a r c c o s } ( u ) ) + \sqrt { 1 - u ^ { 2 } } \right) .
+$$
+
+Note that given a kernel function $\kappa$ , the corresponding activations (3) will generally not be homogeneous, thus the inputs to a random network with such activations need to lie on the sphere (or be appropriately normalized) in order to yield the kernel $\kappa$ .
+
+Extension to deep networks. When considering a deep network with more than two layers and fixed random weights before the last layer, the connection to random features is less direct since the features are correlated through intermediate layers. Nevertheless, when the hidden layers are wide enough, one still approaches a kernel obtained by letting the widths go to infinity (see, e.g., Daniely et al., 2016; Lee et al., 2018; Matthews et al., 2018), which takes a similar form to the multi-layer kernels of Cho $\&$ Saul (2009):
+
+$$
+k ^ { L } ( x , x ^ { \prime } ) = \kappa ^ { L } ( x ^ { \top } x ^ { \prime } ) : = \underbrace { \kappa \circ \cdot \cdot \circ \kappa } _ { L - 1 { \mathrm { ~ t i m e s } } } ( x ^ { \top } x ^ { \prime } ) ,
+$$
+
+for $x , x ^ { \prime }$ on the sphere, where $\kappa$ is obtained as described above for a given activation $\sigma$ , and $L$ is the number of layers. We still refer to this kernel as the random features (RF) kernel in this paper, noting that it is sometimes known as the “conjugate kernel” or NNGP kernel (for neural network Gaussian process). It is usually good to normalize $\kappa$ such that $\kappa ( 1 ) = 1$ , so that we also have $\kappa ^ { L } ( 1 ) = 1$ , avoiding exploding or vanishing behavior for deep networks. In practice, this corresponds to using an activation-dependent scaling in the random weight initialization, which is commonly used by practitioners (He et al., 2015).
+
+Neural tangent kernels. When intermediate layers are trained along with the last layer using gradient methods, the resulting problem is non-convex and the statistical properties of such approaches are not well understood in general, particularly for deep networks. However, in a specific over-parameterized regime, it may be shown that gradient descent can reach a global minimum while keeping weights very close to random initialization. More precisely, for a network $f ( x ; \theta )$ parameterized by $\theta$ with large width $m$ , the model remains close to its linearization around random initialization $\theta _ { 0 }$ throughout training, that is, $f ( x ; \theta ) \approx$ $f ( x ; \theta _ { 0 } ) + \langle \theta - \theta _ { 0 } , \nabla _ { \theta } f ( x ; \theta _ { 0 } ) \rangle$ . This is also known as the lazy training regime (Chizat et al., 2019). Learning is then equivalent to a kernel method with another architecture-specific kernel known as the neural tangent kernel (NTK, Jacot et al., 2018), given by
+
+$$
+k _ { \mathrm { N T K } } ( x , x ^ { \prime } ) = \operatorname* { l i m } _ { m  \infty } \langle \nabla f ( x ; \theta _ { 0 } ) , \nabla f ( x ^ { \prime } ; \theta _ { 0 } ) \rangle .
+$$
+
+For a simple two-layer network with activation $\sigma$ , it is then given by
+
+$$
+k _ { \mathrm { N T K } } ( \boldsymbol { x } , \boldsymbol { x ^ { \prime } } ) = ( x ^ { \top } \boldsymbol { x ^ { \prime } } ) ~ \mathbb { E } _ { w } [ \sigma ^ { \prime } ( w ^ { \top } \boldsymbol { x } ) \sigma ^ { \prime } ( w ^ { \top } \boldsymbol { x ^ { \prime } } ) ] + \mathbb { E } _ { w } [ \sigma ( w ^ { \top } \boldsymbol { x } ) \sigma ( w ^ { \top } \boldsymbol { x ^ { \prime } } ) ] .
+$$
+
+For a ReLU network with $L$ layers with inputs on the sphere, taking appropriate limits on the widths, one can show (Jacot et al., 2018): $k _ { \mathrm { N T K } } ( x , x ^ { \prime } ) = \kappa _ { \mathrm { N T K } } ^ { L } ( x ^ { \top } x ^ { \prime } )$ , with $\kappa _ { \mathrm { N T K } } ^ { 1 } ( u ) =$ $\kappa ^ { 1 } ( u ) = u$ and for $\ell = 2 , \ldots , L$ ,
+
+$$
+\begin{array} { r } { \kappa ^ { \ell } ( u ) = \kappa _ { 1 } ( \kappa ^ { \ell - 1 } ( u ) ) \qquad } \\ { \kappa _ { \mathrm { N T K } } ^ { \ell } ( u ) = \kappa _ { \mathrm { N T K } } ^ { \ell - 1 } ( u ) \kappa _ { 0 } ( \kappa ^ { \ell - 1 } ( u ) ) + \kappa ^ { \ell } ( u ) , } \end{array}
+$$
+
+where $\kappa _ { 0 }$ and $\kappa _ { 1 }$ are given in (4).
+
+# 2.2 Approximation and harmonic analysis with dot-product kernels
+
+In this section, we recall approximation properties of dot-product kernels on the sphere, through spectral decompositions of integral operators in the basis of spherical harmonics. Further background is provided in Appendix A.
+
+Spherical harmonics and description of the RKHS. A standard approach to study the RKHS of a kernel is through the spectral decomposition of an integral operator $T$ given by $\begin{array} { r } { T f ( x ) = \int k ( x , y ) f ( y ) d \tau ( y ) } \end{array}$ for some measure $\tau$ , leading to Mercer’s theorem (e.g., Cucker & Smale, 2002). When inputs lie on the sphere $\mathbb { S } ^ { d - 1 }$ in $d$ dimensions, dot-product kernels of the form $k ( x , x ^ { \prime } ) = \kappa ( x ^ { \prime } x ^ { \prime } )$ are rotationally-invariant, depending only on the angle between $x$ and $x ^ { \prime }$ . Similarly to how translation-invariant kernels are diagonalized in the Fourier basis, rotation-invariant kernels are diagonalized in the basis of spherical harmonics (Smola et al., 2001; Bach, 2017a), which lead to connections between eigenvalue decays and regularity as in the Fourier setting. In particular, if $\tau$ denotes the uniform measure on $\mathbb { S } ^ { d - 1 }$ , then $T Y _ { k , j } = \mu _ { k } Y _ { k , j }$ , where $Y _ { k , j }$ is the $j$ -th spherical harmonic polynomial of degree $k$ , where $k$ plays the role of a frequency as in the Fourier case, and the number of such orthogonal polynomials of degree $k$ is given by $\begin{array} { r } { N ( d , k ) = \frac { 2 k + d - 2 } { k } \binom { k + d - 3 } { d - 2 } } \end{array}$ 2 k+d−3d−2 
, which grows as $k ^ { d - 2 }$ for large $k$ . The eigenvalues $\mu _ { k }$ only depend on the frequency $k$ and are given by
+
+$$
+\mu _ { k } = \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } \int _ { - 1 } ^ { 1 } \kappa ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t ,
+$$
+
+where $P _ { k }$ is the Legendre polynomial of degree $k$ in $d$ dimensions (also known as Gegenbauer polynomial when using a different scaling), and $\omega _ { d - 1 }$ denotes the surface of the sphere $\mathbb { S } ^ { d - 1 }$ . Mercer’s theorem then states that the RKHS $\mathcal { H }$ associated to the kernel is given by
+
+$$
+\mathcal { H } = \left\{ f = \sum _ { k \geq 0 , \mu _ { k } \neq 0 } \sum _ { j = 1 } ^ { N ( d , k ) } a _ { k , j } Y _ { k , j } ( \cdot ) \quad \mathrm { ~ s . t . ~ } \quad \| f \| _ { \mathcal { H } } ^ { 2 } : = \sum _ { k \geq 0 , \mu _ { k } \neq 0 } \sum _ { j = 1 } ^ { N ( d , k ) } \frac { a _ { k , j } ^ { 2 } } { \mu _ { k } } < \infty \right\} .
+$$
+
+In particular, if $\mu _ { k }$ has a fast decay, then the coefficients $u _ { k , j }$ of $f$ must also decay quickly with $k$ in order for $f$ to be in $\mathcal { H }$ , which means $f$ must have a certain level of regularity. Similarly to the Fourier case, an exponential decay of $\mu _ { k }$ implies that the functions in $\mathcal { H }$ are infinitely differentiable, while for polynomial decay $\mathcal { H }$ contains all functions whose derivatives only up to a certain order are bounded, as in Sobolev spaces. If two kernels lead to the same asymptotic decay of $\mu _ { k }$ up to a constant, then by (9) their RKHS norms are equivalent up to a constant, and thus they have the same RKHS. For the specific case of random feature kernels arising from $s$ -positively homogeneous activations, Bach (2017a) shows that $\mu _ { k }$ decays as $k ^ { - d - 2 s }$ for $k$ of the opposite parity of $s$ , and is zero for large enough $k$ of opposite parity, which results in a RKHS that contains even or odd functions (depending on the parity of $s$ ) defined on the sphere with bounded derivatives up to order $\beta : = d / 2 + s$ (note that $\beta$ must be greater than $( d - 1 ) / 2$ in order for the eigenvalues of $T$ to be summable and thus lead to a well-defined RKHS). Bietti $\&$ Mairal (2019b) show that the same decay holds for the NTK of two-layer ReLU networks, with $s = 0$ and a change of parity. Basri et al. (2019) show that the parity constraints may be removed by adding a zero-initialized additive bias term when deriving the NTK. We note that one can also obtain rates of approximation for Lipschitz functions from such decay estimates (Bach, 2017a). Our goal in this paper is to extend this to more general dot-product kernels such as those arising from multi-layer networks, by providing a more general approach for obtaining decay estimates from differentiability properties of the function $\kappa$ .
+
+Non-parametric regression. When the data are uniformly distributed on the sphere, we may also obtain convergence rates for non-parametric regression, which typically depend on the eigenvalue decay of the integral operator associated to the marginal distribution on inputs and on the decomposition of the regression function $f ^ { * } ( x ) = \mathbb { E } [ y | x ]$ on the same basis (e.g., Caponnetto & De Vito, 2007).2 Then one may achieve optimal rates that depend mainly on the regularity of $f ^ { * }$ when using various algorithms with tuned hyperparameters, but the choice of kernel and its decay may have an impact on the rates in some regimes, as well as on the difficulty of the optimization problem (see, e.g., Bach, 2013, Section 4.3).
+
+# 3 Main Result and Applications to Deep Networks
+
+In this section, we present our main results concerning approximation properties of dotproduct kernels on the sphere, and applications to the kernels arising from wide random neural networks. We begin by stating our main theorem, which provides eigenvalue decays for dot-product kernels from differentiability properties of the kernel function $\kappa$ at the endpoints $\pm 1$ . We then present applications of this result to various kernels, including those coming from deep networks, showing in particular that the RKHSs associated to deep and shallow ReLU networks are the same (up to parity constraints).
+
+# 3.1 Statement of our main theorem
+
+We now state our main result regarding the asymptotic eigenvalue decay of dot-product kernels. Recall that we consider a kernel of the form $k ( x , y ) = \kappa ( x ^ { \top } y )$ for $x , y \in \mathbb { S } ^ { d - 1 }$ , and seek to obtain decay estimates on the eigenvalues $\mu _ { k }$ defined in (8). We now state our main theorem, which derives the asymptotic decay of $\mu _ { k }$ with $k$ in terms of differentiability properties of $\kappa$ around $\{ \pm 1 \}$ , assuming that $\kappa$ is infinitely differentiable on $( - 1 , 1 )$ . This latter condition is always verified when $\kappa$ takes the form of a power series (2) with $\kappa ( 1 ) = 1$ , since the radius of convergence is at least 1. We also require a technical condition, namely the ability to “differentiate asymptotic expansions” of $\kappa$ at $\pm 1$ , which holds for the kernels considered in this work.
+
+Theorem 1 (Decay from regularity of $\kappa$ at endpoints, simplified). Let $\kappa : [ - 1 , 1 ] \to \mathbb { R }$ be $a$ function that is $C ^ { \infty }$ on $( - 1 , 1 )$ and has the following asymptotic expansions around $\pm 1$ :
+
+$$
+\begin{array} { c } { { \kappa ( 1 - t ) = p _ { 1 } ( t ) + c _ { 1 } t ^ { \nu } + o ( t ^ { \nu } ) } } \\ { { \kappa ( - 1 + t ) = p _ { - 1 } ( t ) + c _ { - 1 } t ^ { \nu } + o ( t ^ { \nu } ) , } } \end{array}
+$$
+
+for $t \geq 0$ , where $p _ { 1 } , p _ { - 1 }$ are polynomials and $\nu > 0$ is not an integer. Also, assume that the derivatives of $\kappa$ admit similar expansions obtained by differentiating the above ones. Then, there is an absolute constant $C ( d , \nu )$ depending on d and $\nu$ such that:
+
+In the case $| c _ { 1 } | = | c _ { - 1 } |$ , then we have $\mu _ { k } = o ( k ^ { - d - 2 \nu + 1 } )$ for one of the two parities (or both if $c _ { 1 } = c _ { - 1 } = 0$ ). If $\kappa$ is infinitely differentiable on $[ - 1 , 1 ]$ so that no such $\nu$ exists, then $\mu _ { k }$ decays faster than any polynomial.
+
+The full theorem is given in Appendix B along with its proof, and requires an additional mild technical condition on the expansion which is verified for all kernels considered in this paper, namely, a finite number of terms in the expansions with exponents between $\nu$ and $\nu + 1$ . The proof relies on integration by parts using properties of Legendre polynomials, in a way reminiscent of fast decays of Fourier series for differentiable functions, and on precise computations of the decay for simple functions of the form $t \mapsto ( 1 - t ^ { 2 } ) ^ { \nu }$ . This allows us to obtain the asymptotic decay for general kernel functions $\kappa$ as long as the behavior around the endpoints is known, in contrast to previous approaches which rely on the precise form of $\kappa$ , or of the corresponding activation in the case of arc-cosine kernels (Bach, 2017a; Basri et al., 2019; Bietti $\&$ Mairal, 2019b; Geifman et al., 2020). This enables the study of more general and complex kernels, such as those arising from deep networks, as discussed below. When $\kappa$ is of the form $\begin{array} { r } { \kappa ( t ) = \sum _ { k } b _ { k } t ^ { k } } \end{array}$ , the exponent $\nu$ in Theorem 1 is also related to the decay of coefficients $b _ { k }$ . Such coefficients provide a dimension-free description of the kernel which may be useful for instance in the study of kernel methods in certain high-dimensional regimes (see, e.g., El Karoui, 2010; Ghorbani et al., 2019; Liang et al., 2020). We show in Appendix B.1 that the $b _ { k }$ may be recovered from the $\mu _ { k }$ by taking high-dimensional limits $d \to \infty$ , and that they decay as $k ^ { - \nu - 1 }$ .
+
+# 3.2 Consequences for ReLU networks
+
+When considering neural networks with ReLU activations, the corresponding random features and neural tangent kernels depend on the arc-cosine functions $\kappa _ { 1 }$ and $\kappa _ { 0 }$ defined in (4). These have the following expansions (with generalized exponents) near $+ 1$ :
+
+$$
+\begin{array} { l } { { \kappa _ { 0 } ( 1 - t ) = 1 - \displaystyle \frac { \sqrt { 2 } } { \pi } t ^ { 1 / 2 } + O ( t ^ { 3 / 2 } ) } } \\ { { \kappa _ { 1 } ( 1 - t ) = 1 - t + \displaystyle \frac { 2 \sqrt { 2 } } { 3 \pi } t ^ { 3 / 2 } + O ( t ^ { 5 / 2 } ) . } } \end{array}
+$$
+
+Indeed, the first follows from integrating the expansion of the derivative using the relation t arccos(1 ��� t) = 1√2t√1−t/2 and the second follows from the first using the expression of $\kappa _ { 1 }$ in (4). Near $^ { - 1 }$ , we have by symmetry $\begin{array} { r } { \kappa _ { 0 } ( - 1 + t ) = 1 - \kappa _ { 0 } ( 1 - t ) = \frac { \sqrt { 2 } } { \pi } t ^ { 1 / 2 } + O ( t ^ { 3 / 2 } ) } \end{array}$ , and we have different $\begin{array} { r } { \kappa _ { 1 } ( - 1 + t ) = \frac { 2 \sqrt { 2 } } { 3 \pi } t ^ { 3 / 2 } + O ( t ^ { 5 / 3 } ) } \end{array}$ by using (Flajolet $\kappa _ { 1 } ^ { \prime } = \kappa _ { 0 }$ and ick, $\kappa _ { 1 } ( - 1 ) = 0$ . The ability toem VI.8, p.419), $\&$   
+together with a complex-analytic property known as $\Delta$ -analyticity, which was shown to hold for RF and NTK kernels by Chen $\&$ Xu (2021). By Theorem 1, we immediately obtain a decay of $k ^ { - d - 2 }$ for even coefficients for $\kappa _ { 1 }$ , and $k ^ { - d }$ for odd coefficients for $\kappa _ { 0 }$ , recovering results of Bach (2017a). For the two-layer ReLU NTK, we have $\kappa _ { \mathrm { N T K } } ^ { 2 } ( u ) = u \kappa _ { 0 } ( u ) + \kappa _ { 1 } ( u )$ , leading to a similar expansion to $\kappa _ { 0 }$ and thus decay, up to a change of parity due to the factor $u$ which changes signs in the expansion around $^ { - 1 }$ ; this recovers Bietti $\&$ Mairal (2019b). We note that for these specific kernels, Bach (2017a); Bietti & Mairal (2019b) show in addition that coefficients of the opposite parity are exactly zero for large enough $k$ , which imposes parity constraints on functions in the RKHS, although such a constraint may be removed in the NTK case by adding a zero-initialized bias term (Basri et al., 2019), leading to a kernel $\kappa _ { \mathrm { N T K } , b } ( u ) = ( u + 1 ) \kappa _ { 0 } ( u ) + \kappa _ { 1 } ( u )$ .
+
+Deep networks. Recall from Section 2.1 that the RF and NTK kernels for deep ReLU networks may be obtained through compositions and products using the functions $\kappa _ { 1 }$ and $\kappa _ { 0 }$ . Since asymptotic expansions can be composed and multiplied, we can then obtain expansions for the deep RF and NTK kernels. The following results show that such kernels have the same eigenvalue decay as the ones for the corresponding shallow (two-layer) networks.
+
+Corollary 2 (Deep RF decay.). For the random neuron kernel $\kappa _ { R F } ^ { L }$ of an $L$ -layer ReL $U$ network with $L \geq 3$ , we have $\mu _ { k } \sim C ( d , L ) k ^ { - d - 2 }$ , where $C ( d , L )$ is different depending on the parity of $k$ and grows linearly with $L$ .
+
+Corollary 3 (Deep NTK decay.). For the neural tangent kernel $\kappa _ { N T K } ^ { L }$ of an $L$ -layer ReL $U$ network with $L \geq 3$ , we have $\mu _ { k } \sim C ( d , L ) k ^ { - d }$ , where $C ( d , L )$ is different depending on the parity of $k$ and grows quadratically with $L$ (it grows linearly with $L$ when considering the normalized NTK $\kappa _ { N T K } ^ { L } / L$ , which satisfies $\kappa _ { N T K } ^ { L } ( 1 ) / L = 1 ,$ ).
+
+The proofs, given in Appendix C, use the fact that $\kappa _ { 1 } \cup \kappa _ { 1 }$ and $\kappa _ { 1 }$ have the same non-integer exponent factors in their expansions, and similarly for $\kappa _ { 0 } \cup \kappa _ { 1 }$ and $\kappa _ { 0 }$ . One benefit compared to the shallow case is that the odd and even coefficients are both non-zero with the same decay, which removes the parity constraints, but as mentioned before, simple modifications of the shallow kernels can yield the same effect.
+
+The finite neuron case. For two-layer networks with a finite number of neurons, the obtained models correspond to random feature approximations of the limiting kernels (Rahimi & Recht, 2007). Then, one may approximate RKHS functions and achieve optimal rates in non-parametric regression as long as the number of random features exceeds a certain degrees-of-freedom quantity (Bach, 2017b; Rudi & Rosasco, 2017), which is similar to standard such quantities in the analysis of ridge regression (Caponnetto & De Vito, 2007), at least when the data are uniformly distributed on the sphere (otherwise the quantity involved may be larger unless features are sampled non-uniformly). Such a number of random features is optimal for a given eigenvalue decay of the integral operator (Bach, 2017b), which implies that the shallow random feature architectures provides optimal approximation for the multi-layer ReLU kernels as well, since the shallow and deep kernels have the same decay, up to the parity constraint. In order to overcome this constraint for shallow kernels while preserving decay, one may consider vector-valued random features of the form $( \sigma ( w ^ { \top } x ) , x _ { 1 } \sigma ( w ^ { \top } x ) , \ldots , x _ { d } \sigma ( w ^ { \top } x ) )$ with $w \sim \mathcal { N } ( 0 , I )$ , leading to a kernel $\kappa _ { \sigma , b } ( u ) = ( 1 + u ) \kappa _ { \sigma } ( u )$ , where $\kappa _ { \sigma }$ is the random feature kernel corresponding to $\sigma$ . With $\sigma ( u ) = \operatorname* { m a x } ( 0 , u )$ , $\kappa _ { \sigma , b }$ has the same decay as $\kappa _ { \mathrm { R F } } ^ { L }$ , and when $\sigma ( u ) = \mathbb { 1 } \{ u \geq 0 \}$ it has the same decay as κLNTK.
+
+# 3.3 Extensions to other kernels
+
+We now provide other examples of kernels for which Theorem 1 provides approximation properties thanks to its generality.
+
+Laplace kernel and generalizations. The Laplace kernel $k _ { c } ( x , y ) = e ^ { - c \| x - y \| }$ has been found to provide similar empirical behavior to neural networks when fitting randomly labeled data with gradient descent (Belkin et al., 2018). Recently, Geifman et al. (2020) have shown that when inputs are on the sphere, the Laplace kernel has the same decay as the NTK, which may suggest a similar conditioning of the optimization problem as for fully-connected networks, as discussed in Section 2.2. Denoting $\kappa _ { c } ( u ) = e ^ { - c \sqrt { 1 - u } }$ so that $k _ { c } ( x , y ) \mathop { = } \kappa _ { \underline { { c } } \sqrt { 2 } } ( x ^ { \top } y )$ , we may easily recover this result using Theorem 1 by noticing that $\kappa _ { c }$ is infinitely differentiable around $^ { - 1 }$ and satisfies
+
+$$
+\kappa _ { c } ( 1 - t ) = e ^ { - c \sqrt { t } } = 1 - c \sqrt { t } + O ( t ) ,
+$$
+
+which yields the same decay $k ^ { - d }$ as the NTK. Geifman et al. (2020) also consider a heuristic generalization of the Laplace kernel with different exponents, $\kappa _ { c , \gamma } ( u ) = e ^ { - c ( 1 - u ) ^ { \gamma } }$ . Theorem 1 allows us to obtain a precise decay for this kernel as well using $\kappa _ { c , \gamma } ( 1 - t ) =$ $1 - c t ^ { \gamma } + O ( t ^ { 2 \gamma } )$ , which is of the form $k ^ { - d - 2 \gamma + 1 }$ for non-integer $\gamma > 0$ , and in particular approaches the limiting order of smoothness $( d - 1 ) / 2$ when $\gamma \to 0$ .3
+
+Deep kernels with step activations. We saw in Section 3.2 that for ReLU activations, depth does not change the decay of the corresponding kernels. In contrast, when considering step activations $\sigma ( u ) = \mathbb { 1 } \{ u \geq 0 \}$ , we show in Appendix C.3 that approximation properties of the corresponding random neuron kernels (of the form $\kappa _ { 0 } \cup \cdots \cup \kappa _ { 0 }$ ) improve with depth, leading to a decay $k ^ { - d - 2 \nu + 1 }$ with $\nu = 1 / 2 ^ { L - 1 }$ for $L$ layers. This also leads to an RKHS which becomes as large as allowed (order of smoothness close to $( d - 1 ) / 2$ ) when $L  \infty$ . While this may suggest a benefit of depth, note that step activations make optimization hard for anything beyond a linear regime with random weights, since the gradients with respect to inner neurons vanish. Theorem 1 may also be applied to deep kernels with other positively homogeneous activations $\sigma _ { s } ( u ) = \operatorname* { m a x } ( 0 , u ) ^ { s }$ with $s \geq 2$ , for which endpoint expansions easily follow from those of $\kappa _ { 0 }$ or $\kappa _ { 1 }$ through integration.
+
+Infinitely differentiable kernels. Finally, we note that Theorem 1 shows that kernels associated to infinitely differentiable activations (which are themselves infinitely differentiable, see Daniely et al. $( 2 0 1 6 ) ^ { 4 }$ ), as well as Gaussian kernels on the sphere of the form $e ^ { - c ( 1 - x ^ { \top } y ) }$ , have faster decays than any polynomial. This results in a “small” RKHS that only contains smooth functions. See Azevedo $\&$ Menegatto (2014); Minh et al. (2006) for a more precise study of the decay for Gaussian kernels on the sphere.
+
+# 4 Numerical experiments
+
+We now present numerical experiments on synthetic and real data to illustrate our theory.   
+Our code is available at https://github.com/albietz/deep_shallow_kernel.
+
+Synthetic experiments. We consider randomly sampled inputs on the sphere $\mathbb { S } ^ { 3 }$ in 4 dimensions, and outputs generated according to the following target models, for an arbitrary w ∈ S3: f ∗1 (x) = 1{w>x ≥ 0.7} and f ∗2 (x) = e−(1−w>x)3/2 $f _ { 2 } ^ { * } ( x ) = e ^ { - ( 1 - w ^ { \top } x ) ^ { 3 / 2 } } + e ^ { - ( 1 + w ^ { \top } x ) ^ { 3 / 2 } }$ Note that $f _ { 1 } ^ { * }$ is discontinuous and thus not in the RKHS in general, while $f _ { 2 } ^ { * }$ is in the RKHS of $\kappa _ { 1 }$ (since it is even and has the same decay as $\kappa _ { 1 }$ as discussed in Section 3.3). In Figure 1 we compare the quality of approximation for different kernels by examining generalization performance of ridge regression with exact kernels or random features. The regularization parameter $\lambda$ is optimized on 10 000 test datapoints on a logarithmic grid. In order to illustrate the difficulty of optimization due to a small optimal $\lambda$ , which would also indicate slower convergence with gradient methods, we consider grids with $\lambda \geq \lambda _ { \operatorname* { m i n } }$ , for two different choices of $\lambda _ { \mathrm { m i n } }$ . We see that all kernels provide a similar rate of approximation for a large enough grid, but when fixing a smaller optimization budget by taking a larger $\lambda _ { \mathrm { m i n } }$ , the NTK and Laplace kernels can achieve better performance for large sample size $n$ , thanks to a slower eigenvalue decay of the covariance operator. Figure $1 ( \mathrm { r i g h t } )$ shows that when using $m = { \sqrt { n } }$ random features (which can achieve optimal rates in some settings, see Rudi & Rosasco, 2017), the “shallow” ReLU network performs better than a three-layer version, despite having fewer weights. This suggests that in addition to providing no improvements to approximation in the infinite-width case, the kernel regimes for deep ReLU networks may even be worse than their two-layer counterparts in the finite-width setting.
+
+MNIST and Fashion-MNIST. In Table 1, we consider the image classification datasets MNIST and Fashion-MNIST, which both consist of 60k training and 10k test images of size 28x28 with 10 output classes. We evaluate one-versus-all classifiers obtained by using kernel ridge regression by setting $y = 0 . 9$ for the correct label and $y = - 0 . 1$ otherwise. We train on random subsets of 50k examples and use the remaining 10k examples for validation. We find that test accuracy is comparable for different numbers of layers in RF or NTK kernels, with a slightly poorer performance for the two-layer case likely due to parity constraints, in agreement with our theoretical result that the decay is the same for different $L$ . There is a small decrease in accuracy for growing $L$ , which may reflect changes in the decay constants or numerical errors when composing kernels. The slightly better performance of RF compared to NTK may suggest that these problems are relatively easy (e.g., the regression function is smooth), so that a faster decay is preferable due to better adaptivity to smoothness.
+
+![](images/6c8f12eb39c77d53c1f07d01af225d3f2817a6a57c586b89b19762ba0ec058ba.jpg)  
+Figure 1: (left, middle) expected squared error vs sample size $n$ for kernel ridge regression estimators with different kernels on $f _ { 1 } ^ { * }$ and with two different budgets on optimization difficulty $\lambda _ { \mathrm { m i n } }$ (the minimum regularization parameter allowed). (right) ridge regression with one or two layers of random ReLU features on $f _ { 2 } ^ { * }$ , with different scalings of the number of “neurons” at each layer in terms of $n$ .
+
+Table 1: Test accuracies on MNIST (left) and Fashion-MNIST (right) for RF and NTK kernels with varying numbers of layers $L$ . We use kernel ridge regression on 50k samples, with $\lambda$ optimized on a validation set of size 10k, and report mean and standard errors across 5 such random splits of the 60k training samples. For comparison, the Laplace kernel with $c = 1$ yields accuracies $9 8 . 3 9 \pm 0 . 0 2$ on MNIST and $9 0 . 3 8 \pm 0 . 0 6$ on F-MNIST.
+
+MNIST   
+
+<table><tr><td rowspan=1 colspan=1>L</td><td rowspan=1 colspan=1>RF</td><td rowspan=1 colspan=1>NTK</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>98.60 ± 0.03</td><td rowspan=1 colspan=1>98.49± 0.02</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>98.67 ± 0.03</td><td rowspan=2 colspan=1>98.53 ± 0.0298.49 ± 0.01</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>98.66 ± 0.02</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>98.65 ± 0.04</td><td rowspan=1 colspan=1>98.46 ± 0.02</td></tr></table>
+
+F-MNIST   
+
+<table><tr><td>L</td><td>RF</td><td>NTK</td></tr><tr><td>2 3 4 5</td><td>90.75 ± 0.11 90.87 ± 0.16 90.89 ± 0.13 90.88 ± 0.08</td><td>90.65 ± 0.07 90.62 ± 0.08 90.55 ± 0.07 90.50 ± 0.05</td></tr></table>
+
+# 5 Discussion
+
+In this paper, we have analyzed the approximation properties of deep networks in kernel regimes, by studying eigenvalue decays of integral operators through differentiability properties of the kernel function. In particular, the decay is governed by the form of the function’s (generalized) power series expansion around $\pm 1$ , which remains the same for kernels arising from fully-connected ReLU networks of varying depths. This result suggests that the kernel approach is unsatisfactory for understanding the power of depth in fully-connected networks. In particular, it highlights the need to incorporate other regimes in the study of deep networks, such as the mean field regime (Chizat & Bach, 2018; Mei et al., 2018), and other settings with hierarchical structure (see, e.g., Allen-Zhu & Li, 2020; Chen et al., 2020). We note that our results do not rule out benefits of depth for other network architectures in kernel regimes; for instance, depth may improve stability properties of convolutional kernels (Bietti & Mairal, 2019a;b), and a precise study of approximation for such kernels and its dependence on depth would also be of interest.
+
+# Acknowledgments
+
+The authors would like to thank David Holzm¨uller for finding an error in an earlier version of the paper, which led us to include the new assumption on differentiation of asymptotic expansions in Theorem 1. This work was funded in part by the French government under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute). We also acknowledge support of the European Research Council (grant SEQUOIA 724063).
+
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+# A Background on Spherical Harmonics
+
+In this section, we provide some background on spherical harmonics needed for our study of approximation. See (Efthimiou & Frye, 2014; Atkinson & Han, 2012; Ismail, 2005) for references, as well as (Bach, 2017a, Appendix D). We consider inputs on the $d - 1$ sphere $\mathbb { S } ^ { d - 1 } = \{ x \in \mathbb { R } ^ { d } , \| x \| = 1 \}$ .
+
+We recall some properties of the spherical harmonics $Y _ { k , j }$ introduced in Section 2.2. For j = 1, . . . , N (d, k), where N (d, k) = 2k+d−2 k+d−3
, the spherical harmonics $Y _ { k , j }$ are homogeneous harmonic polynomials of degree $k$ that are orthonormal with respect to the uniform distribution $\tau$ on the $d$ –1 sphere. The degree $k$ plays the role of an integer frequency, as in Fourier series, and the collection $\{ Y _ { k , j } , k \geq 0 , j = 1 , \ldots , N ( d , k ) \}$ forms an orthonormal basis of $L ^ { 2 } ( \mathbb { S } ^ { d - 1 } , d \tau )$ . As with Fourier series, there are tight connections between decay of coefficients in this basis w.r.t. $k$ , and regularity/differentiability of functions, in this case differentiability on the sphere. This follows from the fact that spherical harmonics are eigenfunctions of the Laplace-Beltrami operator on the sphere $\Delta _ { \mathbb { S } ^ { d - 1 } }$ (see Efthimiou & Frye, 2014, Proposition 4.5):
+
+$$
+\Delta _ { \mathbb { S } ^ { d - 1 } } Y _ { k , j } = - k ( k + d - 2 ) Y _ { k , j } .
+$$
+
+For a given frequency $k$ , we have the following addition formula:
+
+$$
+\sum _ { j = 1 } ^ { N ( d , k ) } Y _ { k , j } ( x ) Y _ { k , j } ( y ) = N ( d , k ) P _ { k } ( x ^ { \top } y ) ,
+$$
+
+where $P _ { k }$ is the $k$ -th Legendre polynomial in dimension $d$ (also known as Gegenbauer polynomial when using a different scaling), given by the Rodrigues formula:
+
+$$
+P _ { k } ( t ) = ( - 1 / 2 ) ^ { k } \frac { \Gamma ( \frac { d - 1 } { 2 } ) } { \Gamma ( k + \frac { d - 1 } { 2 } ) } ( 1 - t ^ { 2 } ) ^ { ( 3 - d ) / 2 } \left( \frac { d } { d t } \right) ^ { k } ( 1 - t ^ { 2 } ) ^ { k + ( d - 3 ) / 2 } .
+$$
+
+Note that these may also be expressed using the hypergeometric function ${ } _ { 2 } F _ { 1 }$ (see, e.g., Ismail, 2005, Section 4.5), an expression we will use in proof of Theorem 1 (see the proof of Lemma 6).
+
+The polynomials $P _ { k }$ are orthogonal in $L ^ { 2 } ( [ - 1 , 1 ] , d \nu )$ where the measure $d \nu$ is given by the weight function $d \nu ( t ) = ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t$ , and we have
+
+$$
+\int _ { - 1 } ^ { 1 } P _ { k } ^ { 2 } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t = \frac { \omega _ { d - 1 } } { \omega _ { d - 2 } } \frac { 1 } { N ( d , k ) } ,
+$$
+
+where $\begin{array} { r } { \omega _ { p - 1 } = \frac { 2 \pi ^ { p / 2 } } { \Gamma ( p / 2 ) } } \end{array}$ denotes the surface of the sphere $\mathbb { S } ^ { p - 1 }$ in $p$ dimensions. Using the addition formula (15) and orthogonality of spherical harmonics, we can show
+
+$$
+\int P _ { j } ( w ^ { \top } x ) P _ { k } ( w ^ { \top } y ) d \tau ( w ) = \frac { \delta _ { j k } } { N ( d , k ) } P _ { k } ( x ^ { \top } y )
+$$
+
+We will use two other properties of Legendre polynomials, namely the following recurrence relation (Efthimiou & Frye, 2014, Eq. 4.36)
+
+$$
+t P _ { k } ( t ) = \frac { k } { 2 k + d - 2 } P _ { k - 1 } ( t ) + \frac { k + d - 2 } { 2 k + d - 2 } P _ { k + 1 } ( t ) ,
+$$
+
+for $k \geq 1$ , and for $k = 0$ we simply have $t P _ { 0 } ( t ) = P _ { 1 } ( t )$ , as well as the differential equation (see, e.g., Efthimiou $\&$ Frye, 2014, Proposition 4.20):
+
+$$
+( 1 - t ^ { 2 } ) P _ { k } ^ { \prime \prime } ( t ) + ( 1 - d ) t P _ { k } ^ { \prime } ( t ) + k ( k + d - 2 ) P _ { k } ( t ) = 0 .
+$$
+
+The Funk-Hecke formula is helpful for computing Fourier coefficients in the basis of spherical harmonics in terms of Legendre polynomials: for any $j = 1 , \ldots , N ( d , k )$ , we have
+
+$$
+\int f ( x ^ { \top } y ) Y _ { k , j } ( y ) d \tau ( y ) = \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } Y _ { k , j } ( x ) \int _ { - 1 } ^ { 1 } f ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t .
+$$
+
+For example, we may use this to obtain decompositions of dot-product kernels by computing Fourier coefficients of functions $\kappa ( \langle x , \cdot \rangle )$ . Indeed, denoting
+
+$$
+\mu _ { k } = \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } \int _ { - 1 } ^ { 1 } \kappa ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t ,
+$$
+
+writing the decomposition of $\kappa ( \langle x , \cdot \rangle )$ using (21) leads to the following Mercer decomposition of the kernel:
+
+$$
+\kappa ( x ^ { \top } y ) = \sum _ { k = 0 } ^ { \infty } \mu _ { k } \sum _ { j = 1 } ^ { N ( d , k ) } Y _ { k , j } ( x ) Y _ { k , j } ( y ) = \sum _ { k = 0 } ^ { \infty } \mu _ { k } N ( d , k ) P _ { k } ( x ^ { \top } y ) .
+$$
+
+# B Proof of Theorem 1
+
+The proof of Theorem 1, stated below in full as Theorem 7, proceeds as follows. We first derive an upper bound on the decay of $\kappa$ of the form $k ^ { - d - 2 \nu + 3 }$ (Lemma 5), which is weaker than the desired $k ^ { - d - 2 \nu + 1 }$ , by exploiting regularity properties of $\kappa$ through integration by parts. The goal is then to apply this result on a function $\tilde { \kappa } = \kappa - \psi$ , where $\psi$ is a function that allows us to “cancel” the leading terms in the expansions of $\kappa$ , while being simple enough that it allows a precise estimate of its decay. In the proof of Theorem 7, we follow this strategy by considering $\psi$ as a sum of functions of the form $t \mapsto ( 1 - t ^ { 2 } ) ^ { \nu }$ and $t \mapsto t ( 1 - t ^ { 2 } ) ^ { \nu }$ , for which we provide a precise computation of the decay in Lemma 6.
+
+Decay upper bound through regularity. We begin by establishing a weak upper bound on the decay of $\kappa$ (Lemma 5) by leveraging its regularity up to the terms of order $( 1 - t ^ { 2 } ) ^ { \nu }$ . This is achieved by iteratively applying the following integration by parts lemma, which is conceptually similar to integrating by parts on the sphere by leveraging the spherical Laplacian relation (14) in Appendix A, but directly uses properties of $\kappa$ and of Legendre polynomials instead (namely, the differential equation (20)). We note that the final statement in Theorem 1 on infinitely differentiable $\kappa$ directly follows from Lemma 5.
+
+Lemma 4 (Integration by parts lemma). Let $\kappa : [ - 1 , 1 ] \to \mathbb { R }$ be a function that is $C ^ { \infty }$ on $( - 1 , 1 )$ and such that $\kappa ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } = O ( 1 )$ . We have
+
+$$
+\begin{array} { r l r } {  { \int _ { - 1 } ^ { 1 } \kappa ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { \frac { d - 3 } { 2 } } d t = \frac { 1 } { k ( k + d - 2 ) } \Big ( - \kappa ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } P _ { k } ^ { \prime } ( t ) \Big | _ { - 1 } ^ { 1 } } } & { { } } & { { ( 2 3 ) } } \\ { + \kappa ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } P _ { k } ( t ) \Big | _ { - 1 } ^ { 1 } + \int _ { - 1 } ^ { 1 } \tilde { \kappa } ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t } & { { } } & { } \end{array}
+$$
+
+with $\tilde { \kappa } ( t ) = - \kappa ^ { \prime \prime } ( t ) ( 1 - t ^ { 2 } ) + ( d - 1 ) t \kappa ^ { \prime } ( t ) .$
+
+Proof. In order to perform integration by parts, we use the following differential equation satisfied by Legendre polynomials (see, e.g., Efthimiou $\&$ Frye, 2014, Proposition 4.20):
+
+$$
+( 1 - t ^ { 2 } ) P _ { k } ^ { \prime \prime } ( t ) + ( 1 - d ) t P _ { k } ^ { \prime } ( t ) + k ( k + d - 2 ) P _ { k } ( t ) = 0 .
+$$
+
+Using this equation, we may write for $k \geq 1$ ,
+
+$$
+\begin{array} { c } { { \displaystyle { \int _ { - 1 } ^ { 1 } \kappa ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t = \frac { 1 } { k ( k + d - 2 ) } \Big ( ( d - 1 ) \int t \kappa ( t ) P _ { k } ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { \frac { d - 3 } { 2 } } d t } } } \\ { { - \displaystyle { \int \kappa ( t ) P _ { k } ^ { \prime \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } d t } \Big ) . } } \end{array}
+$$
+
+We may integrate the second term by parts using
+
+$$
+\begin{array} { c } { { \displaystyle \frac { d } { d t } \left( \kappa ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } \right) = \kappa ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } - 2 t ( 1 + ( d - 3 ) / 2 ) \kappa ( t ) ( 1 - t ^ { 2 } ) ^ { \frac { d - 3 } { 2 } } } } \\ { { = \kappa ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } - ( d - 1 ) t \kappa ( t ) ( 1 - t ^ { 2 } ) ^ { \frac { d - 3 } { 2 } } . } } \end{array}
+$$
+
+Noting that the first term in (26) cancels out with the integral resulting from the second term in (28), we then obtain
+
+$$
+\begin{array} { r l } { \displaystyle \int _ { - 1 } ^ { 1 } \kappa ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t = \frac { 1 } { k ( k + d - 2 ) } \Big ( - \kappa ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } P _ { k } ^ { \prime } ( t ) \Big | _ { - 1 } ^ { 1 } } & { } \\ { + \displaystyle \int _ { - 1 } ^ { 1 } \kappa ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } P _ { k } ^ { \prime } ( t ) d t \Big ) . } & { } \end{array}
+$$
+
+Integrating by parts once more, the second term becomes
+
+$$
+\begin{array} { r l r } {  { \int _ { - 1 } ^ { 1 } \kappa ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } P _ { k } ^ { \prime } ( t ) d t = \kappa ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } P _ { k } ( t ) \Big | _ { - 1 } ^ { 1 } } } \\ & { } & { \qquad - \int _ { - 1 } ^ { 1 } ( \kappa ^ { \prime \prime } ( t ) ( 1 - t ^ { 2 } ) - ( d - 1 ) t \kappa ^ { \prime } ( t ) ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t . ~ } \end{array}
+$$
+
+The desired result follows.
+
+Lemma 5 (Weak upper bound on the decay). Let $\kappa : [ - 1 , 1 ] \to \mathbb { R }$ be a function that is $C ^ { \infty }$ on $( - 1 , 1 )$ and has the following expansions around $\pm 1$ on its derivatives:
+
+$$
+\begin{array} { l } { { \kappa ^ { ( j ) } ( t ) = p _ { j , 1 } ( 1 - t ) + O ( ( 1 - t ) ^ { \nu - j } ) } } \\ { { \kappa ^ { ( j ) } ( t ) = p _ { j , - 1 } ( 1 + t ) + O ( ( 1 + t ) ^ { \nu - j } ) , } } \end{array}
+$$
+
+for $t \in [ - 1 , 1 ]$ and $j \geq 0$ , where $p _ { j , 1 } , p _ { j , - 1 }$ are polynomials and $\nu$ may be non-integer. Then the Legendre coefficients $\mu _ { k } ( \kappa )$ of $\kappa$ given in (8) satisfy
+
+$$
+\mu _ { k } ( \kappa ) = O ( k ^ { - d - 2 \nu + 3 } ) .
+$$
+
+Proof. Let $f _ { 0 } : = \kappa$ and for $j \geq 1$
+
+$$
+f _ { j } ( t ) : = - f _ { j - 1 } ^ { \prime \prime } ( t ) ( 1 - t ^ { 2 } ) + ( d - 1 ) f _ { j - 1 } ^ { \prime } ( t ) .
+$$
+
+Then $f _ { j }$ is $C ^ { \infty }$ on $( - 1 , 1 )$ and has similar expansions to $\kappa$ of the form
+
+$$
+\begin{array} { l } { { f _ { j } ( t ) = q _ { j , 1 } ( 1 - t ) + O ( ( 1 - t ) ^ { \nu - j } ) } } \\ { { f _ { j } ( t ) = q _ { j , - 1 } ( 1 + t ) + O ( ( 1 + t ) ^ { \nu - j } ) , } } \end{array}
+$$
+
+brackets vanish, until for some polynomials $q _ { j , \pm 1 }$ . We may apply Lemma 4 repeatedly as long as the terms in btain, for $\begin{array} { r } { j = \lceil \nu + \frac { d - 3 } { 2 } \rceil - 1 } \end{array}$ ,
+
+$$
+\begin{array} { l } { \displaystyle \int _ { - 1 } ^ { 1 } \kappa ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t } \\ { = \displaystyle \frac { 1 } { ( k ( k + d - 2 ) ) ^ { j + 1 } } \left( f _ { j } ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } P _ { k } ( t ) \Big | _ { - 1 } ^ { 1 } + \displaystyle \int _ { - 1 } ^ { 1 } f _ { j + 1 } ( t ) P _ { k } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } d t \right) } \end{array}
+$$
+
+Given our choice for $j$ , we have $f _ { j } ^ { \prime } ( t ) ( 1 - t ^ { 2 } ) ^ { 1 + \frac { d - 3 } { 2 } } = O ( 1 )$ , and $f _ { j + 1 } ( t ) ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 } =$ $O ( ( 1 - t ^ { 2 } ) ^ { - 1 + \epsilon } )$ for some $\epsilon > 0$ . Since $P _ { k } ( t ) \in [ - 1 , 1 ]$ for any $t \ \in \ [ - 1 , 1 ]$ , we obtain $\mu _ { k } ( \kappa ) = O ( k ^ { - 2 ( j + 1 ) } ) = O ( k ^ { - d - 2 \nu + 3 } )$ . □
+
+Precise decay for simple function. We now provide precise decay estimates for functions of the form $t \mapsto ( 1 - t ^ { 2 } ) ^ { \nu }$ and $t \mapsto t ( 1 - t ^ { 2 } ) ^ { \nu }$ , which will lead to the dominant terms in the decomposition of $\kappa$ in the main theorem.
+
+Lemma 6 (Decay for simple functions $\phi _ { \nu }$ and $\phi _ { \nu }$ ). Let $\phi _ { \nu } ( t ) = ( 1 - t ^ { 2 } ) ^ { \nu }$ , with $\nu > 0$ noninteger, and let $\mu _ { k } ( \phi _ { \nu } )$ denote its Legendre coefficients in $d$ dimensions given by $\begin{array} { r } { \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } \int _ { - 1 } ^ { 1 } ( 1 - } \end{array}$ $t ^ { 2 } ) ^ { \nu + ( d - 3 ) / 2 } P _ { k } ( t ) d t$ . We have
+
+Analogously, let ${ \bar { \phi } } _ { \nu } ( t ) : = t ( 1 - t ^ { 2 } ) ^ { \nu }$ . We have
+
+Proof. We recall the following representation of Legendre polynomials based on the hypergeometric function (e.g., Ismail, 2005, Section 4.5):5
+
+$$
+P _ { k } ( t ) = { } _ { 2 } F _ { 1 } ( - k , k + d - 2 ; ( d - 1 ) / 2 ; ( 1 - t ) / 2 ) ,
+$$
+
+where the hypergeometric function is given in its generalized form by
+
+$$
+_ p F _ { q } ( a _ { 1 } , \dots , a _ { p } ; b _ { 1 } , \dots , b _ { q } ; x ) = \sum _ { s = 0 } ^ { \infty } { \frac { ( a _ { 1 } ) _ { s } \cdot \cdot \cdot ( a _ { p } ) _ { s } } { ( b _ { 1 } ) _ { s } \cdot \cdot \cdot ( b _ { q } ) _ { s } } } { \frac { x ^ { s } } { s ! } } ,
+$$
+
+where $( a ) _ { s } = \Gamma ( a + s ) / \Gamma ( a )$ is the rising factorial or Pochhammer symbol.
+
+Using the above definitions and the integral representation of Beta functions, we then have
+
+$$
+\begin{array} { r l } { \int _ { - 1 } ^ { 1 } \left( 1 - i ^ { 2 } \right) ^ { n + \frac { \omega ^ { 2 } } { 2 } } P _ { k } ( \psi ) d u = 2 ^ { 2 n + \frac { \omega } { 2 } } \int _ { - 1 } ^ { 1 } \Bigg ( \displaystyle \frac { 1 - i } { 2 } \Bigg ) ^ { n + \frac { \omega } { 2 } } \left( \displaystyle \frac { 1 + i ^ { n } } { 2 } \right) ^ { n + \frac { \omega } { 2 } - 1 } P _ { k } ( \psi ) d u } \\ { = 2 ^ { 2 n + 1 } \ a ^ { 3 } \displaystyle \sum _ { s = 0 } ^ { k } \displaystyle \frac { \left( - i - k \right) \left( i ^ { 2 } - 2 k \right) } { \left( \frac { \omega ^ { 2 } } { 2 } \right) _ { s } , u ^ { 3 } } \int _ { - 1 } ^ { 1 } \left( \displaystyle \frac { 1 - i } { 2 } \right) ^ { s + \frac { \omega } { 2 } + \frac { \omega } { 2 } } \left( \displaystyle \frac { 1 + i } { 2 } \right) ^ { n + \frac { \omega } { 2 } } } \\ { = 2 ^ { 2 n + \frac { \omega } { 2 } } \displaystyle \sum _ { s = 0 } ^ { k } \left( \displaystyle \frac { \left( - k \right) \left( i - 2 + k \right) k } { 2 } \right) _ { s } \int _ { 0 } ^ { 1 } \left( 1 - w \right) ^ { n + \frac { \omega } { 2 } + \frac { \omega } { 2 } } u ^ { \frac { n + \omega } { 2 } } } \\ { = 2 ^ { 2 n + \frac { \omega } { 2 } } \displaystyle \sum _ { s = 0 } ^ { k } \left( \displaystyle \frac { i - k  \left( i - 2 + k \right) k } { \left( \frac { \omega } { 2 } \right) _ { s } , u ^ { 3 } } \int _ { 0 } ^ { 1 } \left( 1 - w \right) ^ { n + \frac { \omega } { 2 } } u ^ { \frac { n + \frac { \omega - 1 } { 2 } } } \right) \left( i ^ { \frac { \omega } { 2 } } + i ^ { \frac { \omega - 1 } { 2 } } \right) } \\  =\right) 2 ^ { 2 n + \frac { \omega - 1 } { 2 } } \displaystyle \sum _ { s = 0 } ^ { k } \frac { \left( i - k \right) \left( i ^ { 2 } - 2 k \right) n ! } { \left( i - 2 \right) _ { s } , u ^ { 3 } } \\  = 2 ^ { 2 n + \frac { \omega } { 2 } - 1 } \displaystyle \sum _ { s = 0 } ^ { k } \frac  \left( i  \end{array}
+$$
+
+Now, we use Watson’s theorem (e.g., Ismail, 2005, Eq. (1.4.12)), which states that
+
+$$
+{ _ 3 F _ { 2 } } \left( { _ { ( a + b + 1 ) / 2 , 2 c } } \Big | 1 \right) = \frac { \Gamma ( \frac { 1 } { 2 } ) \Gamma ( c + \frac { 1 } { 2 } ) \Gamma ( \frac { a + b + 1 } { 2 } ) \Gamma ( c + \frac { 1 - a - b } { 2 } ) } { \Gamma ( \frac { a + 1 } { 2 } ) \Gamma ( \frac { b + 1 } { 2 } ) \Gamma ( c + \frac { 1 - a } { 2 } ) } .
+$$
+
+We remark that with $a = - k , b = k + d - 2 , c = \nu + ( d - 1 ) / 2$ , our expression above is of the form of Watson’s theorem, and we may thus evaluate $\mu _ { k }$ in closed form. Indeed, we have
+
+$$
+{ _ 3 F _ { 2 } } \left( \begin{array} { c } { { - k , k + d - 2 , \nu + ( d - 1 ) / 2 } } \\ { { ( d - 1 ) / 2 , 2 \nu + d - 1 } } \end{array} \biggr | 1 \right) = \frac { \Gamma ( \frac { 1 } { 2 } ) \Gamma ( \nu + \frac { d } { 2 } ) \Gamma ( \frac { d - 1 } { 2 } ) \Gamma ( \nu + 1 ) } { \Gamma ( \frac { 1 - k } { 2 } ) \Gamma ( \frac { d + k - 1 } { 2 } ) \Gamma ( \nu + \frac { k } { 2 } + \frac { d } { 2 } ) \Gamma ( \nu + 1 - \frac { k } { 2 } ) } .
+$$
+
+When $k$ is odd, then $( 1 - k ) / 2$ is a non-positive integer so that the denominator is infinite and thus $\mu _ { k }$ vanishes. We assume from now on that $k$ is even, making the denominator is finite. Using the following relation, for $\epsilon \not \in \mathbb { Z }$ and an integer $n$ :
+
+$$
+{ \frac { \Gamma ( 1 + \epsilon ) } { \Gamma ( \epsilon - n ) } } = \epsilon ( \epsilon - 1 ) \cdot \cdot \cdot ( \epsilon - n ) = ( - 1 ) ^ { n - 1 } { \frac { \Gamma ( n + 1 - \epsilon ) } { \Gamma ( - \epsilon ) } } ,
+$$
+
+we may then rewrite
+
+$$
+{ } _ { 3 } F _ { 2 } \left( { - k , k + d - 2 , \nu + ( d - 1 ) / 2 } \Big | 1 \right) = \frac { \Gamma ( \nu + \frac { d } { 2 } ) \Gamma ( \frac { d - 1 } { 2 } ) \Gamma ( \nu + 1 ) } { \Gamma ( - \frac { 1 } { 2 } ) \Gamma ( \nu + 2 ) \Gamma ( - \nu - 1 ) } \frac { \Gamma ( \frac { k + 1 } { 2 } ) \Gamma ( \frac { k } { 2 } - \nu ) } { \Gamma ( \frac { d + k - 1 } { 2 } ) \Gamma ( \nu + \frac { k } { 2 } + \frac { d } { 2 } ) } \cdot
+$$
+
+When $k  \infty$ , Stirling’s formula $\Gamma ( x ) \sim x ^ { x - { \frac { 1 } { 2 } } } e ^ { - x } \sqrt { 2 \pi }$ yields the equivalent
+
+$$
+\frac { \Gamma ( \frac { k + 1 } { 2 } ) \Gamma ( \frac { k } { 2 } - \nu ) } { \Gamma ( \frac { d + k - 1 } { 2 } ) \Gamma ( \nu + \frac { k } { 2 } + \frac { d } { 2 } ) } \sim \left( \frac { k } { 2 } \right) ^ { - d - 2 \nu + 1 } .
+$$
+
+This yields
+
+$$
+\mu _ { k } \sim C ( d , \nu ) k ^ { - d - 2 \nu + 1 } ,
+$$
+
+with
+
+$$
+C ( d , \nu ) = 2 ^ { 2 \nu + d - 2 } \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } \frac { \Gamma ( \nu + \frac { d - 1 } { 2 } ) ^ { 2 } } { \Gamma ( 2 \nu + d - 1 ) } \frac { \Gamma ( \nu + \frac { d } { 2 } ) \Gamma ( \frac { d - 1 } { 2 } ) \Gamma ( \nu + 1 ) } { \Gamma ( - \frac { 1 } { 2 } ) \Gamma ( \nu + 2 ) \Gamma ( - \nu - 1 ) } ( 1 / 2 ) ^ { - d - 2 \nu + 1 } .
+$$
+
+Decay for $\phi _ { \nu }$ . The decay for $\phi _ { \nu }$ follows from the decay of $\phi _ { \nu }$ and the recurrence relation (Efthimiou $\&$ Frye, 2014, Eq. (4.36))
+
+$$
+t P _ { k } ( t ) = \frac { k } { 2 k + d - 2 } P _ { k - 1 } ( t ) + \frac { k + d - 2 } { 2 k + d - 2 } P _ { k + 1 } ( t ) ,
+$$
+
+which ensures the same decay with a change parity.
+
+Final theorem. We are now ready to prove our main theorem, which differs from the simplified statement of Theorem 1 by the technical assumption that only a finite number $r$ of terms of order between $\nu$ and $\nu + 1$ are present in the series expansions around $\pm 1$ .
+
+Theorem 7 (Main theorem, full version). Let $\kappa : [ - 1 , 1 ] \to \mathbb { R }$ be a function that is $C ^ { \infty }$ on $( - 1 , 1 )$ and has the following expansions around $\pm 1$ :
+
+$$
+\begin{array} { l } { \kappa ( t ) = p _ { 1 } ( 1 - t ) + \displaystyle \sum _ { j = 1 } ^ { r } c _ { j , 1 } ( 1 - t ) ^ { \nu _ { j } } + O ( ( 1 - t ) ^ { \nu _ { 1 } + 1 + \epsilon } ) } \\ { \kappa ( t ) = p _ { - 1 } ( 1 + t ) + \displaystyle \sum _ { j = 1 } ^ { r } c _ { j , - 1 } ( 1 + t ) ^ { \nu _ { j } } + O ( ( 1 + t ) ^ { \nu _ { 1 } + 1 + \epsilon } ) , } \end{array}
+$$
+
+for $t \in [ - 1 , 1 ]$ , where $p _ { 1 } , p _ { - 1 }$ are polynomials and $0 < \nu _ { 1 } < . . . < \nu _ { r }$ are not integers and $0 < \epsilon < \nu _ { 2 } - \nu _ { 1 }$ . We also assume that the derivatives $\kappa ^ { ( s ) }$ of $\kappa$ have the following expansions:
+
+$$
+\begin{array} { l } { { \kappa ^ { ( s ) } ( t ) = p _ { s , 1 } ( 1 - t ) + ( - 1 ) ^ { s } \displaystyle \sum _ { j = 1 } ^ { r } c _ { j , 1 } \displaystyle \frac { \Gamma ( \nu _ { j } + 1 ) } { \Gamma ( \nu _ { j } + 1 - s ) } ( 1 - t ) ^ { \nu _ { j } - s } + { \cal O } ( ( 1 - t ) ^ { \nu _ { 1 } + 1 + \epsilon - s } ) } } \\ { { \kappa ^ { ( s ) } ( t ) = p _ { s , - 1 } ( 1 + t ) + \displaystyle \sum _ { j = 1 } ^ { r } c _ { j , - 1 } \displaystyle \frac { \Gamma ( \nu _ { j } + 1 ) } { \Gamma ( \nu _ { j } + 1 - s ) } ( 1 + t ) ^ { \nu _ { j } - s } + { \cal O } ( ( 1 + t ) ^ { \nu _ { 1 } + 1 + \epsilon - s } ) , } } \end{array}
+$$
+
+for some polynomials $p _ { s , \pm 1 }$ . Then we have, for an absolute constant $C ( d , \nu _ { 1 } )$ depending only on $d$ and $\nu _ { 1 }$ ,
+
+Proof. Define the functions
+
+$$
+\begin{array} { l } { { \displaystyle \psi _ { j } ( t ) = c _ { j , 1 } \frac { \phi _ { \nu _ { j } } ( t ) + \bar { \phi } _ { \nu _ { j } } ( t ) } { 2 ^ { \nu _ { j } + 1 } } + c _ { j , - 1 } \frac { \phi _ { \nu _ { j } } ( t ) - \bar { \phi } _ { \nu _ { j } } ( t ) } { 2 ^ { \nu _ { j } + 1 } } } } \\ { { \displaystyle \quad = \frac { c _ { j , 1 } + c _ { j , - 1 } } { 2 ^ { \nu _ { j } + 1 } } \phi _ { \nu _ { j } } ( t ) + \frac { c _ { j , 1 } - c _ { j , - 1 } } { 2 ^ { \nu _ { j } + 1 } } \bar { \phi } _ { \nu _ { j } } ( t ) , } } \end{array}
+$$
+
+for $j = 1 , \dots , r$ , where $\phi _ { \nu } , \phi _ { \nu }$ are defined in Lemma 6. We have the asymptotic expansions:6
+
+$$
+\begin{array} { l } { \psi _ { 1 } ( t ) = c _ { 1 , 1 } ( 1 - t ) ^ { \nu _ { 1 } } - \displaystyle \frac { ( 1 + \nu _ { 1 } ) c _ { 1 , 1 } + c _ { 1 , - 1 } } { 2 } ( 1 - t ) ^ { \nu _ { 1 } + 1 } + O ( ( 1 - t ) ^ { \nu _ { 1 } + 1 + \epsilon } ) } \\ { \psi _ { 1 } ( t ) = c _ { 1 , - 1 } ( 1 + t ) ^ { \nu _ { 1 } } + \displaystyle \frac { c _ { 1 , 1 } - ( 1 + \nu ) c _ { 1 , - 1 } } { 2 } ( 1 + t ) ^ { \nu _ { 1 } + 1 } + O ( ( 1 + t ) ^ { \nu _ { 1 } + 1 + \epsilon } ) , } \end{array}
+$$
+
+and for $j \geq 2$ ,
+
+$$
+\begin{array} { l } { { \psi _ { j } ( t ) = c _ { j , 1 } ( 1 - t ) ^ { \nu _ { j } } + O ( ( 1 - t ) ^ { \nu _ { 1 } + 1 + \epsilon } ) } } \\ { { \psi _ { j } ( t ) = c _ { j , - 1 } ( 1 + t ) ^ { \nu _ { j } } + O ( ( 1 + t ) ^ { \nu _ { 1 } + 1 + \epsilon } ) . } } \end{array}
+$$
+
+Define additionally $\psi _ { r + 1 }$ the same way as the other $\psi _ { j }$ , with $\nu _ { r + 1 } = \nu _ { 1 } + 1$ , $c _ { r + 1 , 1 } =$ $( ( 1 + \nu _ { 1 } ) c _ { 1 , 1 } + c _ { 1 , - 1 } ) / 2$ , and $c _ { r + 1 , - 1 } = - ( c _ { 1 , 1 } - ( 1 + \nu ) c _ { 1 , - 1 } ) / 2$ , which satisfies a similar asymptotic expansion as the above ones for $j \geq 2$ . One can check that the derivatives $\psi _ { j }$ expanded w, we have for e derivatives of the expansions above. Then, defining, $\begin{array} { r } { \tilde { \kappa } = \kappa - \sum _ { j = 1 } ^ { r + 1 } \psi _ { j } } \end{array}$ $s \geq 0$
+
+$$
+\begin{array} { l } { { \tilde { \kappa } ^ { ( s ) } ( t ) = p _ { s , 1 } ( 1 - t ) + O ( ( 1 - t ) ^ { \nu _ { 1 } + 1 + \epsilon - s } ) } } \\ { { \tilde { \kappa } ^ { ( s ) } ( t ) = p _ { s , - 1 } ( 1 + t ) + O ( ( 1 + t ) ^ { \nu _ { 1 } + 1 + \epsilon - s } ) . } } \end{array}
+$$
+
+The functions $\psi _ { j }$ satisfy
+
+$$
+\mu _ { k } ( \psi _ { j } ) = \left\{ \begin{array} { l l } { \frac { c _ { j , 1 } + c _ { j , - 1 } } { 2 ^ { \nu _ { j } + 1 } } \mu _ { k } ( \phi _ { \nu _ { j } } ) , } & { \mathrm { ~ i f ~ } k \mathrm { ~ e v e n } , } \\ { \frac { c _ { j , 1 } - c _ { j , - 1 } } { 2 ^ { \nu _ { j } + 1 } } \mu _ { k } ( \bar { \phi } _ { \nu _ { j } } ) , } & { \mathrm { ~ i f ~ } k \mathrm { ~ o d d } . } \end{array} \right.
+$$
+
+By Lemma 5, we have
+
+$$
+\begin{array} { l } { \displaystyle \mu _ { k } ( \kappa ) = \mu _ { k } ( \tilde { \kappa } ) + \sum _ { j = 1 } ^ { r } \mu _ { k } ( \psi _ { j } ) } \\ { \displaystyle = \sum _ { j = 1 } ^ { r } \mu _ { k } ( \psi _ { j } ) + O ( k ^ { - d - 2 ( \nu _ { 1 } + 1 + \epsilon ) + 3 } ) } \\ { \displaystyle = \sum _ { j = 1 } ^ { r } \mu _ { k } ( \psi _ { j } ) + o ( k ^ { - d - 2 \nu _ { 1 } + 1 } ) . } \end{array}
+$$
+
+The result then follows from Lemma 6, with a constant $C ( d , \nu _ { 1 } ) / 2 ^ { \nu _ { 1 } + 1 }$ , where $C ( d , \nu _ { 1 } )$ is given by the proof of Lemma 6. $\boxed { \begin{array} { r l } \end{array} }$
+
+# B.1 Dimension-free description
+
+While our above description of the RKHS depends on the dimension $d$ , in some cases a dimension-free description given by Taylor coefficients of the kernel $\kappa$ at 0 may be useful, for instance for the study of kernel methods in certain high-dimensional regimes (e.g., El Karoui, 2010; Ghorbani et al., 2019; Liang et al., 2020). Here we remark that such coefficients and their decay may be recovered from the Legendre coefficients in $d$ dimensions, by taking highdimensional limits $d \to \infty$ . We illustrate this on the functions $\phi _ { \nu } ( t ) = ( 1 - t ^ { 2 } ) ^ { \nu }$ , for which Lemma 6 provides precise estimates of the Legendre coefficients $\mu _ { k , d } ( \phi _ { \nu } )$ in $d$ dimensions (this only serves as an instructive illustration, since in this case Taylor coefficients may be computed directly through a power series expansion of $\phi _ { \nu }$ using the Binomial formula).
+
+Lemma 8 (Recovering Taylor coefficients of $\phi _ { \nu }$ through high-dimensional limits). Let $\begin{array} { r } { b _ { k } ( \phi _ { \nu } ) : = \frac { \phi _ { \nu } ^ { ( k ) } } { k ! } } \end{array}$ for some non-integer $\nu > 0$ . For $k$ even, we have
+
+$$
+b _ { k } ( \phi _ { \nu } ) = C _ { \nu } 2 ^ { k } \frac { \Gamma ( \frac { k + 1 } { 2 } ) \Gamma ( \frac { k } { 2 } - \nu ) } { \Gamma ( k + 1 ) } ,
+$$
+
+for a constant $C _ { \nu }$ depending only on $\nu$ . This leads to an equivalent $b _ { k } \sim C _ { \nu } ^ { \prime } k ^ { - \nu - 1 }$ for $k  \infty$ with $k$ even.
+
+Proof. Assume throughout that $k$ is even. Recall the expression of the Legendre coefficients $\mu _ { k , d } ( \phi _ { \nu } )$ of $\phi _ { \nu }$ in $d$ dimensions (we include $d$ as a subscript for more clarity here) from the proof of Lemma 6:
+
+$$
+\begin{array} { l } { \displaystyle \mu _ { k , d } ( \phi _ { \nu } ) = \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } \int _ { - 1 } ^ { 1 } \kappa ( t ) P _ { k , d } ( t ) ( 1 - t ^ { 2 } ) ^ { \frac { d - 3 } { 2 } } d t } { ( 5 \xi + \frac { d - 1 } { 2 } ) ^ { \lambda } }  \\ { = \displaystyle 2 ^ { 2 \nu + d - 2 } \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } \frac { \Gamma ( \nu + \frac { d - 1 } { 2 } ) ^ { 2 } } { \Gamma ( 2 \nu + d - 1 ) } \frac { \Gamma ( \nu + \frac { d } { 2 } ) \Gamma ( \frac { d - 1 } { 2 } ) \Gamma ( \nu + 1 ) } { \Gamma ( - \frac { 1 } { 2 } ) \Gamma ( \nu + 2 ) \Gamma ( - \nu - 1 ) } \frac { \Gamma ( \frac { k + 1 } { 2 } ) \Gamma ( \frac { k } { 2 } - \nu ) } { \Gamma ( \frac { d + k - 1 } { 2 } ) \Gamma ( \nu + \frac { k } { 2 } + \frac { d } { 2 } ) } . } \end{array}
+$$
+
+Now, note that when $d$ is large enough compared to $k$ , we may use the Rodrigues formula (16) and integration by parts to obtain the following alternative expression:
+
+$$
+\mu _ { k , d } ( \phi _ { \nu } ) = 2 ^ { - k } \frac { \omega _ { d - 2 } } { \omega _ { d - 1 } } \frac { \Gamma ( \frac { d - 1 } { 2 } ) } { \Gamma ( k + \frac { d - 1 } { 2 } ) } \int _ { - 1 } ^ { 1 } \phi _ { \nu } ^ { ( k ) } ( t ) ( 1 - t ^ { 2 } ) ^ { k + \frac { d - 3 } { 2 } } d t
+$$
+
+Following similar arguments to Ghorbani et al. (2019), we may then use dominated convergence to show:
+
+$$
+\frac { \Gamma ( \frac { d } { 2 } ) } { \sqrt { \pi } \Gamma ( \frac { d - 1 } { 2 } ) } \int _ { - 1 } ^ { 1 } \phi _ { \nu } ^ { ( k ) } ( t ) ( 1 - t ^ { 2 } ) ^ { k + \frac { d - 3 } { 2 } } d t  \phi _ { \nu } ^ { ( k ) } ( 0 ) \quad \mathrm { ~ a s ~ } d  \infty .
+$$
+
+Indeed, $\frac { \Gamma ( \frac { d } { 2 } ) } { \sqrt { \pi } \Gamma ( \frac { d - 1 } { 2 } ) } ( 1 - t ^ { 2 } ) ^ { ( d - 3 ) / 2 }$ is a probability density that approaches a Dirac mass at $0$ when $d \to \infty$ . This yields
+
+$$
+b _ { k } ( \phi _ { \nu } ) = \frac { \phi _ { \nu } ^ { ( k ) } } { k ! } = \operatorname * { l i m } _ { d \to \infty } 2 ^ { k } \frac { \omega _ { d - 1 } } { \omega _ { d - 2 } } \frac { \Gamma ( \frac { d } { 2 } ) \Gamma ( k + \frac { d - 1 } { 2 } ) } { \sqrt { \pi } \Gamma ( \frac { d - 1 } { 2 } ) \Gamma ( \frac { d - 1 } { 2 } ) \Gamma ( k + 1 ) } \mu _ { k , d } ( \phi _ { \nu } ) .
+$$
+
+Plugging (59) and using Stirling’s formula to take limits $d \to \infty$ yields
+
+$$
+b _ { k } ( \phi _ { \nu } ) = C _ { \nu } 2 ^ { k } \frac { \Gamma ( \frac { k + 1 } { 2 } ) \Gamma ( \frac { k } { 2 } - \nu ) } { \Gamma ( k + 1 ) } ,
+$$
+
+lent where $b _ { k } ( \phi _ { \nu } ) \sim C _ { \nu } ^ { \prime } k ^ { - \nu - 1 }$ $C _ { \nu }$ only depends on for $\nu$ $k  \infty$ . Using Stirling’s formula once again yields the desired equiva- , $k$ even, with a different constant $C _ { \nu } ^ { \prime }$ . □
+
+We note that a similar asymptotic equivalent holds for $b _ { k } { \left( \phi _ { \nu } \right) }$ for $k$ odd. The next result leverages this to derive asymptotic decays of $b _ { k } ( \kappa )$ for any $\kappa$ of the form $\kappa ( u ) \ =$ $\textstyle \sum _ { k \geq 0 } b _ { k } ( \kappa ) u ^ { k }$ satisfying similar conditions as in Theorem 7.
+
+Corollary 9 (Taylor coefficients of $\kappa$ ). Let $\kappa : [ - 1 , 1 ] \to \mathbb { R }$ be a function admitting a power series expansion $\begin{array} { r } { \kappa ( u ) = \sum _ { k \geq 0 } b _ { k } u ^ { k } } \end{array}$ , with the following expansions around $\pm 1$ :
+
+$$
+\begin{array} { l } { \kappa ( t ) = p _ { 1 } ( 1 - t ) + \displaystyle \sum _ { j = 1 } ^ { r } c _ { j , 1 } ( 1 - t ) ^ { \nu _ { j } } + O ( ( 1 - t ) ^ { \lceil \nu _ { 1 } \rceil + 1 } ) } \\ { \kappa ( t ) = p _ { - 1 } ( 1 + t ) + \displaystyle \sum _ { j = 1 } ^ { r } c _ { j , - 1 } ( 1 + t ) ^ { \nu _ { j } } + O ( ( 1 + t ) ^ { \lceil \nu _ { 1 } \rceil + 1 } ) , } \end{array}
+$$
+
+for $t \in [ - 1 , 1 ]$ , where $p _ { 1 } , p _ { - 1 }$ are polynomials and $0 < \nu _ { 1 } < . . . < \nu _ { r }$ are not integers and $0 < \epsilon < \nu _ { 2 } - \nu _ { 1 }$ . Then we have, for an absolute constant $C ( \nu _ { 1 } )$ depending only on $\nu _ { 1 }$ ,
+
+Proof. As in the proof of Theorem 7, we may construct a function $\begin{array} { r } { \psi = \sum _ { j } \alpha _ { j } \phi _ { \nu _ { j } } + \bar { \alpha } _ { j } \phi _ { \nu _ { j } } } \end{array}$ , with $\begin{array} { r } { \alpha _ { 1 } = \frac { c _ { 1 , 1 } + c _ { 1 , - 1 } } { 2 ^ { \nu _ { 1 } + 1 } } } \end{array}$ , $\begin{array} { r } { \bar { \alpha } _ { 1 } = \frac { c _ { 1 , 1 } - c _ { 1 , - 1 } } { 2 ^ { \nu _ { 1 } + 1 } } } \end{array}$ for $j = 1$ , the other terms being of higher orders $\nu _ { j } >$ $\nu _ { 1 }$ , such that $\tilde { \kappa } : = \kappa - \psi$ (which is also a power series with convergence radius $\geq 1$ ) satisfies
+
+$$
+\begin{array} { l } { { \tilde { \kappa } ( t ) = p _ { 1 } ( 1 - t ) + O ( ( 1 - t ) ^ { \lceil \nu _ { 1 } \rceil + 1 } ) } } \\ { { \tilde { \kappa } ( t ) = p _ { - 1 } ( 1 + t ) + O ( ( 1 + t ) ^ { \lceil \nu _ { 1 } \rceil + 1 } ) , } } \end{array}
+$$
+
+It follows that $\tilde { \kappa } ^ { ( \lceil \nu _ { 1 } \rceil + 1 ) } ( 1 )$ is bounded, so that the Taylor coefficients of $\tilde { \kappa }$ , denoted $b _ { k } ( \tilde { \kappa } )$ , satisfy
+
+$$
+b _ { k } ( \tilde { \kappa } ) = o ( k ^ { - \lceil \nu _ { 1 } \rceil - 1 } ) = o ( k ^ { - \nu _ { 1 } - 1 } ) .
+$$
+
+The result then follows from Lemma 8 by using the decays of $b _ { k } { \left( \phi _ { \nu } \right) }$ and $b _ { k } { \left( \phi _ { \nu } \right) }$ .
+
+# C Other Proofs
+
+In this section, we provide the proofs for results in Section 3.3 related to obtaining power series expansions (with generalized exponents) of kernels arising from deep networks, which leads to the corresponding decays by Theorem 1. We note that for the kernels considered in this section, we can differentiate the expansions since the kernel function is $\Delta$ -analytic (see Chen & Xu, 2021, Theorem 7), so that the technical assumption in Theorem 1 is verified.
+
+# C.1 Proof of Corollary 2
+
+Proof. Let $\kappa ^ { \ell } : = \kappa _ { 1 } \circ \cdot \cdot \cdot \circ \kappa _ { 1 _ { . } } = \kappa _ { \mathrm { R F } } ^ { \ell }$ . We have $\ell - 1$ {z times
+
+$$
+\kappa _ { 1 } ( 1 - t ) = 1 - t + c t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } ) , \quad c : = \frac { 2 \sqrt { 2 } } { 3 \pi } .
+$$
+
+We now show by induction that $\kappa _ { , } ^ { \ell } ( 1 - t ) = 1 - t + a _ { \ell } t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } )$ , with $a _ { \ell } = ( \ell - 1 ) c$ . This is obviously true for $\ell = 2$ since $\kappa ^ { \ell } = \kappa _ { 1 }$ , and for $\ell \geq 3$ we have
+
+$$
+\begin{array} { r l } & { \kappa ^ { \ell } ( 1 - t ) = \kappa _ { 1 } ( \kappa ^ { \ell - 1 } ( 1 - t ) ) } \\ & { \qquad = \kappa _ { 1 } ( 1 - t + a _ { \ell - 1 } t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } ) ) } \\ & { \qquad = 1 - ( t - a _ { \ell - 1 } t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } ) ) + c ( t + O ( t ^ { 3 / 2 } ) ) ^ { 3 / 2 } + o ( O ( t ) ^ { 3 / 2 } ) } \\ & { \qquad = 1 - t + a _ { \ell - 1 } t ^ { 3 / 2 } + c t ^ { 3 / 2 } ( 1 + O ( t ^ { 1 / 2 } ) ) ^ { 3 / 2 } + o ( t ^ { 3 / 2 } ) } \\ & { \qquad = 1 - t + a _ { \ell - 1 } t ^ { 3 / 2 } + c t ^ { 3 / 2 } ( 1 + O ( t ^ { 1 / 2 } ) ) + o ( t ^ { 3 / 2 } ) } \\ & { \qquad = 1 - t + a _ { \ell } t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } ) , } \end{array}
+$$
+
+which proves the result.
+
+Around $^ { - 1 }$ , we know that
+
+$$
+\kappa _ { 1 } ( - 1 + t ) = c t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } ) .
+$$
+
+We then have $\kappa ^ { \ell } ( - 1 + t ) = b _ { \ell } + c _ { \ell } t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } )$ , with $0 \leq b _ { \ell } < 1$ and $0 < c _ { \ell } \le c$ (and the upper bound is strict for $\ell \geq 3$ ). Indeed, this is true for $\ell = 2$ , and for $\ell \geq 3$ we have,
+
+for $t > 0$
+
+$$
+\begin{array} { r l } & { \kappa ^ { \ell } ( - 1 + t ) = \kappa _ { 1 } ( \kappa ^ { \ell - 1 } ( - 1 + t ) ) } \\ & { \qquad = \kappa _ { 1 } \bigl ( b _ { \ell - 1 } + c _ { \ell - 1 } t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } ) \bigr ) } \\ & { \qquad = \kappa _ { 1 } \bigl ( b _ { \ell - 1 } ) + \kappa _ { 1 } ^ { \prime } ( b _ { \ell - 1 } ) c _ { \ell - 1 } t ^ { 3 / 2 } + o ( t ^ { 3 / 2 } ) . } \end{array}
+$$
+
+Now, note that $\kappa _ { 1 }$ and $\kappa _ { 1 } ^ { \prime }$ are both positive and strictly increasing on $[ 0 , 1 ]$ , with $\kappa _ { 1 } ( 1 ) =$ $\kappa _ { 1 } ^ { \prime } ( 1 ) = 1$ . Thus, we have $b _ { \ell } = \kappa _ { 1 } ( b _ { \ell - 1 } ) \in ( 0 , 1 )$ , and $c _ { \ell } = \kappa _ { 1 } ^ { \prime } ( a _ { \ell - 1 } ) c _ { \ell - 1 } < c _ { \ell - 1 }$ , thus completing the proof.
+
+Since $c _ { \ell }$ is bounded while $a _ { \ell }$ grows linearly with $\ell$ , the constants in front of the asymptotic decay $k ^ { - d - 2 }$ grow linearly with $\ell$ .
+
+# C.2 Proof of Corollary 3
+
+Proof. We show by induction that $\kappa _ { \mathrm { N T K } } ^ { \ell }$ as defined in (7) satisfies
+
+$$
+\kappa _ { \mathrm { N T K } } ^ { \ell } ( 1 - t ) = \ell - \left( \sum _ { s = 1 } ^ { \ell - 1 } s \right) c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) , \qquad c : = \frac { \sqrt { 2 } } { \pi } .
+$$
+
+For $\ell = 2$ we have $\kappa _ { \mathrm { N T K } } ^ { \ell } ( u ) = u \kappa _ { 0 } ( u ) + \kappa _ { 1 } ( u )$ , so that
+
+$$
+\kappa _ { \mathrm { N T K } } ^ { 2 } ( 1 - t ) = ( 1 - t ) ( 1 - c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) ) + 1 + O ( t ) = 2 - c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) .
+$$
+
+By induction, for $\ell \geq 3$ , we have $\kappa _ { \mathrm { N T K } } ^ { \ell } ( u ) = \kappa _ { \mathrm { N T K } } ^ { \ell - 1 } ( u ) \kappa _ { 0 } ( \kappa ^ { \ell - 1 } ( u ) ) + \kappa ^ { \ell } ( u )$ , with $\kappa ^ { \ell }$ as in the proof of Corollary 2, which hence satisfies $\kappa ^ { \ell } ( 1 - t ) = 1 - t + o ( t )$ for all $\ell \geq 2$ . We then have
+
+$$
+\begin{array} { r l } & { \kappa _ { 0 } ( \kappa ^ { \ell - 1 } ( 1 - t ) ) = \kappa _ { 0 } ( 1 - t + o ( t ) ) } \\ & { \qquad = 1 - c ( t + o ( t ) ) ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) } \\ & { \qquad = 1 - c t ^ { 1 / 2 } ( 1 + o ( t ^ { 1 / 2 } ) ) + o ( t ^ { 1 / 2 } ) } \\ & { \qquad = 1 - c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) . } \end{array}
+$$
+
+This yields
+
+$$
+\begin{array} { l } { \kappa _ { \mathrm { N T K } } ^ { \ell } ( 1 - t ) = \displaystyle ( \ell - 1 - ( \overset { \ell - 2 } { \underset { s = 1 } { \sum } } s ) c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) ) ( 1 - c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) ) + 1 + O ( t ) } \\ { = \ell - ( \overset { \ell - 1 } { \underset { s = 1 } { \sum } } s ) c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) } \\ { = \ell - \frac { \ell ( \ell - 1 ) } { 2 } c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) , } \end{array}
+$$
+
+which proves the claim for the expansion around $+ 1$ .
+
+Around -1, recall the expansion from the proof of Corollary 2, $\kappa ^ { \ell } ( - 1 + t ) = b _ { \ell } + O ( t ^ { 3 / 2 } )$ , with $0 \leq b _ { \ell } < 1$ . For $\ell = 2$ , we have
+
+$$
+\kappa _ { \mathrm { N T K } } ^ { 2 } ( - 1 + t ) = ( - 1 + t ) ( c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) ) + b _ { 2 } + o ( t ^ { 1 / 2 } ) = b _ { 2 } - c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) .
+$$
+
+Note also that for $\ell \geq 2$ ,
+
+$$
+\kappa _ { 0 } ( \kappa ^ { \ell } ( - 1 + t ) ) = \kappa _ { 0 } ( b _ { \ell } + O ( t ^ { 3 / 2 } ) ) = \kappa _ { 0 } ( b _ { \ell } ) + O ( t ^ { 3 / 2 } ) ,
+$$
+
+since $\kappa _ { 0 } ^ { \prime } ( b _ { \ell } )$ is finite for $b _ { \ell } < 1$ . We also have $\kappa _ { 0 } ( b _ { \ell } ) \in ( 0 , 1 )$ since $\kappa _ { 0 }$ is positive and strictly increasing on $[ 0 , 1 ]$ with $\kappa _ { 0 } ( 1 ) = 1$ . Then, by an easy induction, we obtain
+
+$$
+\kappa _ { \mathrm { N T K } } ^ { \ell } ( - 1 + t ) = a _ { \ell } - c _ { \ell } t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } ) ,
+$$
+
+with $a _ { \ell } \leq \ell$ and $0 < c _ { \ell } < c$ .
+
+Similar to the case of the RF kernel, the constant in front of $t ^ { 1 / 2 }$ grows with $\ell ^ { 2 }$ for the expansion around $+ 1$ but is bounded for the expansion around $^ { - 1 }$ , so that the final constants in front of the asymptotic decay $k ^ { - d }$ grow quadratically with $\ell$ . However, they grow linearly with $\ell$ when considering the NTK normalized by $\ell$ , $\tilde { \kappa } ^ { \ell } = \kappa _ { \mathrm { N T K } } ^ { \ell } / \ell$ , which then satisfies $\tilde { \kappa } ^ { \ell } ( 1 ) = 1$ .
+
+# C.3 Deep networks with step activations
+
+In this section, we study the decay of the random weight kernel arising from deep networks with step activations, as presented in Section 3.3. For an $L$ -layer network, this kernel is of the form $\kappa _ { s } ^ { L } : = \kappa _ { 0 } \circ \cdots \circ \kappa _ { 0 _ { \cdot } }$ .
+
+![](images/6da673c9b85de038e84cc1fc67ec318c7bc9973a95e1484d16c14f82e3c3431c.jpg)
+
+Corollary 10. $\kappa _ { s } ^ { L }$ has a decay $k ^ { - d - 2 \nu _ { L } + 1 }$ with $\nu _ { L } = 1 / 2 ^ { L - 1 }$ for $L$ layers.
+
+Proof. We show by induction that we have, for $\ell \geq 2$ ,
+
+$$
+\begin{array} { r } { \kappa _ { s } ^ { \ell } ( 1 - t ) = 1 - c ^ { \sum _ { j = 0 } ^ { \ell - 1 } 2 ^ { - j } } t ^ { 1 / 2 ^ { \ell - 1 } } + o ( t ^ { 1 / 2 ^ { \ell - 1 } } ) , } \end{array}
+$$
+
+with $\begin{array} { r } { c : = \frac { \sqrt { 2 } } { \pi } } \end{array}$ . This is true for $\ell = 2$ due to the expansion for $\kappa _ { 0 }$ . Now assume it holds for $\ell \geq 2$ . We have
+
+$$
+\begin{array} { r l } & { \kappa _ { s } ^ { \ell + 1 } ( 1 - t ) = \kappa _ { 0 } ( \kappa _ { s } ^ { \ell } ( 1 - t ) ) } \\ & { \qquad = \kappa _ { 0 } \left( 1 - c \sum _ { j = 0 } ^ { \ell - 1 } 2 ^ { - j } t ^ { 1 / 2 ^ { \ell - 1 } } + o ( t ^ { 1 / 2 ^ { \ell - 1 } } ) \right) } \\ & { \qquad = 1 - c \left( c \sum _ { j = 0 } ^ { \ell - 1 } t ^ { 1 / 2 ^ { \ell - 1 } } + o ( t ^ { 1 / 2 ^ { \ell - 1 } } ) \right) ^ { 1 / 2 } + o ( o ( t ^ { 1 / 2 ^ { \ell - 1 } } ) ^ { 1 / 2 } ) } \\ & { \qquad = 1 - c \sum _ { j = 0 } ^ { \ell } 2 ^ { - j } t ^ { 1 / 2 ^ { \ell } } ( 1 + o ( 1 ) ) + o ( t ^ { 1 / 2 ^ { \ell } } ) } \\ & { \qquad = 1 - c \sum _ { j = 0 } ^ { \ell } 2 ^ { - j } t ^ { 1 / 2 ^ { \ell } } + o ( t ^ { 1 / 2 ^ { \ell } } ) , } \end{array}
+$$
+
+proving the desired claim.
+
+Around $^ { - 1 }$ , we have $\kappa _ { 0 } ( - 1 + t ) = c t ^ { 1 / 2 } + o ( t ^ { 1 / 2 } )$ , and for $\ell \geq 3$ , $\kappa _ { s } ^ { \ell } ( - 1 + t ) = a _ { \ell } + O ( t ^ { 1 / 2 } )$ , by an easy induction using the fact that $\kappa _ { 0 } ( [ 0 , 1 ) ) \subset ( 0 , 1 )$ and $\kappa _ { 0 }$ is smooth on $\lfloor 0 , 1 \rfloor$ . Thus the behavior around $^ { - 1 }$ does not affect the decay of $\kappa _ { s } ^ { \ell }$ for $\ell \geq 3$ , and Theorem 1 leads to the desired decay, with a constant that only depends on $\ell$ through $\scriptstyle { C ^ { \sum _ { j = 0 } ^ { \ell - 1 } 2 ^ { - j } } }$ , which lies in the interval $[ c ^ { 2 } , c ]$ for any $\ell$ . □

parse/train/r1kNDlbCb/r1kNDlbCb.md

@@ -0,0 +1,309 @@
+# LEARNING TO ENCODE TEXT AS HUMAN-READABLESUMMARIES USING GENERATIVE ADVERSARIAL NET-WORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Auto-encoders compress input data into a latent-space representation and reconstruct the original data from the representation. This latent representation is not easily interpreted by humans. In this paper, we propose training an auto-encoder that encodes input text into human-readable sentences. The auto-encoder is composed of a generator and a reconstructor. The generator encodes the input text into a shorter word sequence, and the reconstructor recovers the generator input from the generator output. To make the generator output human-readable, a discriminator restricts the output of the generator to resemble human-written sentences. By taking the generator output as the summary of the input text, abstractive summarization is achieved without document-summary pairs as training data. Promising results are shown on both English and Chinese corpora.
+
+# 1 INTRODUCTION
+
+When it comes to learning data representations, a popular approach involves the auto-encoder architecture, which compresses the data into a latent representation without supervision. In this paper we focus on learning text representations. Because text is a sequence of words, to encode a sequence, a sequence-to-sequence (seq2seq) auto-encoder (Li et al., 2015; Kiros et al., 2015) is usually used, in which a RNN is used to encode the input sequence into a fixed-length representation, after which another RNN is used to decode the original input sequence given this representation.
+
+Although the latent representation learned by the auto-encoder can be used in downstream applications, they are usually not human-readable. In this work, we use comprehensible natural language as a latent representation of the input source text in an auto-encoder model. This human-readable latent representation is shorter than the source text; in order to reconstruct the source text, it must reflect the core idea of the source text. Intuitively, the latent representation can be considered a summary of the text.
+
+The idea that using human comprehensible representation as a latent representation has been explored on text summarization (Miao & Blunsom, 2016), but only in a semi-supervised scenario. Previous work uses a prior distribution from a pre-trained language model to constrain the generated sequence to natural language. However, to teach the compressor network to generate text summaries, the model is trained using labeled data. In contrast, in this work we need no labeled data to learn the representations.
+
+The proposed model is inspired from cycle consistency (Zhu et al., 2017; He et al., 2016). As shown in Fig. 1, the proposed model is composed of three components: a generator, a discriminator, and a reconstructor. Together, the generator and reconstructor form a text auto-encoder. The generator acts as an encoder in generating the latent representation from the input text. Instead of using a vector as latent representation, however, the generator generates a word sequence much shorter than the input text. From the shorter text, the reconstructor reconstructs the original input of the generator. By minimizing the reconstruction errors, the generator learns to generate short text segments that contain the main information in the original input. We use the seq2seq model in modeling the generator and reconstructor because both have input and output sequences with different lengths.
+
+However, it is very possible that the generator’s output word sequence can be processed by the reconstructor but is not readable by humans. Here, instead of regularizing the generator output with a pre-trained language model (Miao & Blunsom, 2016), we borrow from adversarial autoencoders (Makhzani et al., 2015) and introduce a third component – the discriminator – to regularize the generator’s output word sequence.
+
+![](images/ded4773453bdbbecb7f20057d902c30e59ccb993a5de3ea27747aab8d1a8e652.jpg)  
+Figure 1: Proposed model. Given long text, the generator produces a shorter text as a summary. The generator is learned by minimizing the reconstruction loss together with the reconstructor and making discriminator regard its output as human-written text.
+
+The discriminator and the generator form a generative adversarial network (GAN) (Goodfellow et al., 2014). GANs are generative models composed of a generator and a discriminator. The discriminator discriminates between the generator output and real data, and the generator produces output as similar as possible to real data to confuse the discriminator. Here, we only have to feed human-written sentences to the discriminator as real data. With the GAN framework, the discriminator teaches the generator how to create human-like summary sentences as a latent representation; however, this only guarantees that the generator produces grammatically correct sentences – not necessarily sentences that represent the input text. It is the reconstructor that teaches the generator how to produce a sentence that captures the core idea of the source text.
+
+However, generating discrete distributions with GAN is challenging, since it is difficult to evaluate the distance between the continuous distribution from the generator and the discrete distribution of the real sample. In addition, if we feed sampled words from the generator output distribution to the discriminator, the process of word selection is non-differentiable, which yields a discriminator gradient that precludes back-propagation to the generator. With GAN, there are two ways to generate language: (1) by training with a policy gradient, which regards words as actions, or (2) by directly feeding the generator’s output layer to the discriminator, which yields a gradient suited to backpropagation to the generator. In this work, we propose new kind of method on training with policy gradient in which the discriminator evaluates the output of generator every time steps. On language generation with GAN, we conduct experiments using both (1) and (2) methods and evaluate their results.
+
+We evaluate the results on an abstractive text summarization task in which the machine generates a text summary in its own words. The model is learned from a set of unpaired documents and summaries1. We use the sentences in the summaries as real data for discriminator2. As the summaries can come from another set of documents not related to the training documents, training is unsupervised. We use the output word sequence of the generator as the summaries of the input text. The results show that the generator generates summaries with reasonable quality on both English and Chinese corpora.
+
+# 2 RELATED WORK
+
+# GAN FOR LANGUAGE GENERATION
+
+The major challenge in applying GAN to sentence generation is the discrete nature of natural language. To generate a word sequence, the generator usually has non-differential parts such as argmax or other sample functions which cause the original GAN to fail. Therefore, new kinds of GANs have been proposed for sentence generation.
+
+SeqGAN (Yu et al., 2017) tackles the sequence generation problem with reinforcement learning. Here, we refer to this approach as adversarial REINFORCE, in which the generator is regarded as an agent, the generated sequence of words is viewed as a sequence of actions, and the current state is defined as the generated sequence to date and the prior input. However, the discriminator only measures the quality of whole sentences, and thus the rewards are extremely sparse and the rewards assigned to all actions in sequence are all the same. To tackle this problem, they propose MC search to evaluate approximate rewards at each time step, but this method suffers from high time complexity. Following this idea, (Li et al., 2017) proposes another approach to evaluate the expected reward at each time step. They break both the generated and real sequences into partial sequences, and the discriminator discriminates between the generated and real partial sequences. Inspired by this idea, we propose the self-critical adversarial REINFORCE algorithm as another way to evaluate the expected reward at each time step.
+
+In (Gulrajani et al., 2017), instead of feeding a discrete word sequence, the authors directly feed the generator output layer to the discriminator. This method works because they use the earth mover’s distance on GAN as proposed in (Arjovsky et al., 2017), which is able to evaluate the distance between a discrete and a continuous distribution. In order to satisfy the requirement of the earth mover’s distance, they use a gradient penalty trick to confine the complexity of discriminator function. Their method achieves an amazing result: it is the first work on GAN training that performs language generation without pre-training. In our work, we also conduct experiments on this method with discriminator settings almost the same as the original paper.
+
+# ABSTRACTIVE TEXT SUMMARIZATION
+
+Recent model architectures for abstractive text summarization basically use the sequence-tosequence (Sutskever et al., 2014) framework in combination with various novel mechanisms. One popular mechanism is attention (Bahdanau et al., 2015), which has been shown helpful for summarization (Nallapati et al., 2016; Rush et al., 2015). It is also possible to directly optimize evaluation metrics such as ROUGE (Lin, 2004) with reinforcement learning (Ranzato et al., 2016; Paulus et al., 2017; Bahdanau et al., 2016). The hybrid pointer-generator network (See et al., 2017) selects words from the original text with a pointer (Vinyals et al., 2015) or from the whole vocabulary with a trained weight. In order to eliminate repetition, a coverage vector (Tu et al., 2016) can be used to keep track of attended words and coverage loss (See et al., 2017) can be used to encourage model focus on diverse words. While most papers focus on supervised learning with novel mechanisms, we explore unsupervised training models.
+
+# 3 PROPOSED METHOD
+
+The overview of the proposed model is shown in Fig. 2. The model is composed of three components: generator $G$ , discriminator $D$ , and reconstructor $R$ . Both $G$ and $R$ are seq2seq hybrid pointer-generator networks (See et al., 2017) which can decide to copy words from encoder input text via pointing or generate from vocabulary.They both take a word sequence as input and output a sequence of word distributions. Discriminator $D$ , on the other hand, takes a sequence as input and outputs a scalar. The model is learned from a set of documents $x$ and human-written sentences $y ^ { r e a l }$ . Although in real implementation, $y ^ { r e a l }$ are the sentences in summaries, we note that the documents and summaries are unpaired.
+
+To train the model, a training document $\boldsymbol { x } = \{ x _ { 1 } , x _ { 2 } , . . . , x _ { t } , . . . , x _ { T } \}$ , where $x _ { t }$ represents a word, is fed to $G$ , which outputs a sequence of word distributions $G ( x ) = \left\{ y _ { 1 } , y _ { 2 } , . . . , y _ { n } , . . . , y _ { N } \right\}$ , where $y _ { n }$ is a distribution over all words in the lexicon. Then we sample a word $y _ { n } ^ { s }$ from each distribution $y _ { n }$ , and a word sequence $y ^ { s } = \{ y _ { 1 } ^ { s } , y _ { 2 } ^ { s } , . . . , y _ { N } ^ { s } \}$ is obtained according to $G ( x )$ . We feed the sampled word sequence $y ^ { s }$ to reconstructor $R$ , which outputs another sequence of word distributions $\hat { x }$ . The reconstructor $R$ reconstructs the original text $x$ from $y ^ { s }$ . That is, we seek an output of reconstructor $\hat { x }$ that is as close to the original text $x$ as possible; hence the loss for training the reconstructor $R$ , $R _ { l o s s }$ , is defined as
+
+$$
+R _ { l o s s } = \sum _ { k = 1 } ^ { K } l _ { s } ( x , \hat { x } ) ,
+$$
+
+![](images/3db872585682f6ac8bfcf5ea68af851d1652db81a2e613ec555ff7ade2f5d1e2.jpg)  
+Figure 2: Architecture of proposed model. The generator network and reconstructor network are a seq2seq hybrid pointer-generator network, but for simplicity, we omit the pointer and the attention parts.
+
+where the reconstruction loss $l _ { s } ( x , \hat { x } )$ is the cross-entropy loss computed between the reconstructor output sequence $\hat { x }$ and the source text $x$ , or the negative conditional log-likelihood of source text $x$ given word sequence $y ^ { s }$ sampled from $G ( x )$ . The reconstructor output sequence $\hat { x }$ is teacher-forced by source text $x$ . The subscript $s$ in $l _ { s } ( x , \hat { x } )$ indicates that $\hat { x }$ is reconstructed from $y ^ { s }$ . $K$ is the number of training examples (documents), and (1) is the summation of the cross-entropy loss over all the training documents $x$ .
+
+In the proposed model, the generator $G$ and reconstructor $R$ form an auto-encoder. However, the reconstructor $R$ does not directly take the generator output distribution $G ( x )$ as input 3. Instead, the reconstructor takes a sampled discrete sequence $y ^ { s }$ as input. Due to the non-differentiable property of discrete sequences, we apply the REINFORCE algorithm, which is described in Section 4.
+
+In addition to reconstruction, we need the discriminator $D$ to discriminate between the real sequence yreal and the generated sequence $y ^ { s }$ to regularize the generated sequence satisfying the summary distribution. $D$ learns to give $y ^ { r e a l }$ higher scores while giving $y ^ { s }$ lower scores. The loss for training the discriminator $D$ is denoted as $D _ { l o s s }$ ; this is further described in Section 5.
+
+$G$ learns to minimize the reconstruction error $R _ { l o s s }$ , while maximizing the loss of the discriminator $D$ by generating a summary sequence $y ^ { s }$ that cannot be differentiated by $D$ from the real thing. The loss when training the generator $G$ , $G _ { l o s s }$ , is
+
+$$
+G _ { l o s s } = \alpha R _ { l o s s } - D _ { l o s s } ^ { \prime }
+$$
+
+where $D _ { l o s s } ^ { \prime }$ is highly related to $D _ { l o s s }$ – but not necessary the same – and $\alpha$ is a hyper-parameter.   
+After obtaining the optimal generator by minimizing (2), we use it to generate summaries.
+
+Generator $G$ and discriminator $D$ together form a GAN. We use two different adversarial training methods to train $D$ and $G$ ; as shown in Fig. 2, these two methods have their own discriminators 1 and 2. Discriminator 1 takes the generator output layer $G ( x )$ as input, whereas discriminator 2 takes the sampled discrete word sequence $y ^ { s }$ as input. The two methods are described respectively in Sections 5.1 and 5.2.
+
+# 4 MINIMIZING RECONSTRUCTION ERROR
+
+Because discrete sequences are non-differentiable, we use the REINFORCE algorithm. The generator is seen as an agent whose reward given the source text $x$ is $- l _ { s } ( x , \hat { x } )$ . Maximizing the reward is equivalent to minimizing the reconstruction loss $R _ { l o s s }$ in (1). However, the reconstruction loss varies widely from sample to sample, and thus the rewards to the generator are not stable either. Hence we add a baseline to reduce their difference. We apply self-critical sequence training (Rennie et al., 2017); the modified reward $r ^ { R } ( x , { \hat { x } } )$ from reconstructor $R$ with the baseline for the generator is
+
+$$
+r ^ { R } ( x , \hat { x } ) = - l _ { s } ( x , \hat { x } ) - ( - l _ { a } ( x , \hat { x } ) - b )
+$$
+
+where $- l _ { a } ( x , { \hat { x } } ) - b$ is the baseline. $l _ { a } ( x , \hat { x } )$ is also the same cross-entropy reconstruction loss as $l _ { s } ( x , \hat { x } )$ , except that $\hat { x }$ is obtained from $y ^ { a }$ instead of $y ^ { s }$ . $y ^ { a }$ is a word sequence $\left\{ y _ { 1 } ^ { a } , y _ { 2 } ^ { a } , . . . , y _ { n } ^ { a } , . . . , y _ { N } ^ { a } \right\}$ , where $y _ { n } ^ { a }$ is selected using the argmax function from the output distribution of generator $y _ { n }$ . As in the early training stage, the sequence $y ^ { s }$ barely yields higher reward than sequence $y ^ { a }$ , to encourage exploration we introduce the second baseline score $b$ , which gradually decreases to zero. Then, the generator is updated using the REINFORCE algorithm with reward $r ^ { R } ( x , { \hat { x } } )$ to minimize $R _ { l o s s }$ .
+
+# 5 GAN TRAINING
+
+With adversarial training, the generator learns to produce sentences as similar to the human-written sentences as possible. Here, we conduct experiments on two kinds of methods of language generation with GAN. In Section 5.1 we directly feed the generator output probability distributions to the discriminator and use a Wasserstein GAN (WGAN) with a gradient penalty. In Section 5.2, we explore adversarial REINFORCE, which feeds sampled discrete word sequences to the discriminator and evaluates the quality of the sequence from the discriminator for use as a reward signal to the generator.
+
+# 5.1 DISCRIMINATOR 1: WASSERSTEIN GAN
+
+In the lower left of Fig. 2, the discriminator model is shown as discriminator1. The discriminator loss $D _ { l o s s }$ is
+
+$$
+D _ { l o s s } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } D ( y ^ { s ( k ) } ) - \frac { 1 } { K } \sum _ { k = 1 } ^ { K } D ( y ^ { r e a l ( k ) } ) + \beta ( \Delta _ { y ^ { i ( k ) } } D ( y ^ { i ( k ) } ) - 1 ) ^ { 2 } ,
+$$
+
+where $K$ denotes the number of training examples in a batch, and $k$ denotes the $k$ -th example. The last term in (4) is the gradient penalty (Gulrajani et al., 2017). We interpolate the generator output layer $G ( x )$ and the real sample $y ^ { r e a \bar { l } }$ , and apply the gradient penalty to the interpolated sequence $y ^ { i }$ . $\beta$ determines the gradient penalty scale. In Equation (2), for WGAN, $D _ { l o s s } ^ { \prime }$ is the score of the generated example:
+
+$$
+D _ { l o s s } ^ { \prime } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } D ( G ( x ^ { ( k ) } ) ) .
+$$
+
+# 5.2 SELF-CRITIC ADVERSARIAL REINFORCE
+
+In this section, we describe in detail the proposed adversarial REINFORCE method. The core idea is we use the LSTM discriminator to evaluate the current quality of the generated sequence $\{ y _ { 1 } ^ { s } , y _ { 2 } ^ { s } , . . . , y _ { i } ^ { s } \}$ at each time step $i$ . Hence, the generator knows that compared to the last time step, as the generated sentence either improves or worsens, it can easily find the problematic generation step in a long sequence, and thus fix the problem easily.
+
+# 5.2.1 DISCRIMINATOR 2
+
+As shown in Fig. 2, the discriminator2 is a one-way LSTM network which takes a discrete word sequence as input. At time step $i$ , given input word $y _ { i } ^ { s }$ it predicts the current score $s _ { i }$ based on the sequence $\left\{ y _ { 1 } , y _ { 2 } , . . . , y _ { i } \right\}$ . The score is viewed as the quality of the current sequence.
+
+In order to compute the discriminator loss $D _ { l o s s }$ , we sum the scores $\left\{ s _ { 1 } , s _ { 2 } , . . . , s _ { N } \right\}$ of the whole sequence $y ^ { s }$ to yield
+
+$$
+\begin{array} { c } { { \displaystyle { D ( y ^ { s } ) = \frac { 1 } { N } \sum _ { n = 1 } ^ { N } s _ { n } . } } } \\ { { \displaystyle { D _ { l o s s } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } D ( y ^ { s ( k ) } ) - \frac { 1 } { K } \sum _ { k = 1 } ^ { K } D ( y ^ { r e a l ( k ) } ) , } } } \end{array}
+$$
+
+where $K$ and $N$ denote the number of training examples and the generated sequence length respectively. With the loss mentioned above, the discriminator attempts to quickly determine whether the current sequence is real or fake. The earlier the timestep discriminator determines whether the current sequence is real or fake, the lower its loss. An example is shown in Fig. 3.
+
+![](images/166009662b8a6fd2ea06cc9753d478913754598d73630c9ea3e6340275b8ceb6.jpg)  
+Figure 3: When the second arrested appears, the discriminator determines that this example came from the generator. Hence, after this time-step, it outputs low scores.
+
+# 5.2.2 SELF-CRITICAL GENERATOR
+
+Since we feed a discrete sequence $y ^ { s }$ to the discriminator, the gradient from the discriminator cannot directly back-propagate to the generator. Here, we use the policy gradient method. At timestep $i$ , we use the $i - 1$ timestep score $s _ { i - 1 }$ from the discriminator as its self-critical baseline. The reward r Di evaluates whether the quality of sequence in timestep $i$ is better or worse than that in timestep $i - 1$ . The generator reward $r _ { i } ^ { D }$ from $D$ is
+
+$$
+r _ { i } ^ { D } = \left\{ \begin{array} { l l } { { s _ { i } \qquad } } & { { \mathrm { i f ~ i = 1 ~ } } } \\ { { s _ { i } - s _ { i - 1 } \qquad } } & { { \mathrm { o t h e r w i s e . } } } \end{array} \right.
+$$
+
+However, some sentences may be judged as bad sentences at the previous timestep, but at later timesteps judged as good sentences, and vice versa. Hence we use the discounted expected reward $d$ with discount factor $\gamma$ to calculate the discounted reward $d _ { i }$ at time step $i$ as
+
+$$
+d _ { i } = \sum _ { j = i } ^ { N } \gamma ^ { j - i } r _ { j } ^ { D } .
+$$
+
+The adversarial REINFORCE score related to discriminator $D _ { l o s s } ^ { \prime }$ in (2) is
+
+$$
+D _ { l o s s } ^ { \prime } = E _ { y _ { i } ^ { s } \sim p _ { G } ( y _ { i } ^ { s } | y _ { 1 } ^ { s } , . . . , y _ { i - 1 } ^ { s } , x ) } ^ { } [ d _ { i } ] .
+$$
+
+We use the likelihood ratio trick to approximate the gradient.
+
+# 6 IMPLEMENTATION
+
+Network Architecture. The model architecture of generator and reconstructor is almost same except the length of input and output sequence. We adapt model architecture for our generator and reconstructor from See et al. (2017) who used hybrid-pointer network with coverage vector for text summarization. The hybrid-pointer networks of generator and reconstructor are all composed of two one-layer LSTMs as its encoder and decoder, respectively, with a hidden layer size of 600. Since we use two kinds of methods on adversarial training, there are two discriminators with different model architecture. In the Section 5.1, the discriminator is composed of four residual blocks with 512 hidden dimensions. While in Section 5.2, we use only one hidden-layer one-way LSTM with a hidden size of 512 as our discriminator.
+
+Details of Training. We set the weight $\alpha$ in (2) controlling $R _ { l o s s }$ to 10 if not specified. We find that the if the value of $\alpha$ is too large, generator will start to generate output unlike human-written sentences. On the other hand, if the value of $\alpha$ is too small, the sentences generated by generator will sometimes become unrelated to input text of generator. For all the experiments, the baseline $b$ in (3) gradually decreases from 0.25 to zero within 10000 updates on generator.
+
+In Section 5.1, we set the weight $\beta$ of the gradient penalty to 10, and used RMSPropOptimizer with a learning rate of 0.00001 and 0.001 on the generator and discriminator, respectively. In Section 5.2, we clip the value of the weights of discriminator to $\pm 0 . 1 5$ , and used RMSPropOptimizer with a learning rate of 0.0001 and 0.001 on the generator and discriminator, respectively. It’s also feasible to apply gradient penalty trick in this method to satisfy requirement of Wasserstein distance. However, in this method, the performance of gradient penalty trick and weights clipping trick is close.
+
+# 7 EXPERIMENT
+
+We evaluate our model on the Chinese Gigaword and English Gigaword datasets. Before jointly training the whole model, we pre-trained the three major components – generator, discriminator, and reconstructor – separately. First, we pre-trained the generator in an unsupervised manner so that the generator would be able to somewhat grasp the semantic meaning of the source text. The details of the pre-training are in Appendix A. We pre-trained the discriminator and reconstructor respectively with the pre-trained generator’s output to ensure that these two critic networks provide good feedback to the generator. During testing, when using the generator to generate summaries, we simply selected the words in a greedy fashion without beam-search, and we eliminated repetition.
+
+# 7.1 CHINESE GIGAWORD
+
+<table><tr><td rowspan=1 colspan=2>Methods</td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=1 colspan=2>(A) Trainingwith paired data (supervised)</td><td rowspan=1 colspan=1>48.664</td><td rowspan=1 colspan=1>33.907</td><td rowspan=1 colspan=1>45.685</td></tr><tr><td rowspan=1 colspan=2>(B) Trivial baselines: lead - 15</td><td rowspan=1 colspan=1>30.077</td><td rowspan=1 colspan=1>18.237</td><td rowspan=1 colspan=1>27.736</td></tr><tr><td rowspan=3 colspan=1>(C) Unsupervised</td><td rowspan=1 colspan=1>(C-1) Pretrained generator</td><td rowspan=1 colspan=1>28.122</td><td rowspan=1 colspan=1>16.656</td><td rowspan=1 colspan=1>26.227</td></tr><tr><td rowspan=1 colspan=1>(C-2) WGAN</td><td rowspan=1 colspan=1>37.803</td><td rowspan=1 colspan=1>24.460</td><td rowspan=1 colspan=1>35.116</td></tr><tr><td rowspan=1 colspan=1>(C-3) Adversarial REINFORCE</td><td rowspan=1 colspan=1>40.053</td><td rowspan=1 colspan=1>26.126</td><td rowspan=1 colspan=1>37.118</td></tr></table>
+
+Table 1: Results on Chinese Gigaword. In row (B), we select the article’s first fifteen words as its summary. Part (C) are the results obtained without paired data.
+
+The Chinese Gigaword corpus is composed of $2 . 2 \mathbf { M }$ paired data of headlines and news. We preprocessed the raw data as following. First, we selected the 4000 most frequent Chinese characters as our vocabulary. We filtered out headline-news pairs with excessively long or short news segments, or that contained too many out-of-vocabulary Chinese characters, yielding 1.1M headline-news pairs from which we randomly selected 5K headline-news pairs as our testing set, 5K headline-news pairs as our validation set, and the remaining pairs as our training set. During training and testing, the generator took only the first 80 Chinese characters of the source text as input.
+
+The results are shown in Table 1. Row (A) lists the results using 1.1 million document-summary pairs to directly train the generator without the reconstructor and discriminator: this is the upper bound of the proposed approach. In row (B), we simply took the first fifteen words in a document as its summary. The number of words was chosen to optimize the evaluation metrics. Part (C) are the results obtained in the unsupervised scenario without paired data. We show the results of the pre-trained generator in row (C-1); rows (C-2) and (C-3) are the results for the two GAN training methods respectively. We find that despite the performance gap between the unsupervised and supervised methods (rows (C-2), (C-3) v.s. (A)), the proposed method yielded much better performance than the trivial baselines (rows (C-2), (C-3) v.s. (B)).
+
+# 7.2 ENGLISH GIGAWORD
+
+On English Gigaword, we set our vocabulary size to 15k, and used the dataset preprocessed by (Rush et al., 2015) for training and testing. We used $3 . 8 \mathbf { M }$ unpaired training data for our training, and we used the whole $2 0 0 \mathrm { k }$ filtered data in validation set for testing.
+
+The results on English Gigaword are shown in Table 2. In row (B-1), we simply took the first eight words in a document as its summary. Row (B-2) is another trivial baseline. With unpaired documents and summaries, we matched documents to the most relevant summaries with unsupervised method. Each document and each summary were represented as tf-idf (term frequency $\&$ inverse document frequency) vectors. The summary whose vector maximized cosine similarity of a document vector was retrieved as summary of the document. With paired data from unsupervised matching, given documents as generator input, the generator was trained to predict retrieved summaries.
+
+Table 2: Results on English Gigaword: In row (B-1), we select the article’s first eight words as its summary. In row (B-2), we match the documents to their most relevant summaries with unsupervised method. Part (C) are the results obtained without paired data. In part (D), we pre-trained the generator on CNN/Diary. In part (E), we not only pre-trained on CNN/Diary but also used the summaries from CNN/Diary as real data for the discriminator.   
+
+<table><tr><td rowspan=1 colspan=4>Methods</td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=1 colspan=4>(A) Training with paired data (supervised)</td><td rowspan=1 colspan=1>37.469</td><td rowspan=1 colspan=1>16.272</td><td rowspan=1 colspan=1>35.175</td></tr><tr><td rowspan=2 colspan=3>(B) Trivial baselines</td><td rowspan=1 colspan=1>(B-1) Lead-8</td><td rowspan=1 colspan=1>27.663</td><td rowspan=1 colspan=1>10.246</td><td rowspan=1 colspan=1>25.852</td></tr><tr><td rowspan=1 colspan=1>(B-2) Unsupervised matching</td><td rowspan=1 colspan=1>29.900</td><td rowspan=1 colspan=1>10.442</td><td rowspan=1 colspan=1>27.379</td></tr><tr><td rowspan=3 colspan=3>(C) Unsupervised</td><td rowspan=1 colspan=1>(C-1) Pre-trained generator</td><td rowspan=1 colspan=1>21.269</td><td rowspan=1 colspan=1>5.608</td><td rowspan=1 colspan=1>18.896</td></tr><tr><td rowspan=1 colspan=1>(C-2) WGAN</td><td rowspan=1 colspan=1>33.043</td><td rowspan=1 colspan=1>12.222</td><td rowspan=1 colspan=1>30.121</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>(C-3) Adversarial REINFORCE</td><td rowspan=1 colspan=1>32.826</td><td rowspan=1 colspan=1>9.332</td><td rowspan=1 colspan=1>28.727</td></tr><tr><td rowspan=3 colspan=3>(D) Transfer learning(Pre-train)</td><td rowspan=1 colspan=1>ng</td><td rowspan=1 colspan=3>(D-1) Pre-trained generator</td></tr><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>(D-2)WGAN</td><td rowspan=1 colspan=1>32.405</td><td rowspan=1 colspan=1>12.313</td><td rowspan=1 colspan=1>29.689</td></tr><tr><td rowspan=1 colspan=1>(D-3) Adversarial REINFORCE</td><td rowspan=1 colspan=1>31.487</td><td rowspan=1 colspan=1>10.495</td><td rowspan=1 colspan=1>28.248</td></tr><tr><td rowspan=2 colspan=3>(E) Transfer learning(Pre-train+Discriminator)</td><td rowspan=1 colspan=1>(E-1) WGAN</td><td rowspan=1 colspan=1>29.912</td><td rowspan=1 colspan=1>10.695</td><td rowspan=1 colspan=1>27.324</td></tr><tr><td rowspan=1 colspan=1>(E-2) Adversarial REINFORCE</td><td rowspan=1 colspan=1>27.755</td><td rowspan=1 colspan=1>9.280</td><td rowspan=1 colspan=1>24.860</td></tr></table>
+
+The results for the pre-trained generator is shown in row (C-1). Compared with the trivial baselines (part (B)), the proposed approach (rows (C-2) and (C-3)) showed good improvement in terms of ROUGE-1. As shown in Fig. 4, the unsupervised method selects key words in the source text and generates the text summary. However, as shown in Table 2, although both unsupervised methods yield ROUGE-1 scores close to that of supervised training, they achieve lower scores on ROUGE-2, especially when training GAN with reinforcement learning. This is because they are extracting the key words from the source text, despite sometimes failing to arrange these words in the correct order. As shown in part (C-3) of Fig. 5, the words in the sentence generated in an unsupervised manner are not arranged correctly: the Italian prime minister, Berlusconi, should not be visiting himself.
+
+# 7.3 TRANSFER LEARNING
+
+In this subsection, we study transfer learning. We used the CNN/Daily Mail dataset (Hermann et al., 2015; Nallapati et al., 2016) preprocessed by the script provided by (See et al., 2017) as our source domain $S$ , and English Gigaword as our target domain $T$ . The data distributions among the two datasets are quite different. In English Gigaword, the articles consist of 32 words on average and the summaries consist of one sentence with 8 words on average, whereas the CNN/Daily Mail dataset is composed of 790-word articles and multi-sentence summaries. We took only the first 30 to 45 words in the original source domain articles as our new source articles $S _ { a }$ . There are 50K source articles in $S _ { a }$ . The 50K source summaries of the source articles are $S _ { t }$ . In contrast to English Gigaword, the summaries in CNN/Daily Mail dataset are composed of more than one sentence. We split the summary of each article into several sentences, and obtained 240K sentences $S _ { r }$ in this way.
+
+Transfer learning was applied in two directions. In the first direction, we pre-trained our generator on the data from the source dataset: we pre-trained the generator with $S _ { a }$ as input, and the generator predicted $S _ { t }$ . We pre-trained the generator in this manner in all of the transfer learning experiments. Then, after pre-training, the generator was further learned jointly with the reconstructor and discriminator on the data from the target domain. The results are shown in part (D) of Table 2. We found that pre-training on the source data set did not degrade performance, and even improved performance in some cases (parts (D) v.s. (C)).
+
+In the second direction, we used the summary from the source domain as our real data. The experiments conducted up to this point required unpaired summary and text data in the target domain. In this experiment, the task was more challenging in that we used summaries $S _ { r }$ from the source domain $S$ as the real data for the discriminator; the generator took the target domain text as input. However, the summaries in each summarization task dataset had a different distribution in terms of the writing style, or in terms of the preferred summary words. To prevent overfitting to $S _ { r }$ , we set the weight $\alpha$ to 50 which was larger than other experiments. With small weight of $\alpha$ , as training progressed, the generator summary diverged more and more from the article, and the ROUGE scores became lower and lower.
+
+The results without using summaries in the target domain are shown in part (E). We find that using sentences $S _ { r }$ from another dataset yields lower ROUGE scores on the target testing set (parts (E) v.s. (D)) due to the mismatch between the summaries of the source and target domains. However, the discriminator still roughly regularizes the language model of generated word sequence. After training, the model still greatly enhanced the ROUGE score of the pre-trained model (rows (E-1), (E-2) v.s. (D-1)).Although the results in part (E) are comparable with the trivial baselines in part (B), in inspecting the real examples, we found that the results in part (E) were in fact better; this is not reflected in the ROUGE scores. In Fig. 4, the results in part (E) are better than the leading 8 words and the pre-trained generator results. To support this idea, we provide more examples in the Appendix C.
+
+Figure 4: Real example from our model in English Gigaword. The proposed method generates summaries that capture the core idea of the article.   
+Figure 5: In part (C-3), some words in the summary sentences are arranged in incorrect order.   
+
+<table><tr><td rowspan=1 colspan=2> Source Text:three stores and markets in beijing &#x27;s fengtai district have been forced to shut down andyesterday each was fined ###,### yuan -lrb- ##,### us dollars -rrb- for violating laws andregulations on fire prevention and control .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth: stores markets punished for lack of fire controls</td><td rowspan=1 colspan=1>(A)Supervised Result:beijing &#x27;s district stores shut down</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:and yesterday prevention have been forced to shut down</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:three stores and markets fined ###,### yuandollars</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:three stores markets in beijing &#x27;s district haveforced to shut down yesterday</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:three stores in beijing forced to shut down forviolating regulations</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :three stores and markets was forced to shutdown</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE:three stores and markets was forced to shutdown and yesterday each was fined</td></tr></table>
+
+![](images/f14fbb1b2c9bb0950c782c5d3666eec5fba5686d3acbf078de5a370af52fe5c0.jpg)
+
+# 7.4 SEMI-SUPERVISED LEARNING
+
+In semi-supervised training, generator was pre-trained with few available labeled data, and during unsupervised training, we conducted teacher-forcing with labeled data on generator every several unsupervised updates. In teacher forcing, given source text as input, the generator was teacherforced to predict the human-written summary of source text. Teacher-forcing can be regarded as regularization of unsupervised training that prevents generator from producing unreasonable summaries of source text. We found that if we teacher-forced generator too frequently, generator would overfit on training data since we only use very few labeled data on semi-supervised training.
+
+The performance of semi-supervised model in English Gigaword regarding available labeled data is shown in Fig. 6. The horizontal axis is the number of labeled documents used in the experiments, while the vertical axis for Fig. 6 (a) and (b) are ROUGE-1 and ROUGE-2 respectively. The green curve is the results of supervised learning, and the red and blue curves are semi-supervised learning with different approaches. With the same amounts of labeled data, the performances of semi-supervised training are always better than supervised training. With only 100K labeled data, the ROUGE score of semi-supervised training using adversarial REINFORCE even slightly outperformed supervised training with whole labeled data. This shows that with the proposed approach, we need only $2 . 6 \%$ of labeled data to achieve the same performance as before (100K v.s. 3.8M). The complete results for semi-supervised learning in both datasets are shown in Appendix B.
+
+![](images/192cbcda93e6638c835881efa30b90cc07dd177b7c02b076ddb36ad66cd8affa.jpg)  
+Figure 6: Semi-supervised results in English Gigaword. With the same amount of labeled data, the performances of semi-supervised training are always better than supervised training.
+
+# 7.5 GAN TRAINING
+
+Table 3: Adversarial REINFORCE with/without self-critic with unsupervised training.   
+
+<table><tr><td rowspan=1 colspan=1>Corpus</td><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=2 colspan=1>Chinese</td><td rowspan=1 colspan=1>with self-critic</td><td rowspan=1 colspan=1>40.053</td><td rowspan=1 colspan=1>26.126</td><td rowspan=1 colspan=1>37.118</td></tr><tr><td rowspan=1 colspan=1>without self-critic</td><td rowspan=1 colspan=1>37.549</td><td rowspan=1 colspan=1>24.181</td><td rowspan=1 colspan=1>35.160</td></tr><tr><td rowspan=2 colspan=1>English</td><td rowspan=1 colspan=1>with self-critic</td><td rowspan=1 colspan=1>32.826</td><td rowspan=1 colspan=1>9.332</td><td rowspan=1 colspan=1>28.727</td></tr><tr><td rowspan=1 colspan=1>without self-critic</td><td rowspan=1 colspan=1>31.104</td><td rowspan=1 colspan=1>9.249</td><td rowspan=1 colspan=1>28.592</td></tr></table>
+
+The two GAN training methods are not comparable as their settings are quite different, but we can still discuss the advantages and disadvantages of these two methods. When training with feeding output layer to discriminator, convergence is faster. This method sharpens the distribution at an early stage in training because it directly evaluates the distance between the generator’s continuous distribution and the real data’s discrete distribution data. However, this cause generator to converge to a not very good place.
+
+In fact, adversarial REINFORCE is sensitive to initialization parameters. Adversarial REINFORCE requires better initialization for exploration; otherwise, with so many actions whose number is equal to vocabulary size to choose, it is extremely difficult to train generator from scratch. In semisupervised training, since we pre-trained generator with labeled data, the generator was better initialized, therefore adversarial REINFORCE performed better. To support this idea, in Fig. 6, we compare the performance of two methods regarding labeled data. The result implies that with more labeled data, our proposed adversarial REINFORCE method performs better. In order to evaluate the performance of proposed self-critic baseline trick mentioned in Section 5.2.2, we compared the performance of our model with and without this baseline trick in Table 3. For the experiments without the self-critic baseline trick, we replaced $s _ { i } - s _ { i - 1 }$ in Section 5.2.2 with $s _ { i }$ . We found that the performance degraded without self-critic.
+
+# 8 CONCLUSION AND FUTURE WORK
+
+Using GAN, we propose a model that encodes text as a human-readable summary, learned without document-summary pairs. Promising results are obtained on both Chinese and English corpora. In future work, we hope to explore more techniques for natural language generation using GAN. Moreover, we hope to use extra discriminators to control the style and sentiment of the generated summaries.
+
+# REFERENCES
+
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+
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+
+Dzmitry Bahdanau, Philemon Brakel, Kelvin Xu, Anirudh Goyal, Ryan Lowe, Joelle Pineau, Aaron Courville, and Yoshua Bengio. An actor-critic algorithm for sequence prediction. arXiv preprint arXiv:1607.07086, 2016.
+
+Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks. arXiv preprint arXiv:1406.2661, 2014.
+
+Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of wasserstein gans. arXiv preprint arXiv:1704.00028, 2017.
+
+Di He, Yingce Xia, Tao Qin, Liwei Wang, Nenghai Yu, Tie-Yan Liu, and Wei-Ying Ma. Dual learning for machine translation. NIPS, 2016.
+
+Karl Moritz Hermann, Tom Koisk, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom. Teaching machines to read and comprehend. NIPS, 2015.
+
+Ryan Kiros, Yukun Zhu, Ruslan Salakhutdinov, Richard S. Zemel, Antonio Torralba, Raquel Urtasun, and Sanja Fidler. Skip-thought vectors. NIPS, 2015.
+
+Jiwei Li, Minh-Thang Luong, and Dan Jurafsky. A hierarchical neural autoencoder for paragraphs and documents. ACL, 2015.
+
+Jiwei Li, Will Monroe, Tianlin Shi, Sbastien Jean, Alan Ritter, and Dan Jurafsky. Adversarial learning for neural dialogue generation. arXiv preprint arXiv:1701.06547, 2017.
+
+Chin-Yew Lin. Rouge: A package for automatic evaluation of summaries. In Text summarization branches out: ACL workshop, 2004.
+
+Alireza Makhzani, Jonathon Shlens, Navdeep Jaitly, Ian Goodfellow, and Brendan Frey. Adversarial autoencoders. arXiv preprint arXiv:1511.05644, 2015.
+
+Yishu Miao and Phil Blunsom. Language as a latent variable: Discrete generative models for sentence compression. EMNLP, 2016.
+
+Ramesh Nallapati, Bowen Zhou, Cicero Nogueira dos santos, Caglar Gulcehre, and Bing Xiang. Abstractive text summarization using sequence-to-sequence rnns and beyond. EMNLP, 2016.
+
+Romain Paulus, Caiming Xiong, and Richard Socher. A deep reinforced model for abstractive summarization. arXiv preprint arXiv:1705.04304, 2017.
+
+Marc’Aurelio Ranzato, Sumit Chopra, Michael Auli, and Wojciech Zaremba. Sequence level training with recurrent neural networks. ICLR, 2016.
+
+Steven J. Rennie, Etienne Marcheret, Youssef Mroueh, Jarret Ross, and Vaibhava Goel. Self-critical sequence training for image captioning. CVPR, 2017.
+
+Alexander M. Rush, Sumit Chopra, and Jason Weston. A neural attention model for abstractive sentence summarization. EMNLP, 2015.
+
+Abigail See, Peter J. Liu, and Christopher D. Manning. Get to the point: Summarization with pointer-generator networks. ACL, 2017.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V. Le. Sequence to sequence learning with neural networks. NIPS, 2014.
+
+Zhaopeng Tu, Zhengdong Lu, Yang Liu, Xiaohua Liu, and Hang Li. Modeling coverage for neural machine translation. ACL, 2016.
+
+Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. NIPS, 2015.
+
+Lantao Yu, Weinan Zhang, Jun Wang, and Yong Yu. Seqgan: Sequence generative adversarial nets with policy gradient. AAAI, 2017.
+
+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint arXiv:1703.10593, 2017.
+
+# A MODEL PRE-TRAINING
+
+As we found that the different pre-training methods for the generator influenced final performance dramatically in all of the experiments, we felt it was important to find a proper unsupervised pretraining method to help the machine grasp semantic meaning. We used the different pre-training strategies described below.
+
+• Chinese Gigaword: Given the previous $i - 1$ sentences $s e n t _ { 0 } , s e n t _ { 1 } , . . . , s e n t _ { i - 1 }$ from the source text, the generator predicted the next sentence $s e n t _ { i }$ in the source text as its pretraining target. If more than $50 \%$ of the words in sentence $s e n t _ { i }$ did not appear in the given text, we filtered out this pre-training sample pair. This pre-training method allowed the generator to capture the important semantic meanings of the source text. English Gigaword: As the length of the source texts in English Gigaword dataset is comparatively short, it is difficult to split the last sentence from the source text; hence the previous pre-training method on Chinese Gigaword is not appropriate for this dataset. To properly initialize the set, we randomly selected 6 to 11 consecutive words in the source text, after which we randomly swapped $70 \%$ of the words in the source text. Given text with incorrect word arrangements, the generator predicted the selected words in the correct arrangement. We pre-trained in this way because we expect the generator to initialize with a rough language model. In Chinese Gigaword we also conducted experiments on pre-training in this manner, but the results were not as good as those shown in the part (C) of Table 1. We also used the retrieved paired data in row (B-1) in Table 2 to pre-train generator. However, pre-training generator with this method doesn’t yield results better than those in Table 2.
+
+# B SEMI-SUPERVISED LEARNING
+
+Semi-supervised learning experiments were conducted with 10K, 50K, 100K labeled data in both datasets. We conducted teacher-forcing on generator every 30, 12, 7 unsupervised updates with 10K, 50K, 100K labeled data respectively. The complete results for semi-supervised learning are shown in Tables 4 and 5.
+
+Table 4: Semi-supervised learning in Chinese Gigaword with different amounts of labeled data (10K, 50K, 100K).   
+
+<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=2 colspan=1>Unsupervised</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>37.803</td><td rowspan=1 colspan=1>24.460</td><td rowspan=1 colspan=1>35.116</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>40.053</td><td rowspan=1 colspan=1>26.126</td><td rowspan=1 colspan=1>37.118</td></tr><tr><td rowspan=2 colspan=1>Semi-supervised(10K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>42.359</td><td rowspan=1 colspan=1>27.192</td><td rowspan=1 colspan=1>38.467</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>43.109</td><td rowspan=1 colspan=1>28.626</td><td rowspan=1 colspan=1>40.202</td></tr><tr><td rowspan=2 colspan=1>Semi-supervised(50K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>43.989</td><td rowspan=1 colspan=1>29.012</td><td rowspan=1 colspan=1>40.764</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>44.706</td><td rowspan=1 colspan=1>29.872</td><td rowspan=1 colspan=1>41.737</td></tr><tr><td rowspan=2 colspan=1>Semi-supervised(100K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>45.642</td><td rowspan=1 colspan=1>31.475</td><td rowspan=1 colspan=1>42.711</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>46.216</td><td rowspan=1 colspan=1>31.980</td><td rowspan=1 colspan=1>43.522</td></tr><tr><td rowspan=1 colspan=2>Supervised</td><td rowspan=1 colspan=1>48.664</td><td rowspan=1 colspan=1>33.907</td><td rowspan=1 colspan=1>45.685</td></tr></table>
+
+Table 5: Semi-supervised learning in English Gigaword with different amounts of labeled data (10K, 50K, 100K).   
+
+<table><tr><td rowspan=1 colspan=4></td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=2 colspan=3>Unsupervised</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>33.043</td><td rowspan=1 colspan=1>12.222</td><td rowspan=1 colspan=1>30.121</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>32.826</td><td rowspan=1 colspan=1>9.332</td><td rowspan=1 colspan=1>28.727</td></tr><tr><td rowspan=2 colspan=3>Semi-supervised(10K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>34.127</td><td rowspan=1 colspan=1>13.087</td><td rowspan=1 colspan=1>31.451</td></tr><tr><td rowspan=1 colspan=1>led）</td><td rowspan=1 colspan=1>AdversarialREINFORCE</td><td rowspan=1 colspan=1>34.239</td><td rowspan=1 colspan=1>12.570</td><td rowspan=1 colspan=1>31.834</td></tr><tr><td rowspan=1 colspan=2>Semi-supervised</td><td rowspan=1 colspan=1>Semi-supervised</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>34.937</td><td rowspan=1 colspan=1>13.838</td><td rowspan=1 colspan=1>32.372</td></tr><tr><td rowspan=1 colspan=3>(50K labeled)</td><td rowspan=1 colspan=1>AdversarialREINFORCE</td><td rowspan=1 colspan=1>35.642</td><td rowspan=1 colspan=1>14.057</td><td rowspan=1 colspan=1>32.983</td></tr><tr><td rowspan=2 colspan=3>Semi-supervised(100K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>36.615</td><td rowspan=1 colspan=1>15.363</td><td rowspan=1 colspan=1>33.682</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>38.213</td><td rowspan=1 colspan=1>16.279</td><td rowspan=1 colspan=1>35.137</td></tr><tr><td rowspan=1 colspan=4>Supervised</td><td rowspan=1 colspan=1>37.469</td><td rowspan=1 colspan=1>16.272</td><td rowspan=1 colspan=1>35.175</td></tr></table>
+
+# C EXAMPLES
+
+From Fig. 7 to 12, we show more examples.
+
+Figure 7: Real example from our model in English Gigaword. In part (E-2), due to transfer learning, the summary sentence begins with word he, which never appears in the English Gigaword summary sentences.   
+
+<table><tr><td rowspan=1 colspan=2>Source Text: former zambian president kenneth kaunda appeared in court monday on charges of holding an illegal ally ， declaring that he would continue to fight the “ oppressive regime &quot; of presidentfredrick chiluba .</td></tr><tr><td rowspan=1 colspan=1> Ground Truth:former zambian president in court for illegal assembly</td><td rowspan=1 colspan=1>(A)Supervised Result:kaunda to continue to fight chiluba</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator: president kenneth chiluba appeared in courtmonday on charges of</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:kaunda declaring that he would continue tofight the regime says</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN: zambian kenneth kaunda in court charges of holding illegal rally</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:former zambian president to continue illegalrally fight</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :former zambian president kenneth kaundaappeared in court of illegal rally</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE:he appeared in court he would continue to fight the regime</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=2>Source Text:hong kong tourist association -Irb- hkta -rrb- said wednesday it regretted having placed an advertisement thanking sponsors of last week &#x27;s lunar new year parade which killed one man andleft ## others injured .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth:hong kong tourist body expresses regret overadvertisement</td><td rowspan=1 colspan=1>(A)Supervised Result:hong kong tourist regrets having placedadvertisement sponsors</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:hong kong said wednesday it regretted anadvertisement</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator: one man was killed in the head of thesponsors of the said</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN: hong kong tourist association regretted havingplaced advertisement sponsors</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:hong kong tourist killed advertisement sponsors of last year &#x27;s lunar parade</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN : hong kong tourist association regretted havingplaced an advertisement sponsors</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE: it regretted having placed advertisement sponsors of last week &#x27;s lunar new year</td></tr></table>
+
+Figure 9: Real example from our model in English Gigaword.   
+
+<table><tr><td rowspan=1 colspan=2>Source Text:experts from iran and the un nuclear watchdog met thursday in vienna to discuss tehran &#x27;s plans toresume atomic fuel research ，an official from the agency said .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth: iranian experts meet with un nuclear watchdog</td><td rowspan=1 colspan=1>(A)Supervised Result: iran un nuclear watchdog discuss tehran</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:to discuss tehran &#x27;s nuclear research watchdog</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:. official says experts from iran &#x27;s president &#x27;swatchdog met</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:experts iran to un nuclear watchdog in vienna</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE: experts discuss un nuclear watchdog plans toresume tehran</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :experts from iran and the un nuclear watchdogmet</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE: experts from iran and the un nuclear watchdogmet</td></tr></table>
+
+Figure 10: Real example from our model in English Gigaword.   
+
+<table><tr><td rowspan=1 colspan=2> Source Text:european shares fell monday， pressured by higher crude prices after oil giant bp said it will shutdown a key production field and on caution before a u.s. interest-rate decision .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth:european stocks end lower</td><td rowspan=1 colspan=1>(A)Supervised Result:european stocks fall on bp decision</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:prices fell monday after caution on caution</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:oil giant bp says higher crude prices oncaution</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:european shares shut down higher after oil</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:european shares shut down after higher oilproduction</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :european shares fell by higher crude prices afteroil</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE:by higher crude prices after oil giant and oncaution before it will shut</td></tr></table>
+
+Figure 11: Real example from our model in English Gigaword.   
+
+<table><tr><td rowspan=1 colspan=2>Source Text: russian foreign minister igor ivanov urged iran to be open about its nuclear programs during a meeting with his iranian counterpart at the united nations , the foreign ministry said tuesday .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth:russian minister urges iran to be open about nuclear programs</td><td rowspan=1 colspan=1>(A)Supervised Result:russian fm urges iran to be open about nuclearprograms</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator: ivanov urged iran to be open its nuclear programs during</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:russian foreign minister igor ivanov saysmeeting will be open</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:russian igor to be open about nuclear programs during meeting</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE: russian foreign minister to open iran meetingabout its nuclear programs</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :russian foreign minister igor ivanov urged iran tobe open about nuclear programs</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE: new foreign feature urges iran to be open about its nuclear programs</td></tr></table>
+
+Figure 12: Real example from our model in English Gigaword.   
+
+<table><tr><td rowspan=1 colspan=2> Source Text: the bewildering fight between the government and telemarketers over the national do-not-call list took another turn when a second federal agency said it would enforce the program , promising thatconsumers would soon see some reduction in telephone sales pitches .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth: fcc steps in to enforce do-not-callist ; bush signs new law to support program</td><td rowspan=1 colspan=1>(A)Supervised Result:fight against consumers</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:the second list in telephone sales that wouldenforce</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:， promising that consumers would enforce turn</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:fight between government over national list took turn</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:fight between government pitches see anotherreduction</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN : the fight between the government and over thenational list took another turn</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE: fight between government and over nationallist took another turn when a second federal</td></tr></table>

parse/train/r1kNDlbCb/r1kNDlbCb_content_list.json

@@ -0,0 +1,1638 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING TO ENCODE TEXT AS HUMAN-READABLESUMMARIES USING GENERATIVE ADVERSARIAL NET-WORKS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            260,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Auto-encoders compress input data into a latent-space representation and reconstruct the original data from the representation. This latent representation is not easily interpreted by humans. In this paper, we propose training an auto-encoder that encodes input text into human-readable sentences. The auto-encoder is composed of a generator and a reconstructor. The generator encodes the input text into a shorter word sequence, and the reconstructor recovers the generator input from the generator output. To make the generator output human-readable, a discriminator restricts the output of the generator to resemble human-written sentences. By taking the generator output as the summary of the input text, abstractive summarization is achieved without document-summary pairs as training data. Promising results are shown on both English and Chinese corpora. ",
+        "bbox": [
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+            294,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            336,
+            488
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When it comes to learning data representations, a popular approach involves the auto-encoder architecture, which compresses the data into a latent representation without supervision. In this paper we focus on learning text representations. Because text is a sequence of words, to encode a sequence, a sequence-to-sequence (seq2seq) auto-encoder (Li et al., 2015; Kiros et al., 2015) is usually used, in which a RNN is used to encode the input sequence into a fixed-length representation, after which another RNN is used to decode the original input sequence given this representation. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although the latent representation learned by the auto-encoder can be used in downstream applications, they are usually not human-readable. In this work, we use comprehensible natural language as a latent representation of the input source text in an auto-encoder model. This human-readable latent representation is shorter than the source text; in order to reconstruct the source text, it must reflect the core idea of the source text. Intuitively, the latent representation can be considered a summary of the text. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The idea that using human comprehensible representation as a latent representation has been explored on text summarization (Miao & Blunsom, 2016), but only in a semi-supervised scenario. Previous work uses a prior distribution from a pre-trained language model to constrain the generated sequence to natural language. However, to teach the compressor network to generate text summaries, the model is trained using labeled data. In contrast, in this work we need no labeled data to learn the representations. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The proposed model is inspired from cycle consistency (Zhu et al., 2017; He et al., 2016). As shown in Fig. 1, the proposed model is composed of three components: a generator, a discriminator, and a reconstructor. Together, the generator and reconstructor form a text auto-encoder. The generator acts as an encoder in generating the latent representation from the input text. Instead of using a vector as latent representation, however, the generator generates a word sequence much shorter than the input text. From the shorter text, the reconstructor reconstructs the original input of the generator. By minimizing the reconstruction errors, the generator learns to generate short text segments that contain the main information in the original input. We use the seq2seq model in modeling the generator and reconstructor because both have input and output sequences with different lengths. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, it is very possible that the generator’s output word sequence can be processed by the reconstructor but is not readable by humans. Here, instead of regularizing the generator output with a pre-trained language model (Miao & Blunsom, 2016), we borrow from adversarial autoencoders (Makhzani et al., 2015) and introduce a third component – the discriminator – to regularize the generator’s output word sequence. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/ded4773453bdbbecb7f20057d902c30e59ccb993a5de3ea27747aab8d1a8e652.jpg",
+        "image_caption": [
+            "Figure 1: Proposed model. Given long text, the generator produces a shorter text as a summary. The generator is learned by minimizing the reconstruction loss together with the reconstructor and making discriminator regard its output as human-written text. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            302,
+            159,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The discriminator and the generator form a generative adversarial network (GAN) (Goodfellow et al., 2014). GANs are generative models composed of a generator and a discriminator. The discriminator discriminates between the generator output and real data, and the generator produces output as similar as possible to real data to confuse the discriminator. Here, we only have to feed human-written sentences to the discriminator as real data. With the GAN framework, the discriminator teaches the generator how to create human-like summary sentences as a latent representation; however, this only guarantees that the generator produces grammatically correct sentences – not necessarily sentences that represent the input text. It is the reconstructor that teaches the generator how to produce a sentence that captures the core idea of the source text. ",
+        "bbox": [
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+            352,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, generating discrete distributions with GAN is challenging, since it is difficult to evaluate the distance between the continuous distribution from the generator and the discrete distribution of the real sample. In addition, if we feed sampled words from the generator output distribution to the discriminator, the process of word selection is non-differentiable, which yields a discriminator gradient that precludes back-propagation to the generator. With GAN, there are two ways to generate language: (1) by training with a policy gradient, which regards words as actions, or (2) by directly feeding the generator’s output layer to the discriminator, which yields a gradient suited to backpropagation to the generator. In this work, we propose new kind of method on training with policy gradient in which the discriminator evaluates the output of generator every time steps. On language generation with GAN, we conduct experiments using both (1) and (2) methods and evaluate their results. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the results on an abstractive text summarization task in which the machine generates a text summary in its own words. The model is learned from a set of unpaired documents and summaries1. We use the sentences in the summaries as real data for discriminator2. As the summaries can come from another set of documents not related to the training documents, training is unsupervised. We use the output word sequence of the generator as the summaries of the input text. The results show that the generator generates summaries with reasonable quality on both English and Chinese corpora. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            739,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "GAN FOR LANGUAGE GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            763,
+            426,
+            779
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The major challenge in applying GAN to sentence generation is the discrete nature of natural language. To generate a word sequence, the generator usually has non-differential parts such as argmax or other sample functions which cause the original GAN to fail. Therefore, new kinds of GANs have been proposed for sentence generation. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "SeqGAN (Yu et al., 2017) tackles the sequence generation problem with reinforcement learning. Here, we refer to this approach as adversarial REINFORCE, in which the generator is regarded as an agent, the generated sequence of words is viewed as a sequence of actions, and the current state is defined as the generated sequence to date and the prior input. However, the discriminator only measures the quality of whole sentences, and thus the rewards are extremely sparse and the rewards assigned to all actions in sequence are all the same. To tackle this problem, they propose MC search to evaluate approximate rewards at each time step, but this method suffers from high time complexity. Following this idea, (Li et al., 2017) proposes another approach to evaluate the expected reward at each time step. They break both the generated and real sequences into partial sequences, and the discriminator discriminates between the generated and real partial sequences. Inspired by this idea, we propose the self-critical adversarial REINFORCE algorithm as another way to evaluate the expected reward at each time step. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In (Gulrajani et al., 2017), instead of feeding a discrete word sequence, the authors directly feed the generator output layer to the discriminator. This method works because they use the earth mover’s distance on GAN as proposed in (Arjovsky et al., 2017), which is able to evaluate the distance between a discrete and a continuous distribution. In order to satisfy the requirement of the earth mover’s distance, they use a gradient penalty trick to confine the complexity of discriminator function. Their method achieves an amazing result: it is the first work on GAN training that performs language generation without pre-training. In our work, we also conduct experiments on this method with discriminator settings almost the same as the original paper. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACTIVE TEXT SUMMARIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            430,
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+            444
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recent model architectures for abstractive text summarization basically use the sequence-tosequence (Sutskever et al., 2014) framework in combination with various novel mechanisms. One popular mechanism is attention (Bahdanau et al., 2015), which has been shown helpful for summarization (Nallapati et al., 2016; Rush et al., 2015). It is also possible to directly optimize evaluation metrics such as ROUGE (Lin, 2004) with reinforcement learning (Ranzato et al., 2016; Paulus et al., 2017; Bahdanau et al., 2016). The hybrid pointer-generator network (See et al., 2017) selects words from the original text with a pointer (Vinyals et al., 2015) or from the whole vocabulary with a trained weight. In order to eliminate repetition, a coverage vector (Tu et al., 2016) can be used to keep track of attended words and coverage loss (See et al., 2017) can be used to encourage model focus on diverse words. While most papers focus on supervised learning with novel mechanisms, we explore unsupervised training models. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            622,
+            375,
+            638
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The overview of the proposed model is shown in Fig. 2. The model is composed of three components: generator $G$ , discriminator $D$ , and reconstructor $R$ . Both $G$ and $R$ are seq2seq hybrid pointer-generator networks (See et al., 2017) which can decide to copy words from encoder input text via pointing or generate from vocabulary.They both take a word sequence as input and output a sequence of word distributions. Discriminator $D$ , on the other hand, takes a sequence as input and outputs a scalar. The model is learned from a set of documents $x$ and human-written sentences $y ^ { r e a l }$ . Although in real implementation, $y ^ { r e a l }$ are the sentences in summaries, we note that the documents and summaries are unpaired. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To train the model, a training document $\\boldsymbol { x } = \\{ x _ { 1 } , x _ { 2 } , . . . , x _ { t } , . . . , x _ { T } \\}$ , where $x _ { t }$ represents a word, is fed to $G$ , which outputs a sequence of word distributions $G ( x ) = \\left\\{ y _ { 1 } , y _ { 2 } , . . . , y _ { n } , . . . , y _ { N } \\right\\}$ , where $y _ { n }$ is a distribution over all words in the lexicon. Then we sample a word $y _ { n } ^ { s }$ from each distribution $y _ { n }$ , and a word sequence $y ^ { s } = \\{ y _ { 1 } ^ { s } , y _ { 2 } ^ { s } , . . . , y _ { N } ^ { s } \\}$ is obtained according to $G ( x )$ . We feed the sampled word sequence $y ^ { s }$ to reconstructor $R$ , which outputs another sequence of word distributions $\\hat { x }$ . The reconstructor $R$ reconstructs the original text $x$ from $y ^ { s }$ . That is, we seek an output of reconstructor $\\hat { x }$ that is as close to the original text $x$ as possible; hence the loss for training the reconstructor $R$ , $R _ { l o s s }$ , is defined as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7256d20fa140f9a33b997def11c5bfc826382e02a7d6800f1d122e634a825684.jpg",
+        "text": "$$\nR _ { l o s s } = \\sum _ { k = 1 } ^ { K } l _ { s } ( x , \\hat { x } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/3db872585682f6ac8bfcf5ea68af851d1652db81a2e613ec555ff7ade2f5d1e2.jpg",
+        "image_caption": [
+            "Figure 2: Architecture of proposed model. The generator network and reconstructor network are a seq2seq hybrid pointer-generator network, but for simplicity, we omit the pointer and the attention parts. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            99,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the reconstruction loss $l _ { s } ( x , \\hat { x } )$ is the cross-entropy loss computed between the reconstructor output sequence $\\hat { x }$ and the source text $x$ , or the negative conditional log-likelihood of source text $x$ given word sequence $y ^ { s }$ sampled from $G ( x )$ . The reconstructor output sequence $\\hat { x }$ is teacher-forced by source text $x$ . The subscript $s$ in $l _ { s } ( x , \\hat { x } )$ indicates that $\\hat { x }$ is reconstructed from $y ^ { s }$ . $K$ is the number of training examples (documents), and (1) is the summation of the cross-entropy loss over all the training documents $x$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the proposed model, the generator $G$ and reconstructor $R$ form an auto-encoder. However, the reconstructor $R$ does not directly take the generator output distribution $G ( x )$ as input 3. Instead, the reconstructor takes a sampled discrete sequence $y ^ { s }$ as input. Due to the non-differentiable property of discrete sequences, we apply the REINFORCE algorithm, which is described in Section 4. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In addition to reconstruction, we need the discriminator $D$ to discriminate between the real sequence yreal and the generated sequence $y ^ { s }$ to regularize the generated sequence satisfying the summary distribution. $D$ learns to give $y ^ { r e a l }$ higher scores while giving $y ^ { s }$ lower scores. The loss for training the discriminator $D$ is denoted as $D _ { l o s s }$ ; this is further described in Section 5. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$G$ learns to minimize the reconstruction error $R _ { l o s s }$ , while maximizing the loss of the discriminator $D$ by generating a summary sequence $y ^ { s }$ that cannot be differentiated by $D$ from the real thing. The loss when training the generator $G$ , $G _ { l o s s }$ , is ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8fe9d222853587ac8c23b5fe3a0ab02904346db40787e0dc3957a19ec637eec0.jpg",
+        "text": "$$\nG _ { l o s s } = \\alpha R _ { l o s s } - D _ { l o s s } ^ { \\prime }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $D _ { l o s s } ^ { \\prime }$ is highly related to $D _ { l o s s }$ – but not necessary the same – and $\\alpha$ is a hyper-parameter.   \nAfter obtaining the optimal generator by minimizing (2), we use it to generate summaries. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Generator $G$ and discriminator $D$ together form a GAN. We use two different adversarial training methods to train $D$ and $G$ ; as shown in Fig. 2, these two methods have their own discriminators 1 and 2. Discriminator 1 takes the generator output layer $G ( x )$ as input, whereas discriminator 2 takes the sampled discrete word sequence $y ^ { s }$ as input. The two methods are described respectively in Sections 5.1 and 5.2. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 MINIMIZING RECONSTRUCTION ERROR ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Because discrete sequences are non-differentiable, we use the REINFORCE algorithm. The generator is seen as an agent whose reward given the source text $x$ is $- l _ { s } ( x , \\hat { x } )$ . Maximizing the reward is equivalent to minimizing the reconstruction loss $R _ { l o s s }$ in (1). However, the reconstruction loss varies widely from sample to sample, and thus the rewards to the generator are not stable either. Hence we add a baseline to reduce their difference. We apply self-critical sequence training (Rennie et al., 2017); the modified reward $r ^ { R } ( x , { \\hat { x } } )$ from reconstructor $R$ with the baseline for the generator is ",
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+        "type": "equation",
+        "img_path": "images/82d2d58a68f608baff6e6de406b3105433c8bb8f88d267c830c7eff1d5b0644d.jpg",
+        "text": "$$\nr ^ { R } ( x , \\hat { x } ) = - l _ { s } ( x , \\hat { x } ) - ( - l _ { a } ( x , \\hat { x } ) - b )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            220,
+            633,
+            239
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $- l _ { a } ( x , { \\hat { x } } ) - b$ is the baseline. $l _ { a } ( x , \\hat { x } )$ is also the same cross-entropy reconstruction loss as $l _ { s } ( x , \\hat { x } )$ , except that $\\hat { x }$ is obtained from $y ^ { a }$ instead of $y ^ { s }$ . $y ^ { a }$ is a word sequence $\\left\\{ y _ { 1 } ^ { a } , y _ { 2 } ^ { a } , . . . , y _ { n } ^ { a } , . . . , y _ { N } ^ { a } \\right\\}$ , where $y _ { n } ^ { a }$ is selected using the argmax function from the output distribution of generator $y _ { n }$ . As in the early training stage, the sequence $y ^ { s }$ barely yields higher reward than sequence $y ^ { a }$ , to encourage exploration we introduce the second baseline score $b$ , which gradually decreases to zero. Then, the generator is updated using the REINFORCE algorithm with reward $r ^ { R } ( x , { \\hat { x } } )$ to minimize $R _ { l o s s }$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "5 GAN TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With adversarial training, the generator learns to produce sentences as similar to the human-written sentences as possible. Here, we conduct experiments on two kinds of methods of language generation with GAN. In Section 5.1 we directly feed the generator output probability distributions to the discriminator and use a Wasserstein GAN (WGAN) with a gradient penalty. In Section 5.2, we explore adversarial REINFORCE, which feeds sampled discrete word sequences to the discriminator and evaluates the quality of the sequence from the discriminator for use as a reward signal to the generator. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.1 DISCRIMINATOR 1: WASSERSTEIN GAN ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the lower left of Fig. 2, the discriminator model is shown as discriminator1. The discriminator loss $D _ { l o s s }$ is ",
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+    {
+        "type": "equation",
+        "img_path": "images/946970613a85237719a8584f56b8baf3db897352bba4faf50f0c487677aa1391.jpg",
+        "text": "$$\nD _ { l o s s } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } D ( y ^ { s ( k ) } ) - \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } D ( y ^ { r e a l ( k ) } ) + \\beta ( \\Delta _ { y ^ { i ( k ) } } D ( y ^ { i ( k ) } ) - 1 ) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $K$ denotes the number of training examples in a batch, and $k$ denotes the $k$ -th example. The last term in (4) is the gradient penalty (Gulrajani et al., 2017). We interpolate the generator output layer $G ( x )$ and the real sample $y ^ { r e a \\bar { l } }$ , and apply the gradient penalty to the interpolated sequence $y ^ { i }$ . $\\beta$ determines the gradient penalty scale. In Equation (2), for WGAN, $D _ { l o s s } ^ { \\prime }$ is the score of the generated example: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/43b17f8a33d7efc60838917b0dcedf7b95ea685779c601906eac6940e74f46ca.jpg",
+        "text": "$$\nD _ { l o s s } ^ { \\prime } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } D ( G ( x ^ { ( k ) } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.2 SELF-CRITIC ADVERSARIAL REINFORCE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe in detail the proposed adversarial REINFORCE method. The core idea is we use the LSTM discriminator to evaluate the current quality of the generated sequence $\\{ y _ { 1 } ^ { s } , y _ { 2 } ^ { s } , . . . , y _ { i } ^ { s } \\}$ at each time step $i$ . Hence, the generator knows that compared to the last time step, as the generated sentence either improves or worsens, it can easily find the problematic generation step in a long sequence, and thus fix the problem easily. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.2.1 DISCRIMINATOR 2 ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As shown in Fig. 2, the discriminator2 is a one-way LSTM network which takes a discrete word sequence as input. At time step $i$ , given input word $y _ { i } ^ { s }$ it predicts the current score $s _ { i }$ based on the sequence $\\left\\{ y _ { 1 } , y _ { 2 } , . . . , y _ { i } \\right\\}$ . The score is viewed as the quality of the current sequence. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In order to compute the discriminator loss $D _ { l o s s }$ , we sum the scores $\\left\\{ s _ { 1 } , s _ { 2 } , . . . , s _ { N } \\right\\}$ of the whole sequence $y ^ { s }$ to yield ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05ac9e30152dfb857d337bb27a6204daeb208c88e47e4884c526ffff929b464a.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\displaystyle { D ( y ^ { s } ) = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } s _ { n } . } } } \\\\ { { \\displaystyle { D _ { l o s s } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } D ( y ^ { s ( k ) } ) - \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } D ( y ^ { r e a l ( k ) } ) , } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            136,
+            655,
+            226
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $K$ and $N$ denote the number of training examples and the generated sequence length respectively. With the loss mentioned above, the discriminator attempts to quickly determine whether the current sequence is real or fake. The earlier the timestep discriminator determines whether the current sequence is real or fake, the lower its loss. An example is shown in Fig. 3. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/166009662b8a6fd2ea06cc9753d478913754598d73630c9ea3e6340275b8ceb6.jpg",
+        "image_caption": [
+            "Figure 3: When the second arrested appears, the discriminator determines that this example came from the generator. Hence, after this time-step, it outputs low scores. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2.2 SELF-CRITICAL GENERATOR ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            444,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Since we feed a discrete sequence $y ^ { s }$ to the discriminator, the gradient from the discriminator cannot directly back-propagate to the generator. Here, we use the policy gradient method. At timestep $i$ , we use the $i - 1$ timestep score $s _ { i - 1 }$ from the discriminator as its self-critical baseline. The reward r Di evaluates whether the quality of sequence in timestep $i$ is better or worse than that in timestep $i - 1$ . The generator reward $r _ { i } ^ { D }$ from $D$ is ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b986fa83260372e9b4518f5ac64dd4b9aa030ba253ee2e0322a0f5930eb11634.jpg",
+        "text": "$$\nr _ { i } ^ { D } = \\left\\{ \\begin{array} { l l } { { s _ { i } \\qquad } } & { { \\mathrm { i f ~ i = 1 ~ } } } \\\\ { { s _ { i } - s _ { i - 1 } \\qquad } } & { { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            535,
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+            569
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "However, some sentences may be judged as bad sentences at the previous timestep, but at later timesteps judged as good sentences, and vice versa. Hence we use the discounted expected reward $d$ with discount factor $\\gamma$ to calculate the discounted reward $d _ { i }$ at time step $i$ as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d18fca4c787d7bf932ae66df847d39156073ed403cb2c7c7833185148de86b0.jpg",
+        "text": "$$\nd _ { i } = \\sum _ { j = i } ^ { N } \\gamma ^ { j - i } r _ { j } ^ { D } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            617,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The adversarial REINFORCE score related to discriminator $D _ { l o s s } ^ { \\prime }$ in (2) is ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/105f91988c39d7e576b333f4bbaf7178842060037b05506ae6c205795dfb8013.jpg",
+        "text": "$$\nD _ { l o s s } ^ { \\prime } = E _ { y _ { i } ^ { s } \\sim p _ { G } ( y _ { i } ^ { s } | y _ { 1 } ^ { s } , . . . , y _ { i - 1 } ^ { s } , x ) } ^ { } [ d _ { i } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use the likelihood ratio trick to approximate the gradient. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6 IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            739,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Network Architecture. The model architecture of generator and reconstructor is almost same except the length of input and output sequence. We adapt model architecture for our generator and reconstructor from See et al. (2017) who used hybrid-pointer network with coverage vector for text summarization. The hybrid-pointer networks of generator and reconstructor are all composed of two one-layer LSTMs as its encoder and decoder, respectively, with a hidden layer size of 600. Since we use two kinds of methods on adversarial training, there are two discriminators with different model architecture. In the Section 5.1, the discriminator is composed of four residual blocks with 512 hidden dimensions. While in Section 5.2, we use only one hidden-layer one-way LSTM with a hidden size of 512 as our discriminator. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Details of Training. We set the weight $\\alpha$ in (2) controlling $R _ { l o s s }$ to 10 if not specified. We find that the if the value of $\\alpha$ is too large, generator will start to generate output unlike human-written sentences. On the other hand, if the value of $\\alpha$ is too small, the sentences generated by generator will sometimes become unrelated to input text of generator. For all the experiments, the baseline $b$ in (3) gradually decreases from 0.25 to zero within 10000 updates on generator. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Section 5.1, we set the weight $\\beta$ of the gradient penalty to 10, and used RMSPropOptimizer with a learning rate of 0.00001 and 0.001 on the generator and discriminator, respectively. In Section 5.2, we clip the value of the weights of discriminator to $\\pm 0 . 1 5$ , and used RMSPropOptimizer with a learning rate of 0.0001 and 0.001 on the generator and discriminator, respectively. It’s also feasible to apply gradient penalty trick in this method to satisfy requirement of Wasserstein distance. However, in this method, the performance of gradient penalty trick and weights clipping trick is close. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "7 EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate our model on the Chinese Gigaword and English Gigaword datasets. Before jointly training the whole model, we pre-trained the three major components – generator, discriminator, and reconstructor – separately. First, we pre-trained the generator in an unsupervised manner so that the generator would be able to somewhat grasp the semantic meaning of the source text. The details of the pre-training are in Appendix A. We pre-trained the discriminator and reconstructor respectively with the pre-trained generator’s output to ensure that these two critic networks provide good feedback to the generator. During testing, when using the generator to generate summaries, we simply selected the words in a greedy fashion without beam-search, and we eliminated repetition. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "7.1 CHINESE GIGAWORD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/c68c8403943f785cf4ec57247ed3baeb99f7b66b34cc8ac9f612e433fc1a19eb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>Methods</td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=1 colspan=2>(A) Trainingwith paired data (supervised)</td><td rowspan=1 colspan=1>48.664</td><td rowspan=1 colspan=1>33.907</td><td rowspan=1 colspan=1>45.685</td></tr><tr><td rowspan=1 colspan=2>(B) Trivial baselines: lead - 15</td><td rowspan=1 colspan=1>30.077</td><td rowspan=1 colspan=1>18.237</td><td rowspan=1 colspan=1>27.736</td></tr><tr><td rowspan=3 colspan=1>(C) Unsupervised</td><td rowspan=1 colspan=1>(C-1) Pretrained generator</td><td rowspan=1 colspan=1>28.122</td><td rowspan=1 colspan=1>16.656</td><td rowspan=1 colspan=1>26.227</td></tr><tr><td rowspan=1 colspan=1>(C-2) WGAN</td><td rowspan=1 colspan=1>37.803</td><td rowspan=1 colspan=1>24.460</td><td rowspan=1 colspan=1>35.116</td></tr><tr><td rowspan=1 colspan=1>(C-3) Adversarial REINFORCE</td><td rowspan=1 colspan=1>40.053</td><td rowspan=1 colspan=1>26.126</td><td rowspan=1 colspan=1>37.118</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Results on Chinese Gigaword. In row (B), we select the article’s first fifteen words as its summary. Part (C) are the results obtained without paired data. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The Chinese Gigaword corpus is composed of $2 . 2 \\mathbf { M }$ paired data of headlines and news. We preprocessed the raw data as following. First, we selected the 4000 most frequent Chinese characters as our vocabulary. We filtered out headline-news pairs with excessively long or short news segments, or that contained too many out-of-vocabulary Chinese characters, yielding 1.1M headline-news pairs from which we randomly selected 5K headline-news pairs as our testing set, 5K headline-news pairs as our validation set, and the remaining pairs as our training set. During training and testing, the generator took only the first 80 Chinese characters of the source text as input. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results are shown in Table 1. Row (A) lists the results using 1.1 million document-summary pairs to directly train the generator without the reconstructor and discriminator: this is the upper bound of the proposed approach. In row (B), we simply took the first fifteen words in a document as its summary. The number of words was chosen to optimize the evaluation metrics. Part (C) are the results obtained in the unsupervised scenario without paired data. We show the results of the pre-trained generator in row (C-1); rows (C-2) and (C-3) are the results for the two GAN training methods respectively. We find that despite the performance gap between the unsupervised and supervised methods (rows (C-2), (C-3) v.s. (A)), the proposed method yielded much better performance than the trivial baselines (rows (C-2), (C-3) v.s. (B)). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "7.2 ENGLISH GIGAWORD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "On English Gigaword, we set our vocabulary size to 15k, and used the dataset preprocessed by (Rush et al., 2015) for training and testing. We used $3 . 8 \\mathbf { M }$ unpaired training data for our training, and we used the whole $2 0 0 \\mathrm { k }$ filtered data in validation set for testing. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results on English Gigaword are shown in Table 2. In row (B-1), we simply took the first eight words in a document as its summary. Row (B-2) is another trivial baseline. With unpaired documents and summaries, we matched documents to the most relevant summaries with unsupervised method. Each document and each summary were represented as tf-idf (term frequency $\\&$ inverse document frequency) vectors. The summary whose vector maximized cosine similarity of a document vector was retrieved as summary of the document. With paired data from unsupervised matching, given documents as generator input, the generator was trained to predict retrieved summaries. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/37bd99d7ade69bc2df608ebd8d8cee45db3439d71e99305978456d7aa615dd68.jpg",
+        "table_caption": [
+            "Table 2: Results on English Gigaword: In row (B-1), we select the article’s first eight words as its summary. In row (B-2), we match the documents to their most relevant summaries with unsupervised method. Part (C) are the results obtained without paired data. In part (D), we pre-trained the generator on CNN/Diary. In part (E), we not only pre-trained on CNN/Diary but also used the summaries from CNN/Diary as real data for the discriminator. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=4>Methods</td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=1 colspan=4>(A) Training with paired data (supervised)</td><td rowspan=1 colspan=1>37.469</td><td rowspan=1 colspan=1>16.272</td><td rowspan=1 colspan=1>35.175</td></tr><tr><td rowspan=2 colspan=3>(B) Trivial baselines</td><td rowspan=1 colspan=1>(B-1) Lead-8</td><td rowspan=1 colspan=1>27.663</td><td rowspan=1 colspan=1>10.246</td><td rowspan=1 colspan=1>25.852</td></tr><tr><td rowspan=1 colspan=1>(B-2) Unsupervised matching</td><td rowspan=1 colspan=1>29.900</td><td rowspan=1 colspan=1>10.442</td><td rowspan=1 colspan=1>27.379</td></tr><tr><td rowspan=3 colspan=3>(C) Unsupervised</td><td rowspan=1 colspan=1>(C-1) Pre-trained generator</td><td rowspan=1 colspan=1>21.269</td><td rowspan=1 colspan=1>5.608</td><td rowspan=1 colspan=1>18.896</td></tr><tr><td rowspan=1 colspan=1>(C-2) WGAN</td><td rowspan=1 colspan=1>33.043</td><td rowspan=1 colspan=1>12.222</td><td rowspan=1 colspan=1>30.121</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>(C-3) Adversarial REINFORCE</td><td rowspan=1 colspan=1>32.826</td><td rowspan=1 colspan=1>9.332</td><td rowspan=1 colspan=1>28.727</td></tr><tr><td rowspan=3 colspan=3>(D) Transfer learning(Pre-train)</td><td rowspan=1 colspan=1>ng</td><td rowspan=1 colspan=3>(D-1) Pre-trained generator</td></tr><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>(D-2)WGAN</td><td rowspan=1 colspan=1>32.405</td><td rowspan=1 colspan=1>12.313</td><td rowspan=1 colspan=1>29.689</td></tr><tr><td rowspan=1 colspan=1>(D-3) Adversarial REINFORCE</td><td rowspan=1 colspan=1>31.487</td><td rowspan=1 colspan=1>10.495</td><td rowspan=1 colspan=1>28.248</td></tr><tr><td rowspan=2 colspan=3>(E) Transfer learning(Pre-train+Discriminator)</td><td rowspan=1 colspan=1>(E-1) WGAN</td><td rowspan=1 colspan=1>29.912</td><td rowspan=1 colspan=1>10.695</td><td rowspan=1 colspan=1>27.324</td></tr><tr><td rowspan=1 colspan=1>(E-2) Adversarial REINFORCE</td><td rowspan=1 colspan=1>27.755</td><td rowspan=1 colspan=1>9.280</td><td rowspan=1 colspan=1>24.860</td></tr></table>",
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+        "type": "text",
+        "text": "The results for the pre-trained generator is shown in row (C-1). Compared with the trivial baselines (part (B)), the proposed approach (rows (C-2) and (C-3)) showed good improvement in terms of ROUGE-1. As shown in Fig. 4, the unsupervised method selects key words in the source text and generates the text summary. However, as shown in Table 2, although both unsupervised methods yield ROUGE-1 scores close to that of supervised training, they achieve lower scores on ROUGE-2, especially when training GAN with reinforcement learning. This is because they are extracting the key words from the source text, despite sometimes failing to arrange these words in the correct order. As shown in part (C-3) of Fig. 5, the words in the sentence generated in an unsupervised manner are not arranged correctly: the Italian prime minister, Berlusconi, should not be visiting himself. ",
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+    {
+        "type": "text",
+        "text": "7.3 TRANSFER LEARNING ",
+        "text_level": 1,
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+        "text": "In this subsection, we study transfer learning. We used the CNN/Daily Mail dataset (Hermann et al., 2015; Nallapati et al., 2016) preprocessed by the script provided by (See et al., 2017) as our source domain $S$ , and English Gigaword as our target domain $T$ . The data distributions among the two datasets are quite different. In English Gigaword, the articles consist of 32 words on average and the summaries consist of one sentence with 8 words on average, whereas the CNN/Daily Mail dataset is composed of 790-word articles and multi-sentence summaries. We took only the first 30 to 45 words in the original source domain articles as our new source articles $S _ { a }$ . There are 50K source articles in $S _ { a }$ . The 50K source summaries of the source articles are $S _ { t }$ . In contrast to English Gigaword, the summaries in CNN/Daily Mail dataset are composed of more than one sentence. We split the summary of each article into several sentences, and obtained 240K sentences $S _ { r }$ in this way. ",
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+        "text": "Transfer learning was applied in two directions. In the first direction, we pre-trained our generator on the data from the source dataset: we pre-trained the generator with $S _ { a }$ as input, and the generator predicted $S _ { t }$ . We pre-trained the generator in this manner in all of the transfer learning experiments. Then, after pre-training, the generator was further learned jointly with the reconstructor and discriminator on the data from the target domain. The results are shown in part (D) of Table 2. We found that pre-training on the source data set did not degrade performance, and even improved performance in some cases (parts (D) v.s. (C)). ",
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+        "type": "text",
+        "text": "In the second direction, we used the summary from the source domain as our real data. The experiments conducted up to this point required unpaired summary and text data in the target domain. In this experiment, the task was more challenging in that we used summaries $S _ { r }$ from the source domain $S$ as the real data for the discriminator; the generator took the target domain text as input. However, the summaries in each summarization task dataset had a different distribution in terms of the writing style, or in terms of the preferred summary words. To prevent overfitting to $S _ { r }$ , we set the weight $\\alpha$ to 50 which was larger than other experiments. With small weight of $\\alpha$ , as training progressed, the generator summary diverged more and more from the article, and the ROUGE scores became lower and lower. ",
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+        "text": "The results without using summaries in the target domain are shown in part (E). We find that using sentences $S _ { r }$ from another dataset yields lower ROUGE scores on the target testing set (parts (E) v.s. (D)) due to the mismatch between the summaries of the source and target domains. However, the discriminator still roughly regularizes the language model of generated word sequence. After training, the model still greatly enhanced the ROUGE score of the pre-trained model (rows (E-1), (E-2) v.s. (D-1)).Although the results in part (E) are comparable with the trivial baselines in part (B), in inspecting the real examples, we found that the results in part (E) were in fact better; this is not reflected in the ROUGE scores. In Fig. 4, the results in part (E) are better than the leading 8 words and the pre-trained generator results. To support this idea, we provide more examples in the Appendix C. ",
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+        "table_caption": [
+            "Figure 4: Real example from our model in English Gigaword. The proposed method generates summaries that capture the core idea of the article. ",
+            "Figure 5: In part (C-3), some words in the summary sentences are arranged in incorrect order. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2> Source Text:three stores and markets in beijing &#x27;s fengtai district have been forced to shut down andyesterday each was fined ###,### yuan -lrb- ##,### us dollars -rrb- for violating laws andregulations on fire prevention and control .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth: stores markets punished for lack of fire controls</td><td rowspan=1 colspan=1>(A)Supervised Result:beijing &#x27;s district stores shut down</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:and yesterday prevention have been forced to shut down</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:three stores and markets fined ###,### yuandollars</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:three stores markets in beijing &#x27;s district haveforced to shut down yesterday</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:three stores in beijing forced to shut down forviolating regulations</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :three stores and markets was forced to shutdown</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE:three stores and markets was forced to shutdown and yesterday each was fined</td></tr></table>",
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+        "type": "image",
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+    {
+        "type": "text",
+        "text": "7.4 SEMI-SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In semi-supervised training, generator was pre-trained with few available labeled data, and during unsupervised training, we conducted teacher-forcing with labeled data on generator every several unsupervised updates. In teacher forcing, given source text as input, the generator was teacherforced to predict the human-written summary of source text. Teacher-forcing can be regarded as regularization of unsupervised training that prevents generator from producing unreasonable summaries of source text. We found that if we teacher-forced generator too frequently, generator would overfit on training data since we only use very few labeled data on semi-supervised training. ",
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+    {
+        "type": "text",
+        "text": "The performance of semi-supervised model in English Gigaword regarding available labeled data is shown in Fig. 6. The horizontal axis is the number of labeled documents used in the experiments, while the vertical axis for Fig. 6 (a) and (b) are ROUGE-1 and ROUGE-2 respectively. The green curve is the results of supervised learning, and the red and blue curves are semi-supervised learning with different approaches. With the same amounts of labeled data, the performances of semi-supervised training are always better than supervised training. With only 100K labeled data, the ROUGE score of semi-supervised training using adversarial REINFORCE even slightly outperformed supervised training with whole labeled data. This shows that with the proposed approach, we need only $2 . 6 \\%$ of labeled data to achieve the same performance as before (100K v.s. 3.8M). The complete results for semi-supervised learning in both datasets are shown in Appendix B. ",
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+    {
+        "type": "image",
+        "img_path": "images/192cbcda93e6638c835881efa30b90cc07dd177b7c02b076ddb36ad66cd8affa.jpg",
+        "image_caption": [
+            "Figure 6: Semi-supervised results in English Gigaword. With the same amount of labeled data, the performances of semi-supervised training are always better than supervised training. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "7.5 GAN TRAINING ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/0f14e9178c496df740ebc9c2f2b0bc9a534502b2183a8e42ff9414ee5c01ab11.jpg",
+        "table_caption": [
+            "Table 3: Adversarial REINFORCE with/without self-critic with unsupervised training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Corpus</td><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=2 colspan=1>Chinese</td><td rowspan=1 colspan=1>with self-critic</td><td rowspan=1 colspan=1>40.053</td><td rowspan=1 colspan=1>26.126</td><td rowspan=1 colspan=1>37.118</td></tr><tr><td rowspan=1 colspan=1>without self-critic</td><td rowspan=1 colspan=1>37.549</td><td rowspan=1 colspan=1>24.181</td><td rowspan=1 colspan=1>35.160</td></tr><tr><td rowspan=2 colspan=1>English</td><td rowspan=1 colspan=1>with self-critic</td><td rowspan=1 colspan=1>32.826</td><td rowspan=1 colspan=1>9.332</td><td rowspan=1 colspan=1>28.727</td></tr><tr><td rowspan=1 colspan=1>without self-critic</td><td rowspan=1 colspan=1>31.104</td><td rowspan=1 colspan=1>9.249</td><td rowspan=1 colspan=1>28.592</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The two GAN training methods are not comparable as their settings are quite different, but we can still discuss the advantages and disadvantages of these two methods. When training with feeding output layer to discriminator, convergence is faster. This method sharpens the distribution at an early stage in training because it directly evaluates the distance between the generator’s continuous distribution and the real data’s discrete distribution data. However, this cause generator to converge to a not very good place. ",
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+    },
+    {
+        "type": "text",
+        "text": "In fact, adversarial REINFORCE is sensitive to initialization parameters. Adversarial REINFORCE requires better initialization for exploration; otherwise, with so many actions whose number is equal to vocabulary size to choose, it is extremely difficult to train generator from scratch. In semisupervised training, since we pre-trained generator with labeled data, the generator was better initialized, therefore adversarial REINFORCE performed better. To support this idea, in Fig. 6, we compare the performance of two methods regarding labeled data. The result implies that with more labeled data, our proposed adversarial REINFORCE method performs better. In order to evaluate the performance of proposed self-critic baseline trick mentioned in Section 5.2.2, we compared the performance of our model with and without this baseline trick in Table 3. For the experiments without the self-critic baseline trick, we replaced $s _ { i } - s _ { i - 1 }$ in Section 5.2.2 with $s _ { i }$ . We found that the performance degraded without self-critic. ",
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+    {
+        "type": "text",
+        "text": "8 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Using GAN, we propose a model that encodes text as a human-readable summary, learned without document-summary pairs. Promising results are obtained on both Chinese and English corpora. In future work, we hope to explore more techniques for natural language generation using GAN. Moreover, we hope to use extra discriminators to control the style and sentiment of the generated summaries. ",
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+    {
+        "type": "text",
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+    },
+    {
+        "type": "text",
+        "text": "Lantao Yu, Weinan Zhang, Jun Wang, and Yong Yu. Seqgan: Sequence generative adversarial nets with policy gradient. AAAI, 2017. ",
+        "bbox": [
+            173,
+            127,
+            821,
+            156
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint arXiv:1703.10593, 2017. ",
+        "bbox": [
+            169,
+            165,
+            825,
+            194
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A MODEL PRE-TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            403,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As we found that the different pre-training methods for the generator influenced final performance dramatically in all of the experiments, we felt it was important to find a proper unsupervised pretraining method to help the machine grasp semantic meaning. We used the different pre-training strategies described below. ",
+        "bbox": [
+            176,
+            133,
+            825,
+            189
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Chinese Gigaword: Given the previous $i - 1$ sentences $s e n t _ { 0 } , s e n t _ { 1 } , . . . , s e n t _ { i - 1 }$ from the source text, the generator predicted the next sentence $s e n t _ { i }$ in the source text as its pretraining target. If more than $50 \\%$ of the words in sentence $s e n t _ { i }$ did not appear in the given text, we filtered out this pre-training sample pair. This pre-training method allowed the generator to capture the important semantic meanings of the source text. English Gigaword: As the length of the source texts in English Gigaword dataset is comparatively short, it is difficult to split the last sentence from the source text; hence the previous pre-training method on Chinese Gigaword is not appropriate for this dataset. To properly initialize the set, we randomly selected 6 to 11 consecutive words in the source text, after which we randomly swapped $70 \\%$ of the words in the source text. Given text with incorrect word arrangements, the generator predicted the selected words in the correct arrangement. We pre-trained in this way because we expect the generator to initialize with a rough language model. In Chinese Gigaword we also conducted experiments on pre-training in this manner, but the results were not as good as those shown in the part (C) of Table 1. We also used the retrieved paired data in row (B-1) in Table 2 to pre-train generator. However, pre-training generator with this method doesn’t yield results better than those in Table 2. ",
+        "bbox": [
+            215,
+            202,
+            825,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B SEMI-SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            457,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Semi-supervised learning experiments were conducted with 10K, 50K, 100K labeled data in both datasets. We conducted teacher-forcing on generator every 30, 12, 7 unsupervised updates with 10K, 50K, 100K labeled data respectively. The complete results for semi-supervised learning are shown in Tables 4 and 5. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            189
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/6aba29804b7149bb0c055c20af1ad879ddd1d7a725c2c8def2d5360e673def20.jpg",
+        "table_caption": [
+            "Table 4: Semi-supervised learning in Chinese Gigaword with different amounts of labeled data (10K, 50K, 100K). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=2 colspan=1>Unsupervised</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>37.803</td><td rowspan=1 colspan=1>24.460</td><td rowspan=1 colspan=1>35.116</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>40.053</td><td rowspan=1 colspan=1>26.126</td><td rowspan=1 colspan=1>37.118</td></tr><tr><td rowspan=2 colspan=1>Semi-supervised(10K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>42.359</td><td rowspan=1 colspan=1>27.192</td><td rowspan=1 colspan=1>38.467</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>43.109</td><td rowspan=1 colspan=1>28.626</td><td rowspan=1 colspan=1>40.202</td></tr><tr><td rowspan=2 colspan=1>Semi-supervised(50K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>43.989</td><td rowspan=1 colspan=1>29.012</td><td rowspan=1 colspan=1>40.764</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>44.706</td><td rowspan=1 colspan=1>29.872</td><td rowspan=1 colspan=1>41.737</td></tr><tr><td rowspan=2 colspan=1>Semi-supervised(100K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>45.642</td><td rowspan=1 colspan=1>31.475</td><td rowspan=1 colspan=1>42.711</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>46.216</td><td rowspan=1 colspan=1>31.980</td><td rowspan=1 colspan=1>43.522</td></tr><tr><td rowspan=1 colspan=2>Supervised</td><td rowspan=1 colspan=1>48.664</td><td rowspan=1 colspan=1>33.907</td><td rowspan=1 colspan=1>45.685</td></tr></table>",
+        "bbox": [
+            202,
+            199,
+            794,
+            347
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/cb0f7f54096977e0d93d8b63e5ae5f10d0cc6191f3010a6921367ea62ecb65b3.jpg",
+        "table_caption": [
+            "Table 5: Semi-supervised learning in English Gigaword with different amounts of labeled data (10K, 50K, 100K). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=4></td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=2 colspan=3>Unsupervised</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>33.043</td><td rowspan=1 colspan=1>12.222</td><td rowspan=1 colspan=1>30.121</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>32.826</td><td rowspan=1 colspan=1>9.332</td><td rowspan=1 colspan=1>28.727</td></tr><tr><td rowspan=2 colspan=3>Semi-supervised(10K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>34.127</td><td rowspan=1 colspan=1>13.087</td><td rowspan=1 colspan=1>31.451</td></tr><tr><td rowspan=1 colspan=1>led）</td><td rowspan=1 colspan=1>AdversarialREINFORCE</td><td rowspan=1 colspan=1>34.239</td><td rowspan=1 colspan=1>12.570</td><td rowspan=1 colspan=1>31.834</td></tr><tr><td rowspan=1 colspan=2>Semi-supervised</td><td rowspan=1 colspan=1>Semi-supervised</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>34.937</td><td rowspan=1 colspan=1>13.838</td><td rowspan=1 colspan=1>32.372</td></tr><tr><td rowspan=1 colspan=3>(50K labeled)</td><td rowspan=1 colspan=1>AdversarialREINFORCE</td><td rowspan=1 colspan=1>35.642</td><td rowspan=1 colspan=1>14.057</td><td rowspan=1 colspan=1>32.983</td></tr><tr><td rowspan=2 colspan=3>Semi-supervised(100K labeled)</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>36.615</td><td rowspan=1 colspan=1>15.363</td><td rowspan=1 colspan=1>33.682</td></tr><tr><td rowspan=1 colspan=1>Adversarial REINFORCE</td><td rowspan=1 colspan=1>38.213</td><td rowspan=1 colspan=1>16.279</td><td rowspan=1 colspan=1>35.137</td></tr><tr><td rowspan=1 colspan=4>Supervised</td><td rowspan=1 colspan=1>37.469</td><td rowspan=1 colspan=1>16.272</td><td rowspan=1 colspan=1>35.175</td></tr></table>",
+        "bbox": [
+            202,
+            401,
+            794,
+            549
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            302,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "From Fig. 7 to 12, we show more examples. ",
+        "bbox": [
+            176,
+            133,
+            462,
+            148
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/f1eec366fb1fb912521cfc972e089f6fd9a38945b64cf84dd9f797eb8033e188.jpg",
+        "table_caption": [
+            "Figure 7: Real example from our model in English Gigaword. In part (E-2), due to transfer learning, the summary sentence begins with word he, which never appears in the English Gigaword summary sentences. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>Source Text: former zambian president kenneth kaunda appeared in court monday on charges of holding an illegal ally ， declaring that he would continue to fight the “ oppressive regime &quot; of presidentfredrick chiluba .</td></tr><tr><td rowspan=1 colspan=1> Ground Truth:former zambian president in court for illegal assembly</td><td rowspan=1 colspan=1>(A)Supervised Result:kaunda to continue to fight chiluba</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator: president kenneth chiluba appeared in courtmonday on charges of</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:kaunda declaring that he would continue tofight the regime says</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN: zambian kenneth kaunda in court charges of holding illegal rally</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:former zambian president to continue illegalrally fight</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :former zambian president kenneth kaundaappeared in court of illegal rally</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE:he appeared in court he would continue to fight the regime</td></tr></table>",
+        "bbox": [
+            209,
+            162,
+            787,
+            407
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/a784a3c0bdbbabb20d9212cca4e11ff5684dd036ae9b0828b29bae659e91b4a5.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>Source Text:hong kong tourist association -Irb- hkta -rrb- said wednesday it regretted having placed an advertisement thanking sponsors of last week &#x27;s lunar new year parade which killed one man andleft ## others injured .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth:hong kong tourist body expresses regret overadvertisement</td><td rowspan=1 colspan=1>(A)Supervised Result:hong kong tourist regrets having placedadvertisement sponsors</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:hong kong said wednesday it regretted anadvertisement</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator: one man was killed in the head of thesponsors of the said</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN: hong kong tourist association regretted havingplaced advertisement sponsors</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:hong kong tourist killed advertisement sponsors of last year &#x27;s lunar parade</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN : hong kong tourist association regretted havingplaced an advertisement sponsors</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE: it regretted having placed advertisement sponsors of last week &#x27;s lunar new year</td></tr></table>",
+        "bbox": [
+            210,
+            493,
+            787,
+            736
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/af252569f01412268147360416b18fd04888805dabacd73cebd7c772e25925af.jpg",
+        "table_caption": [
+            "Figure 9: Real example from our model in English Gigaword. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>Source Text:experts from iran and the un nuclear watchdog met thursday in vienna to discuss tehran &#x27;s plans toresume atomic fuel research ，an official from the agency said .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth: iranian experts meet with un nuclear watchdog</td><td rowspan=1 colspan=1>(A)Supervised Result: iran un nuclear watchdog discuss tehran</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:to discuss tehran &#x27;s nuclear research watchdog</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:. official says experts from iran &#x27;s president &#x27;swatchdog met</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:experts iran to un nuclear watchdog in vienna</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE: experts discuss un nuclear watchdog plans toresume tehran</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :experts from iran and the un nuclear watchdogmet</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE: experts from iran and the un nuclear watchdogmet</td></tr></table>",
+        "bbox": [
+            209,
+            174,
+            787,
+            395
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/57c9ee357852f78f17b3d81d35bec15f98c233ab5e9b352f54bdc4a03210d993.jpg",
+        "table_caption": [
+            "Figure 10: Real example from our model in English Gigaword. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2> Source Text:european shares fell monday， pressured by higher crude prices after oil giant bp said it will shutdown a key production field and on caution before a u.s. interest-rate decision .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth:european stocks end lower</td><td rowspan=1 colspan=1>(A)Supervised Result:european stocks fall on bp decision</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:prices fell monday after caution on caution</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:oil giant bp says higher crude prices oncaution</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:european shares shut down higher after oil</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:european shares shut down after higher oilproduction</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :european shares fell by higher crude prices afteroil</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE:by higher crude prices after oil giant and oncaution before it will shut</td></tr></table>",
+        "bbox": [
+            210,
+            580,
+            785,
+            816
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/6f5d19613ef5709798bfe4618b7a2a160795eed8d2307bc6197d22e9129d5817.jpg",
+        "table_caption": [
+            "Figure 11: Real example from our model in English Gigaword. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>Source Text: russian foreign minister igor ivanov urged iran to be open about its nuclear programs during a meeting with his iranian counterpart at the united nations , the foreign ministry said tuesday .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth:russian minister urges iran to be open about nuclear programs</td><td rowspan=1 colspan=1>(A)Supervised Result:russian fm urges iran to be open about nuclearprograms</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator: ivanov urged iran to be open its nuclear programs during</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:russian foreign minister igor ivanov saysmeeting will be open</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:russian igor to be open about nuclear programs during meeting</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE: russian foreign minister to open iran meetingabout its nuclear programs</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN :russian foreign minister igor ivanov urged iran tobe open about nuclear programs</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE: new foreign feature urges iran to be open about its nuclear programs</td></tr></table>",
+        "bbox": [
+            209,
+            104,
+            787,
+            330
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/59164f46b2ff38b8e63df91724ec4af041a52b1dd49cf9648ae24027f22efbd5.jpg",
+        "table_caption": [
+            "Figure 12: Real example from our model in English Gigaword. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2> Source Text: the bewildering fight between the government and telemarketers over the national do-not-call list took another turn when a second federal agency said it would enforce the program , promising thatconsumers would soon see some reduction in telephone sales pitches .</td></tr><tr><td rowspan=1 colspan=1>Ground Truth: fcc steps in to enforce do-not-callist ; bush signs new law to support program</td><td rowspan=1 colspan=1>(A)Supervised Result:fight against consumers</td></tr><tr><td rowspan=1 colspan=1>(C-1)Pretrained Generator:the second list in telephone sales that wouldenforce</td><td rowspan=1 colspan=1>(D-1)Pretrained Generator:， promising that consumers would enforce turn</td></tr><tr><td rowspan=1 colspan=1>(C-2)WGAN:fight between government over national list took turn</td><td rowspan=1 colspan=1>(C-3)Adversarial REINFORCE:fight between government pitches see anotherreduction</td></tr><tr><td rowspan=1 colspan=1>(E-1)WGAN : the fight between the government and over thenational list took another turn</td><td rowspan=1 colspan=1>(E-2)Adversarial REINFORCE: fight between government and over nationallist took another turn when a second federal</td></tr></table>",
+        "bbox": [
+            209,
+            390,
+            787,
+            633
+        ],
+        "page_idx": 16
+    }
+]

parse/train/tPCrkaLa9Y5ld/tPCrkaLa9Y5ld.md

@@ -0,0 +1,200 @@
+# One-Shot Adaptation of Supervised Deep Convolutional Models
+
+Judy Hoffman, Eric Tzeng, Jeff Donahue UC Berkeley, EECS & ICSI {jhoffman,etzeng,jdonahue}@eecs.berkeley.edu
+
+Yangqing Jia∗ Google Research jiayq@google.com
+
+Kate Saenko UMass Lowell, CS & ICSI saenko@cs.uml.edu
+
+Trevor Darrell UC Berkeley, EECS & ICSI trevor@eecs.berkeley.edu
+
+# Abstract
+
+Dataset bias remains a significant barrier towards solving real world computer vision tasks. Though deep convolutional networks have proven to be a competitive approach for image classification, a question remains: have these models have solved the dataset bias problem? In general, training or fine-tuning a state-ofthe-art deep model on a new domain requires a significant amount of data, which for many applications is simply not available. Transfer of models directly to new domains without adaptation has historically led to poor recognition performance. In this paper, we pose the following question: is a single image dataset, much larger than previously explored for adaptation, comprehensive enough to learn general deep models that may be effectively applied to new image domains? In other words, are deep CNNs trained on large amounts of labeled data as susceptible to dataset bias as previous methods have been shown to be? We show that a generic supervised deep CNN model trained on a large dataset reduces, but does not remove, dataset bias. Furthermore, we propose several methods for adaptation with deep models that are able to operate with little (one example per category) or no labeled domain specific data. Our experiments show that adaptation of deep models on benchmark visual domain adaptation datasets can provide a significant performance boost.
+
+# 1 Introduction
+
+Supervised deep convolutional neural networks (CNNs) trained on large-scale classification tasks have been shown to learn impressive mid-level structures and obtain high levels of performance on contemporary classification challenges [3, 23]. These models generally assume extensive training using labeled data, and testing is limited to data from the same domain. In practice, however, the images we would like to classify are often produced under different imaging conditions or drawn from a different distribution, leading to a domain shift. Scaling such models to new domains remains an open challenge.
+
+Deep CNNs require large amounts of training data to learn good mid-level convolutional models and final fully-connected classifier stages. While the continuing expansion of web-based datasets like ImageNet [3] promises to produce labeled data for almost any desired category, such large-scale supervised datasets may not include images of the category across all domains of practical interest. Earlier deep learning efforts addressed this challenge by learning layers in an unsupervised fashion using unlabeled data to discover salient mid-level structures [6, 8]. While such approaches are appealing, they have heretofore been unable to match the level of performance of supervised models, and unsupervised training of networks with the same level of depth as [17] remains a challenge.
+
+Unfortunately, image datasets are inherently biased [21]. Theoretical [2, 4] and practical results from [20, 21] have shown that supervised methods’ test error increases in proportion to the difference between the test and training input distribution. Many visual domain adaptation methods have been put forth to compensate for dataset bias [7, 22, 1, 20, 18, 16, 13, 12, 14, 15], but are limited to shallow models. Evaluation for image category classification across visually distinct domains has focused on the Office dataset, which contains 31 image categories and 3 domains [20]. Recently, [9] showed that using the deep mid-level features learned on ImageNet, instead of the more conventional bag-of-words features, effectively removed the bias in some of the domain adaptation settings in the Office dataset [20]. However, [9] limited their experiments to small-scale source domains found only in Office, and evaluated on only a subset of relevant layers.
+
+Yet until now, almost none of the previous domain adaptation studies used ImageNet as the source domain, nor utilized the full set of parameters of a deep CNN trained on source data. Recent work by Rodner et al. [19] attempted to adapt from ImageNet to the SUN dataset, but did not take advantage of deep convolutional features.
+
+In this paper, we ask the question: will deep models still suffer from dataset bias when trained with all layers of the CNN and a truly large scale source dataset? Here, we provide the first evaluation of domain adaptation with deep learned representations in its most natural setting, in which all of ImageNet is used as source data for a target category. We use the 1.2 million labeled images available in the 2012 ImageNet 1000-way classification dataset [3] to train the model in [17] and evaluate its generalization to the Office dataset. This constitutes a three orders of magnitude increase in source data compared to the several thousand images available for the largest domain in Office.
+
+We find that it is easier to adapt from ImageNet than from previous smaller source domains, but that dataset bias remains a major issue. Fine-tuning the parameters on the small amount of labeled target data (we consider one-shot adaptation) turns out to be unsurprisingly problematic. Instead, we propose a simple yet intuitive adaptation method: train a final domain-adapted classification “layer” using various layers of the pre-trained network as features, without any fine-tuning its parameters. We provide a comprehensive evaluation of existing methods for classifier adaptation as applied to each of the fully connected layers of the network, including the last, task-specific classification layer. When adapting from ImageNet to Office, it turns out to be possible to achieve target domain performance on par with source domain performance using only a single labeled example per target category.
+
+We examine both the setting where there are a few labeled examples from the target domain (supervised adaptation) and the setting where there are no labeled target examples (unsupervised adaptation). We also describe practical solutions for choosing between the various adaptation methods based on experimental constraints such as limited computation time.
+
+# 2 Background: Deep Domain Adaptation Approaches
+
+For our task we consider adapting between a large source domain and a target domain with few or or no labeled examples. A typical approach to domain adaptation or transfer learning with deep architectures is to take the representation learned via back-propagation on a large dataset, and then transfer the representation to a smaller dataset by fine-tuning, i.e. backpropagation at a lower learning rate [11, 23]. However, fine-tuning requires an ample amount of labeled target data and so should not be expected to work well when we consider the very sparse label condition, such as the one-shot learning scenario we evaluate below, where we have just one labeled example per category in the target domain.
+
+In fact, in our experiments under this setting, fine-tuning actually reduces performance. Specifically, on the ImageNet Webcam task reported in Section 4, using the final output layer as a predictor in the target domain received $6 6 \%$ accuracy, while using the final output layer after fine tuning produced a degraded accuracy of $61 \%$ .
+
+A separate method that was recently proposed for deep adaptation is called Deep Learning for domain adaptation by Interpolating between Domains (DLID) [5]. This method learns multiple unsupervised deep models directly on the source, target, and combined datasets and uses a representation which is the concatenation of the outputs of each model as its adaptation approach. While this was shown to be an interesting approach, it is limited by its use of unsupervised deep structures.
+
+In general, unsupervised deep convolutional models have been unable to achieve the performance of supervised deep CNNs. However, training a supervised deep model requires sufficient labeled data. Our insight is that the extensive labeled data available in the source domain can be exploited using a supervised model without requiring a significant amount of labeled target data.
+
+Therefore, we propose using a supervised deep source model with supervised or unsupervised adaptation algorithms that are applied to models learned on the target data directly. This hybrid approach will utilize the strong representation available from the supervised deep model trained on a large source dataset while requiring only enough target labeled data to train a shallow model with far fewer parameters. Specifically, we consider training a convolutional neural network (CNN) on the source domain and using that network to extract features on the target data that can then be used to train an auxiliary shallow learner. For extracting features from the deep source model, we follow the setup of Donahue et al. [9], which extracts a visual feature $D e C A F$ from the ImageNet-trained architecture of [17].
+
+# 3 Adapting Deep CNNs with Few Labeled Target Examples
+
+We propose a general framework for selectively adapting the parameters of a convolutional neural network (CNN) whose representation and classifier weights are trained on a large-scale source domain, such as ImageNet. Our framework adds a final domain-adaptive classification “layer” that takes the activations of one of the existing network’s layers as input features. Note that the network cannot be effectively fine-tuned without access to more labeled target data. This adapted layer is a linear classifier that combines source and target training data using an adaptation method. To demonstrate the generality of our framework, we select a representative set of popular linear classifier adaptation approaches that we empirically evaluate in Section 4. We separate our discussion into the set of supervised and unsupervised adaptation settings.
+
+Below we denote the features extracted over the source domain as $\boldsymbol { X }$ and the features extracted over the target domain as $\tilde { X }$ . Similarly, we denote the source domain image classifier as $\pmb { \theta }$ and the target domain image classifier as $\tilde { \theta }$ .
+
+# 3.1 Unsupervised Adaptation
+
+Many unsupervised adaptation techniques seek to minimize the distance between subspaces that represent the source and target domains. We denote these subspaces as $U$ and $\tilde { U }$ , respectively.
+
+GFK [12] The Geodesic Flow Kernel (GFK) method [12] is an unsupervised domain adaptation approach which seeks embeddings for the source and target points that minimize domain shift. Inputs to the method are $U$ and $\tilde { U }$ , lower-dimensional embeddings of the source and target domains (e.g. from principal component analysis). The method constructs the geodesic flow $\phi ( t )$ along the manifold of subspaces such that $U = \phi ( 0 )$ and $\tilde { U } = \phi ( 1 )$ . Finally, a transformation $G$ is constructed by computing $\begin{array} { r } { G = \int _ { 0 } ^ { 1 } \phi ( t ) \phi ( t ) ^ { \intercal } d t } \end{array}$ using a closed-form solution, and classification is performed by training an SVM on the source data $\boldsymbol { X }$ and transformed target data $G \tilde { X }$ .
+
+SA [10] The Subspace Alignment (SA) method [10] also begins with low-dimensional embeddings of the source and target domains $U$ and $\tilde { U }$ , respectively. It seeks to minimize in $M$ , a transformation matrix, the objective $\| U M - \tilde { U } \| _ { F } ^ { 2 }$ . The analytical solution to this objective is $M ^ { * } = U ^ { \boldsymbol { \mathsf { T } } } \tilde { U }$ . Given $M ^ { * }$ , an SVM is trained on source data $\boldsymbol { X }$ and transformed target data $U M ^ { * } \tilde { U } ^ { \dagger } \tilde { X }$ .
+
+# 3.2 Supervised Adaptation
+
+Late Fusion Perhaps the simplest supervised adaptation method is to independently train a source and target classifier and combine the scores of the two to create a final scoring function. We call this approach Late Fusion. It has been explored by many for a simple adaptation approach. Let us denote the score from the source classifier as $v _ { s }$ and the score from the target classifier as $v _ { t }$ . For our experiments we explore two methods of combining these scores, which are described below:
+
+• Max: Produce the scores of both the source and target classifier and simply choose the max of the two as the final score for each example. Therefore, $v _ { \mathrm { a d a p t } } = \operatorname* { m a x } ( v _ { s } , v _ { t } )$ . • Linear Interpolation: Set the score for a particular example to equal the convex combination of the source and target classifier scores, $v _ { \mathrm { a d a p t } } = ( 1 - \alpha ) v _ { s } + \alpha v _ { t }$ . This method requires setting a hyperparameter, $\alpha$ , which determines the weights of the source and target classifiers.
+
+Late Fusion has two major advantages: it is easy to implement, and the source classifier it uses may be precomputed to make adaptation very fast. In the case of the linear interpolation combination rule, however, this method can potentially suffer from having a sensitive hyperparameter. We show a hyperparameter analysis in Section 4.
+
+Daume III [7] ´ This simple feature replication method was proposed for domain adaptation by [7]. The method augments feature vectors with a source component, a target component, and a shared component. Each source data point $_ { \textbf { \em x } }$ is augmented to $\bar { \mathbf { x ^ { \prime } } } = ( \mathbf { x } ; x ; \mathbf { 0 } )$ , and each target data point $\tilde { \pmb { x } }$ is augmented to $\tilde { \pmb { x } } ^ { \prime } = ( \tilde { \pmb { x } } ; \mathbf { 0 } ; \tilde { \pmb { x } } )$ . Finally, an SVM is trained on the augmented source and target data—a relatively expensive procedure given the potentially large size of the source domain and the tripled augmented feature dimensionality.
+
+PMT [1] This classifier adaptation method, Projective Model Transfer (PMT), proposed by [1], is a variant of adaptive SVM. It takes as input a classifier $\pmb \theta$ pre-trained on the source domain. PMTSVM learns a target domain classifier $\tilde { \pmb { \theta } }$ by adding an extra term to the usual SVM objective which regularizes the angle $\begin{array} { r } { \alpha ( \tilde { \theta } , \theta ) = \cos ^ { - 1 } \left( \frac { \theta ^ { \top } \tilde { \theta } } { \lVert \theta \rVert \lVert \tilde { \theta } \rVert } \right) } \end{array}$ between the target and source hyperplanes. This results in the following loss function:
+
+$$
+\mathcal { L } _ { P M T } ( \tilde { \pmb { \theta } } ) = \frac { 1 } { 2 } \| \tilde { \pmb { \theta } } \| _ { 2 } ^ { 2 } + \frac { \Gamma } { 2 } \| \tilde { \pmb { \theta } } \| _ { 2 } ^ { 2 } \sin ^ { 2 } \alpha ( \tilde { \pmb { \theta } } , \pmb { \theta } ) + \ell _ { h i n g e } ( \tilde { \pmb { X } } , \tilde { \pmb { Y } } ; \tilde { \pmb { \theta } } ) \ ,
+$$
+
+where $\ell _ { h i n g e } ( X , Y ; \pmb \theta )$ denotes the SVM hinge loss of a data matrix $\boldsymbol { X }$ , label vector $\mathbf { Y }$ , and classifier hyperplane $\pmb \theta$ , and $\Gamma$ is a hyperparameter which, as it increases, enforces more transfer from the source classifier.
+
+MMDT [15] The Max-margin Domain Transforms (MMDT) method from [15] jointly optimizes an SVM-like objective over a feature transformation matrix $A$ mapping target points to the source feature space and classifier parameters $\pmb { \theta }$ in the source feature space. In particular, MMDT minimizes the following loss function (assuming a binary classification task to simplify notation, and with $\ell _ { h i n g e }$ defined as in PMT):
+
+$$
+\mathcal { L } _ { M M D T } ( \pmb { \theta } , A ) = \frac { 1 } { 2 } \| \pmb { \theta } \| _ { 2 } ^ { 2 } + \frac { 1 } { 2 } \| A - I \| _ { F } ^ { 2 } + C _ { s } \ell _ { h i n g e } ( \pmb { X } , \pmb { Y } ; \pmb { \theta } ) + C _ { t } \ell _ { h i n g e } ( A \tilde { \pmb { X } } , \tilde { \pmb { Y } } ; \pmb { \theta } ) ,
+$$
+
+where $C _ { s }$ and $C _ { t }$ are hyperparameters controlling the importance of correctly classifying the source and target points (respectively).
+
+# 4 Evaluation
+
+# 4.1 Datasets
+
+The Office [20] dataset is a collection of images from three distinct domains: Amazon, DSLR, and Webcam. The 31 categories in the dataset consist of objects commonly encountered in office settings, such as keyboards, file cabinets, and laptops. Of these 31 categories, 16 overlap with the categories present in the 1000-category ImageNet classification task1. Thus, for our experiments, we limit ourselves to these 16 classes. In our experiments using Amazon as a source domain, we follow the standard training protocol for this dataset of using 20 source examples per category [20, 12], for a total of 320 images.
+
+ImageNet [3] is the largest available dataset of image category labels. We use 1000 categories’ worth of data (1.2M images) to train the network, and use the 16 categories that overlap with Office (approximately 1200 examples per category or ${ \approx } 2 0 \mathrm { K }$ images total) as labeled source classifier data.
+
+# 4.2 Experimental Setup & Baselines
+
+For our experiments, we use the fully trained deep CNN model described in Section 2, extracting feature representations from three different layers of the CNN. We then train a source classifier using these features on one of two source domains, and adapt to the target domain.
+
+The source domains we consider are either the Amazon domain, or the corresponding 16-category ImageNet subset where each category has many more examples. We focus on the Webcam domain as our target (test) domain, as Amazon-to-Webcam was shown to be the only challenging shift in [9] (the DSLR domain is much more similar to Webcam and did not require adaptation when using deep mid-level features). This combination exemplifies the shift from online web images to realworld images taken in typical office/home environments. Note that, regardless of the source domain chosen to learn the classifier, ImageNet data from all 1000 categories was used to train the network.
+
+In addition, for the supervised adaptation setting we assume access to only a single example per category from the target domain (Webcam).
+
+Each method is then evaluated across 20 random train/test splits, and we report averages and standard errors for each setting. For each random train/test split we choose one example for training and 10 other examples for testing (so there is a balanced test set across categories). Therefore, each test split has 160 examples. The unsupervised adaptation methods operate in a transductive setting, so the target subspaces are learned from the unlabeled test data.
+
+Non-adaptive Baselines In addition to the adaptation methods outlined in Section 3, we also evaluate using the following non-adaptive baselines.
+
+• SVM (source only): A support vector machine trained only on source data. • SVM (target only): A support vector machine trained only on target data. • SVM (source and target): A support vector machine trained on both source and target data. To account for the large discrepancy between the number of training data points in the source and target domains, we weighted the data points such that the constraints from the source and target domains effectively contribute equally to the optimization problem. Specifically, each source data point receives a weight of $\frac { n _ { t } } { n _ { s } + n _ { t } }$ , and each target data point receives a weight of $\frac { n _ { s } } { n _ { s } + n _ { t } }$ , where $n _ { s } , n _ { t }$ denote the number of data points in the source and target, respectively.
+
+Many of the adaptation methods we evaluate have hyperparameters that must be cross-validated for use in practice, so we set the parameters of the adaptation techniques as follows.
+
+First, the C value used for C-SVM in the classifier for all methods is set to $C = 1$ . Without any validation data we are not able to tune this parameter properly, so we choose to leave it as the default value. Since all methods we report require setting of this parameter, we feel that the relative comparisons between methods is sound even if the absolute numbers could be improved with a new setting for C. For Daume III and MMDT, which look at the source and target data simultaneously, ´ we use the same weighting scheme as we did for the source and target SVM. Late Fusion with the linear interpolation combination rule is reported across hyperparameter settings in Figure 1(a) to help understand how performance varies as we trade off emphasis between the learned classifiers from the source and target domains. Again, we do not have the validation data to tune this parameter so we report in the tables the performance averaged across parameter settings. The plot vs $\alpha$ indicates that there is usually a best parameter setting that could be learned with more available data. For PMT, we choose $\Gamma = 1 0 0 0$ , which corresponds to allowing a large amount of transfer from the source classifier to the target classifier. We do this because the source-only classifier is stronger than the target-only classifier (with ImageNet source). For the unsupervised methods GFK and SA, again we evaluated a variety of subspace dimensionalities and Figure 1(b) shows that the overall method performance does not vary significantly with the dimensionality choice.
+
+<table><tr><td>Adaptation Method</td><td>Training Data</td><td>DeCAF6</td><td>DeCAF7</td></tr><tr><td>SVM (source only)</td><td>Amazon</td><td>50.28 ± 1.8</td><td>54.08 ± 1.7</td></tr><tr><td>SVM (target only)</td><td>Webcam</td><td>62.28 ± 1.8</td><td>64.97 ± 1.8</td></tr><tr><td>GFK[12]</td><td>Amazon</td><td>53.13 ±1.1</td><td>53.39 ± 1.1</td></tr><tr><td>SA [10]</td><td>Amazon</td><td>51.74 ± 1.2</td><td>53.86 ± 1.0</td></tr><tr><td>SVM (source and target)</td><td>Amazon+Webcam</td><td>62.91 ±1.8</td><td>65.82 ± 1.4</td></tr><tr><td>Late Fusion (Max)</td><td>Amazon+Webcam</td><td>65.35 ± 1.7</td><td>58.42 ± 1.1</td></tr><tr><td>Late Fusion (Lin. Int. Avg)</td><td>Amazon+Webcam</td><td>63.23 ± 1.4</td><td>64.29 ± 1.3</td></tr><tr><td>Daumé III [7]</td><td>Amazon+Webcam</td><td>68.89 ± 1.9</td><td>72.09 ± 1.4</td></tr><tr><td>PMT[1]</td><td>Amazon+Webcam</td><td>64.84 ± 1.5</td><td>65.63 ± 1.8</td></tr><tr><td>MMDT[15]</td><td>Amazon+Webcam</td><td>65.47 ± 1.8</td><td>68.10 ± 1.5</td></tr><tr><td>Late Fusion (Lin. Int. Oracle)</td><td>Amazon+Webcam</td><td>71.1 ± 1.7</td><td>72.82 ±1.4</td></tr></table>
+
+Table 1: Amazon Webcam adaptation experiment. We show here multiclass accuracy on the target domain test set for both supervised and unsupervised adaptation experiments across the two fully connected layer features (similar to [9], but with one labeled target example). The best performing unsupervised adaptation algorithms are shown in blue and the best performing supervised adaptation algorithms are shown in red.
+
+# 4.3 Effect of Source Domain Size
+
+Previous studies considered source domains from the Office dataset. In this section, we ask what happens when an orders-of-magnitute larger source dataset is used.
+
+For completeness we begin by evaluating Amazon as a source domain. Preliminary results on this setting are reported in [9], here we extend the comparison here by presenting the results with more adaptation algorithms and more complete evaluation of hyperparameter settings. Table 1 presents multiclass accuracies for each algorithm using either layer 6 or 7 from the deep network, which corresponds to the output from each of the fully connected layers.
+
+An SVM trained using only Amazon data achieves $7 8 . 6 \%$ in-domain accuracy (tested on the same domain) when using the ${ \mathrm { D e C A F } } _ { 6 }$ feature and $8 0 . 2 \%$ in-domain accuracy when using the ${ \mathrm { D e C A F } } _ { 7 }$ feature. These numbers are significantly higher than the performance of the same classifier on Webcam test data, indicating that even with the DeCAF features, there is a still a domain shift between the Amazon and Webcam datasets.
+
+Next, we consider an unsupervised adaptation setting where no labeled examples are available from the target dataset. In this scenario, we apply two state-of-the-art unsupervised adaptation methods, GFK [12] and SA [10]. Both of these methods make use of a subspace dimensionality hyperparameter. We show the results using a 100-dimensional subspace and leave the discussion of setting this parameter until Section 4.6. For this shift the adaptation algorithms increase performance when using the layer 6 feature, but offer no additional improvement when using the layer 7 feature.
+
+We finally assume that a single example per category is available in the target domain. As the bottom rows of Table 1 show, supervised adaptation algorithms are able to provide significant improvement regardless of the feature space chosen, even in the one-shot scenario. For this experiment we noticed that using the second fully connected layer $( \mathrm { D e C A F } _ { 7 }$ ) was a stronger overall feature in general.
+
+# 4.4 Adapting with a Large Scale Source Domain
+
+We next address one of the main questions of this paper: Is there still a domain shift when using a large source dataset such as ImageNet? To begin to answer this question we follow the same experimental paradigm as the previous experiment, but use ImageNet as our source dataset. The results are shown in Table 2.
+
+Again, we first verify that the source only SVM achieves higher performance when tested on indomain data than on Webcam data. Indeed, for the 16 overlapping labels, the source SVM produces $6 2 . 5 0 \%$ accuracy on ImageNet data using ${ \mathrm { D e C A F } } _ { 6 }$ features and $7 4 . 5 0 \%$ accuracy when using ${ \mathrm { D e C A F } } _ { 7 }$ features. Compare this to the $54 \%$ and $59 \%$ for Webcam evaluation and a dataset bias is still clearly evident.
+
+Table 2: ImageNet Webcam adaptation experiment. Comparison of unsupervised and supervised adaptation algorithms on the ImageNet to Webcam domain shift. Results are computed using the outputs of each of the fully connected layers as features. The best supervised adaptation performance is indicated in red and the best unsupervised adaptation performance is highlighted in blue.   
+
+<table><tr><td>Adaptation Method</td><td>Training Data</td><td>DeCAF6</td><td>DeCAF7</td></tr><tr><td>SVM (source only)</td><td>ImageNet</td><td>53.51 ± 1.1</td><td>59.15 ± 1.1</td></tr><tr><td>SVM (target only)</td><td>Webcam</td><td>62.28 ± 1.8</td><td>64.97 ± 1.8</td></tr><tr><td>GFK [12]</td><td>ImageNet</td><td>65.16 ± 1.1</td><td>67.97 ± 1.4</td></tr><tr><td>SA [10]</td><td>ImageNet</td><td>59.30 ± 1.4</td><td>66.08 ± 1.4</td></tr><tr><td>SVM (source and target)</td><td>ImageNet+Webcam</td><td>56.68 ± 1.2</td><td>66.93 ± 1.3</td></tr><tr><td>Late Fusion (Max)</td><td>ImageNet+Webcam</td><td>59.59 ± 1.3</td><td>68.86 ± 1.2</td></tr><tr><td>Late Fusion (Lin. Int. Avg)</td><td>ImageNet+Webcam</td><td>60.64 ± 1.3</td><td>66.45 ± 1.1</td></tr><tr><td>Daumé III [7]</td><td>ImageNet+Webcam</td><td>59.21 ± 1.7</td><td>71.39 ± 1.5</td></tr><tr><td>PMT[1]</td><td>ImageNet+Webcam</td><td>66.30 ± 2.1</td><td>69.81 ± 1.8</td></tr><tr><td>MMDT[15]</td><td>ImageNet+Webcam</td><td>59.21 ± 1.3</td><td>67.75 ± 1.4</td></tr><tr><td>Late Fusion (Lin. Int. Oracle)</td><td>ImageNet+Webcam</td><td>71.65 ± 2.0</td><td>76.76 ± 1.3</td></tr></table>
+
+Table 3: ImageNet Webcam and Amazon Webcam adaptation experiments using $\mathrm { D e C A F _ { 8 } }$ , the label activations of the CNN trained on the full ImageNet data. Again, we compare multiclass accuracy of various unsupervised and supervised adaptation methods. The best performing unsupervised adaptation algorithm is shown in blue and the best performing supervised adaptation algorithms are shown in red.   
+
+<table><tr><td>Adaptation Method</td><td>Training Data</td><td>Source=ImageNet</td><td>Source=Amazon</td></tr><tr><td>SVM (source only)</td><td>Source</td><td>66.23 ± 0.8</td><td>53.23 ± 1.6</td></tr><tr><td>SVM (target only)</td><td>Webcam</td><td>63.13 ± 1.9</td><td>63.13 ± 1.9</td></tr><tr><td>GFK[12]</td><td>Source</td><td>68.73 ±1.1</td><td>54.56 ± 1.2</td></tr><tr><td>SA[10]</td><td>Source</td><td>66.08 ± 1.1</td><td>55.98 ± 1.0</td></tr><tr><td> SVM (source and target)</td><td>Source+Webcam</td><td>75.13 ± 1.1</td><td>63.20 ± 1.7</td></tr><tr><td>Late Fusion (Max)</td><td>Source+Webcam</td><td>71.77 ± 1.4</td><td>62.25 ± 0.8</td></tr><tr><td>Late Fusion (LinInt Avg)</td><td>Source+Webcam</td><td>70.56 ± 1.2</td><td>64.56 ± 1.3</td></tr><tr><td>Daumé III [7]</td><td>Source+Webcam</td><td>77.15 ± 1.1</td><td>70.51 ± 1.7</td></tr><tr><td>PMT[1]</td><td>Source+Webcam</td><td>70.28 ± 1.8</td><td>66.77 ± 2.1</td></tr><tr><td>MMDT[15]</td><td>Source+Webcam</td><td>73.96 ± 1.2</td><td>66.23 ± 1.4</td></tr><tr><td>Late Fusion (Lin. Int. Oracle)</td><td>Source+Webcam</td><td>76.61 ± 1.5</td><td>71.49 ± 1.3</td></tr></table>
+
+Note that when using ImageNet as a source domain, overall performance of all algorithms improves. In addition, unsupervised adaptation approaches are more effective than for the smaller source domain experiment.
+
+# 4.5 Adapting a Pre-trained Classifier to a New Label Set
+
+$\mathrm { D e C A F _ { 8 } }$ differs from the other DeCAF features in that it constitutes the 1000 activations corresponding to the 1000 labels in the ImageNet classification task. In the CNN proposed by [17], these activations are fed into a softmax unit to compute the label probabilities. We instead experiment with using the $\mathrm { D e C A F _ { 8 } }$ activations directly as a feature representation, which is akin to training another classifier using the output of the 1000-way CNN classifier.
+
+Table 3 shows results for various adaptation techniques using both ImageNet and Amazon as source domains. We use the same setup as before, but instead use $\mathrm { D e C A F _ { 8 } }$ as the feature representation.
+
+The ImageNet results are uniformly better with $\mathrm { D e C A F _ { 8 } }$ than with ${ \mathrm { D e C A F } } _ { 6 }$ or ${ \mathrm { D e C A F } } _ { 7 }$ , likely due to the fact that $\mathrm { D e C A F _ { 8 } }$ was explicitly trained on ImageNet data to effectively discriminate between ImageNet categories. Because it can more effectively classify images from the source domain, it is able to better adapt from the source domain to the target domain.
+
+However, we see a negligible difference in performance for Amazon, with performance actually decreasing with respect to ${ \mathrm { D e C A F } } _ { 7 }$ for certain adaptation methods. We believe this is because the final activation vector is too specific to the 1000-way ImageNet task, and that ${ \mathrm { D e C A F } } _ { 7 }$ provides a more general representation that is better suited to the Amazon domain. This, in turn, results in improved adaptation. In general, however, the difference between the various DeCAF representations with Amazon as a source are small enough to be insignificant.
+
+# 4.6 Analysis and Practical Considerations
+
+Our adaptation experiments show that, despite its large size, even ImageNet is not large enough to cover all domains, and that traditional domain adaptation methods go a long way in increasing performance and mitigating the effects of this shift. Depending on the characteristics of the problem at hand, our results suggest different methods may be most suitable.
+
+If no labels exist in the target domain, then there are unsupervised adaptation algorithms that are easy to use and fast to compute at adaptation time, yet still achieve increased performance over sourceonly methods. For this scenario, we experimented with two subspace alignment based methods that both require setting a parameter that indicates the dimensionality of the input subspaces. Figure 1(b) shows the effect that changing the subspace dimensionality has on the overall method performance. In general, we noticed that these methods were not particularly sensitive to this parameter so long as the dimensionality remains larger than the number of categories in our label set. Below this threshold, the subspace is less likely to capture all important discriminative information needed for classification.
+
+In the case where we have a large source dataset and a limited number of labeled target examples, it may be preferable to compute source classifier parameters in advance, then examine only the source parameters and the target data at adaptation time. Examples of these kinds of methods are Late Fusion and PMT. These methods are unaffected by the number of data points in the source domain at adaptation time, and can thus be applied quickly. In our experiments, we found that a properly tuned Late Fusion classifier with linear interpolation was the fastest and most effective approach. Figure 1(a) shows the performance of linear interpolation Late Fusion as we vary the hyperparameter $\alpha$ . Although the method is sensitive to $\alpha$ , we found that for both source domains, the basic strategy of setting $\alpha$ around 0.8 provides a close approximation to optimal performance. This setting can be interpreted as trusting the target classifier more than the source, but not so much as to completely discount the information available from the source classifier. In each table we report both the performance of linear interpolation both averaged across hyper parameter settings $\bar { \alpha \in [ 0 , 1 ] }$ as well as the performance of linear interpolation with the best possible setting of $\alpha$ per experiment – this is denoted as “Oracle” performance.
+
+If there are no computational constraints and there are very few labels in the target domain, the best-performing method seems to be the “frustratingly easy” approach originally proposed by Daume III [7] and applied again for deep models in [5]. ´
+
+Finally, we found that feature representation can have a significant impact on adaptation performance. Our results show that ImageNet as source performs best with the $\mathrm { D e C A F _ { 8 } }$ representation, whereas Amazon as source performs best with the ${ \mathrm { D e C A F } } _ { 7 }$ representation. This, combined with our intuition, seems to indicate that for adaptation from source domains other than ImageNet, an intermediate representation other than $\mathrm { D e C A F _ { 8 } }$ is more powerful for adaptation, whereas ImageNet classification works best with the full representation that was trained on it.
+
+# 5 Conclusion
+
+In this paper, we presented the first evaluation of domain adaptation from a large-scale source dataset with deep features. We demonstrated that, although using ImageNet as a source domain generalizes better than other smaller source domains, there is still a domain shift when adapting to other visual domains.
+
+![](images/3bc68fe9e1a528a694c293fb0f4369506b121f899b4f01134593f6e12b3afc0f.jpg)  
+Figure 1: Evaluation of hyperparameters for domain adaptation methods. (a) Analysis of the combination hyperparameter $\alpha$ for Late Fusion with linear interpolation. (b) Analysis of the subspace dimensionality for the unsupervised adaptation algorithms
+
+Our experimental results show that deep adaptation methods can go a long way in mitigating the effects of this domain shift. Based on our results, we also provided a set of practical recommendations for choosing a feature representation and adaptation method accounting for constraints on runtime and accuracy.
+
+There are a number of interesting directions to take given our results. First we notice that though $\mathrm { D e C A F _ { 8 } }$ is the strongest feature to use for learning a classifier on ImageNet data, ${ \mathrm { D e C A F } } _ { 7 }$ is actually a better feature to use with the Amazon source domain and the Webcam target domain. This could lead to a hybrid approach where one uses different feature representations for the various domains and produces a combined adapted model. Another interesting direction that should be explored is to integrate the adaption algorithms into the deep models explicitly and even allow for feedback between the two stages. Current deep models although allow information flow between the final classifier and the representation learning architecture. We feel that the next step is to have a separate task specific adaptable layer that does not simply learn a new final layer, but instead learns a separate, but equivalent final layer, that is regularized by the final layer learned on the source dataset.
+
+This future work is a natural extension of the result we have shown in this paper: that pre-trained deep representations with large source domains can be effectively adapted to new target domains using only shallow, linear adaptation methods, and that in cases where the target data is limited, this approach is the best way to mitigate dataset bias.
+
+# References
+
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+[2] Shai Ben-David, John Blitzer, Koby Crammer, Fernando Pereira, et al. Analysis of representations for domain adaptation. Proc. NIPS, 2007.   
+[3] A. Berg, J. Deng, and L. Fei-Fei. ImageNet Large Scale Visual Recognition Challenge 2012. 2012.   
+[4] John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman. Learning bounds for domain adaptation. In Proc. NIPS, 2007.   
+[5] S. Chopra, S. Balakrishnan, and R. Gopalan. DLID: Deep learning for domain adaptation by interpolating between domains. In ICML Workshop on Challenges in Representation Learning, 2013.   
+[6] A. Coates, A. Karpathy, and A. Ng. Emergence of object-selective features in unsupervised feature learning. In Proc. NIPS, 2012.   
+[7] H. Daume III. Frustratingly easy domain adaptation. In ´ ACL, 2007.   
+[8] J. Dean, G. Corrado, R. Monga, K. Chen, M. Devin, Q. Le, M. Mao, M. Ranzato, A. Senior, P. Tucker, K. Yang, and A. Ng. Large scale distributed deep networks. In Proc. NIPS, 2012.   
+[9] J. Donahue, Y. Jia, O. Vinyals, J. Hoffman, N. Zhang, E. Tzeng, and T. Darrell. DeCAF: A Deep Convolutional Activation Feature for Generic Visual Recognition. arXiv e-prints, 2013.   
+[10] B. Fernando, A. Habrard, M. Sebban, and T. Tuytelaars. Unsupervised visual domain adaptation using subspace alignment. In Proc. ICCV, 2013.   
+[11] R. Girshick, J. Donahue, T. Darrell, and J. Malik. Rich feature hierarchies for accurate object detection and semantic segmentation. arXiv e-prints, 2013.   
+[12] B. Gong, Y. Shi, F. Sha, and K. Grauman. Geodesic flow kernel for unsupervised domain adaptation. In Proc. CVPR, 2012.   
+[13] R. Gopalan, R. Li, and R. Chellappa. Domain adaptation for object recognition: An unsupervised approach. In Proc. ICCV, 2011.   
+[14] J. Hoffman, B. Kulis, T. Darrell, and K. Saenko. Discovering latent domains for multisource domain adaptation. In Proc. ECCV, 2012.   
+[15] J. Hoffman, E. Rodner, J. Donahue, K. Saenko, and T. Darrell. Efficient learning of domain-invariant image representations. In Proc. ICLR, 2013.   
+[16] A. Khosla, T. Zhou, T. Malisiewicz, A. Efros, and A. Torralba. Undoing the damage of dataset bias. In Proc. ECCV, 2012.   
+[17] A. Krizhevsky, I. Sutskever, and G. E. Hinton. ImageNet classification with deep convolutional neural networks. In Proc. NIPS, 2012.   
+[18] B. Kulis, K. Saenko, and T. Darrell. What you saw is not what you get: Domain adaptation using asymmetric kernel transforms. In Proc. CVPR, 2011.   
+[19] Erik Rodner, Judy Hoffman, Jeff Donahue, Trevor Darrell, and Kate Saenko. Towards adapting imagenet to reality: Scalable domain adaptation with implicit low-rank transformations. CoRR, abs/1308.4200, 2013.   
+[20] K. Saenko, B. Kulis, M. Fritz, and T. Darrell. Adapting visual category models to new domains. In Proc. ECCV, 2010.   
+[21] A. Torralba and A. Efros. Unbiased look at dataset bias. In Proc. CVPR, 2011.   
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+[23] M. Zeiler and R. Fergus. Visualizing and Understanding Convolutional Networks. ArXiv e-prints, 2013.

parse/train/tPCrkaLa9Y5ld/tPCrkaLa9Y5ld_content_list.json

@@ -0,0 +1,1040 @@
+[
+    {
+        "type": "text",
+        "text": "One-Shot Adaptation of Supervised Deep Convolutional Models ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Judy Hoffman, Eric Tzeng, Jeff Donahue UC Berkeley, EECS & ICSI {jhoffman,etzeng,jdonahue}@eecs.berkeley.edu ",
+        "bbox": [
+            207,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yangqing Jia∗ Google Research jiayq@google.com ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kate Saenko UMass Lowell, CS & ICSI saenko@cs.uml.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Trevor Darrell UC Berkeley, EECS & ICSI trevor@eecs.berkeley.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            376,
+            535,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dataset bias remains a significant barrier towards solving real world computer vision tasks. Though deep convolutional networks have proven to be a competitive approach for image classification, a question remains: have these models have solved the dataset bias problem? In general, training or fine-tuning a state-ofthe-art deep model on a new domain requires a significant amount of data, which for many applications is simply not available. Transfer of models directly to new domains without adaptation has historically led to poor recognition performance. In this paper, we pose the following question: is a single image dataset, much larger than previously explored for adaptation, comprehensive enough to learn general deep models that may be effectively applied to new image domains? In other words, are deep CNNs trained on large amounts of labeled data as susceptible to dataset bias as previous methods have been shown to be? We show that a generic supervised deep CNN model trained on a large dataset reduces, but does not remove, dataset bias. Furthermore, we propose several methods for adaptation with deep models that are able to operate with little (one example per category) or no labeled domain specific data. Our experiments show that adaptation of deep models on benchmark visual domain adaptation datasets can provide a significant performance boost. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Supervised deep convolutional neural networks (CNNs) trained on large-scale classification tasks have been shown to learn impressive mid-level structures and obtain high levels of performance on contemporary classification challenges [3, 23]. These models generally assume extensive training using labeled data, and testing is limited to data from the same domain. In practice, however, the images we would like to classify are often produced under different imaging conditions or drawn from a different distribution, leading to a domain shift. Scaling such models to new domains remains an open challenge. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep CNNs require large amounts of training data to learn good mid-level convolutional models and final fully-connected classifier stages. While the continuing expansion of web-based datasets like ImageNet [3] promises to produce labeled data for almost any desired category, such large-scale supervised datasets may not include images of the category across all domains of practical interest. Earlier deep learning efforts addressed this challenge by learning layers in an unsupervised fashion using unlabeled data to discover salient mid-level structures [6, 8]. While such approaches are appealing, they have heretofore been unable to match the level of performance of supervised models, and unsupervised training of networks with the same level of depth as [17] remains a challenge. ",
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+        "text": "",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Unfortunately, image datasets are inherently biased [21]. Theoretical [2, 4] and practical results from [20, 21] have shown that supervised methods’ test error increases in proportion to the difference between the test and training input distribution. Many visual domain adaptation methods have been put forth to compensate for dataset bias [7, 22, 1, 20, 18, 16, 13, 12, 14, 15], but are limited to shallow models. Evaluation for image category classification across visually distinct domains has focused on the Office dataset, which contains 31 image categories and 3 domains [20]. Recently, [9] showed that using the deep mid-level features learned on ImageNet, instead of the more conventional bag-of-words features, effectively removed the bias in some of the domain adaptation settings in the Office dataset [20]. However, [9] limited their experiments to small-scale source domains found only in Office, and evaluated on only a subset of relevant layers. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Yet until now, almost none of the previous domain adaptation studies used ImageNet as the source domain, nor utilized the full set of parameters of a deep CNN trained on source data. Recent work by Rodner et al. [19] attempted to adapt from ImageNet to the SUN dataset, but did not take advantage of deep convolutional features. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In this paper, we ask the question: will deep models still suffer from dataset bias when trained with all layers of the CNN and a truly large scale source dataset? Here, we provide the first evaluation of domain adaptation with deep learned representations in its most natural setting, in which all of ImageNet is used as source data for a target category. We use the 1.2 million labeled images available in the 2012 ImageNet 1000-way classification dataset [3] to train the model in [17] and evaluate its generalization to the Office dataset. This constitutes a three orders of magnitude increase in source data compared to the several thousand images available for the largest domain in Office. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We find that it is easier to adapt from ImageNet than from previous smaller source domains, but that dataset bias remains a major issue. Fine-tuning the parameters on the small amount of labeled target data (we consider one-shot adaptation) turns out to be unsurprisingly problematic. Instead, we propose a simple yet intuitive adaptation method: train a final domain-adapted classification “layer” using various layers of the pre-trained network as features, without any fine-tuning its parameters. We provide a comprehensive evaluation of existing methods for classifier adaptation as applied to each of the fully connected layers of the network, including the last, task-specific classification layer. When adapting from ImageNet to Office, it turns out to be possible to achieve target domain performance on par with source domain performance using only a single labeled example per target category. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We examine both the setting where there are a few labeled examples from the target domain (supervised adaptation) and the setting where there are no labeled target examples (unsupervised adaptation). We also describe practical solutions for choosing between the various adaptation methods based on experimental constraints such as limited computation time. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 Background: Deep Domain Adaptation Approaches ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For our task we consider adapting between a large source domain and a target domain with few or or no labeled examples. A typical approach to domain adaptation or transfer learning with deep architectures is to take the representation learned via back-propagation on a large dataset, and then transfer the representation to a smaller dataset by fine-tuning, i.e. backpropagation at a lower learning rate [11, 23]. However, fine-tuning requires an ample amount of labeled target data and so should not be expected to work well when we consider the very sparse label condition, such as the one-shot learning scenario we evaluate below, where we have just one labeled example per category in the target domain. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In fact, in our experiments under this setting, fine-tuning actually reduces performance. Specifically, on the ImageNet Webcam task reported in Section 4, using the final output layer as a predictor in the target domain received $6 6 \\%$ accuracy, while using the final output layer after fine tuning produced a degraded accuracy of $61 \\%$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A separate method that was recently proposed for deep adaptation is called Deep Learning for domain adaptation by Interpolating between Domains (DLID) [5]. This method learns multiple unsupervised deep models directly on the source, target, and combined datasets and uses a representation which is the concatenation of the outputs of each model as its adaptation approach. While this was shown to be an interesting approach, it is limited by its use of unsupervised deep structures. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "In general, unsupervised deep convolutional models have been unable to achieve the performance of supervised deep CNNs. However, training a supervised deep model requires sufficient labeled data. Our insight is that the extensive labeled data available in the source domain can be exploited using a supervised model without requiring a significant amount of labeled target data. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we propose using a supervised deep source model with supervised or unsupervised adaptation algorithms that are applied to models learned on the target data directly. This hybrid approach will utilize the strong representation available from the supervised deep model trained on a large source dataset while requiring only enough target labeled data to train a shallow model with far fewer parameters. Specifically, we consider training a convolutional neural network (CNN) on the source domain and using that network to extract features on the target data that can then be used to train an auxiliary shallow learner. For extracting features from the deep source model, we follow the setup of Donahue et al. [9], which extracts a visual feature $D e C A F$ from the ImageNet-trained architecture of [17]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Adapting Deep CNNs with Few Labeled Target Examples ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            340,
+            686,
+            358
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose a general framework for selectively adapting the parameters of a convolutional neural network (CNN) whose representation and classifier weights are trained on a large-scale source domain, such as ImageNet. Our framework adds a final domain-adaptive classification “layer” that takes the activations of one of the existing network’s layers as input features. Note that the network cannot be effectively fine-tuned without access to more labeled target data. This adapted layer is a linear classifier that combines source and target training data using an adaptation method. To demonstrate the generality of our framework, we select a representative set of popular linear classifier adaptation approaches that we empirically evaluate in Section 4. We separate our discussion into the set of supervised and unsupervised adaptation settings. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Below we denote the features extracted over the source domain as $\\boldsymbol { X }$ and the features extracted over the target domain as $\\tilde { X }$ . Similarly, we denote the source domain image classifier as $\\pmb { \\theta }$ and the target domain image classifier as $\\tilde { \\theta }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Unsupervised Adaptation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            568,
+            390,
+            582
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Many unsupervised adaptation techniques seek to minimize the distance between subspaces that represent the source and target domains. We denote these subspaces as $U$ and $\\tilde { U }$ , respectively. ",
+        "bbox": [
+            174,
+            594,
+            823,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "GFK [12] The Geodesic Flow Kernel (GFK) method [12] is an unsupervised domain adaptation approach which seeks embeddings for the source and target points that minimize domain shift. Inputs to the method are $U$ and $\\tilde { U }$ , lower-dimensional embeddings of the source and target domains (e.g. from principal component analysis). The method constructs the geodesic flow $\\phi ( t )$ along the manifold of subspaces such that $U = \\phi ( 0 )$ and $\\tilde { U } = \\phi ( 1 )$ . Finally, a transformation $G$ is constructed by computing $\\begin{array} { r } { G = \\int _ { 0 } ^ { 1 } \\phi ( t ) \\phi ( t ) ^ { \\intercal } d t } \\end{array}$ using a closed-form solution, and classification is performed by training an SVM on the source data $\\boldsymbol { X }$ and transformed target data $G \\tilde { X }$ . ",
+        "bbox": [
+            174,
+            632,
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+            741
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "SA [10] The Subspace Alignment (SA) method [10] also begins with low-dimensional embeddings of the source and target domains $U$ and $\\tilde { U }$ , respectively. It seeks to minimize in $M$ , a transformation matrix, the objective $\\| U M - \\tilde { U } \\| _ { F } ^ { 2 }$ . The analytical solution to this objective is $M ^ { * } = U ^ { \\boldsymbol { \\mathsf { T } } } \\tilde { U }$ . Given $M ^ { * }$ , an SVM is trained on source data $\\boldsymbol { X }$ and transformed target data $U M ^ { * } \\tilde { U } ^ { \\dagger } \\tilde { X }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 Supervised Adaptation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            853,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Late Fusion Perhaps the simplest supervised adaptation method is to independently train a source and target classifier and combine the scores of the two to create a final scoring function. We call this approach Late Fusion. It has been explored by many for a simple adaptation approach. Let us denote the score from the source classifier as $v _ { s }$ and the score from the target classifier as $v _ { t }$ . For our experiments we explore two methods of combining these scores, which are described below: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Max: Produce the scores of both the source and target classifier and simply choose the max of the two as the final score for each example. Therefore, $v _ { \\mathrm { a d a p t } } = \\operatorname* { m a x } ( v _ { s } , v _ { t } )$ . • Linear Interpolation: Set the score for a particular example to equal the convex combination of the source and target classifier scores, $v _ { \\mathrm { a d a p t } } = ( 1 - \\alpha ) v _ { s } + \\alpha v _ { t }$ . This method requires setting a hyperparameter, $\\alpha$ , which determines the weights of the source and target classifiers. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Late Fusion has two major advantages: it is easy to implement, and the source classifier it uses may be precomputed to make adaptation very fast. In the case of the linear interpolation combination rule, however, this method can potentially suffer from having a sensitive hyperparameter. We show a hyperparameter analysis in Section 4. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Daume III [7] ´ This simple feature replication method was proposed for domain adaptation by [7]. The method augments feature vectors with a source component, a target component, and a shared component. Each source data point $_ { \\textbf { \\em x } }$ is augmented to $\\bar { \\mathbf { x ^ { \\prime } } } = ( \\mathbf { x } ; x ; \\mathbf { 0 } )$ , and each target data point $\\tilde { \\pmb { x } }$ is augmented to $\\tilde { \\pmb { x } } ^ { \\prime } = ( \\tilde { \\pmb { x } } ; \\mathbf { 0 } ; \\tilde { \\pmb { x } } )$ . Finally, an SVM is trained on the augmented source and target data—a relatively expensive procedure given the potentially large size of the source domain and the tripled augmented feature dimensionality. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "PMT [1] This classifier adaptation method, Projective Model Transfer (PMT), proposed by [1], is a variant of adaptive SVM. It takes as input a classifier $\\pmb \\theta$ pre-trained on the source domain. PMTSVM learns a target domain classifier $\\tilde { \\pmb { \\theta } }$ by adding an extra term to the usual SVM objective which regularizes the angle $\\begin{array} { r } { \\alpha ( \\tilde { \\theta } , \\theta ) = \\cos ^ { - 1 } \\left( \\frac { \\theta ^ { \\top } \\tilde { \\theta } } { \\lVert \\theta \\rVert \\lVert \\tilde { \\theta } \\rVert } \\right) } \\end{array}$ between the target and source hyperplanes. This results in the following loss function: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b99d14d5372910db55db88924c62fa62ecad3618069c447831362caea3468ca1.jpg",
+        "text": "$$\n\\mathcal { L } _ { P M T } ( \\tilde { \\pmb { \\theta } } ) = \\frac { 1 } { 2 } \\| \\tilde { \\pmb { \\theta } } \\| _ { 2 } ^ { 2 } + \\frac { \\Gamma } { 2 } \\| \\tilde { \\pmb { \\theta } } \\| _ { 2 } ^ { 2 } \\sin ^ { 2 } \\alpha ( \\tilde { \\pmb { \\theta } } , \\pmb { \\theta } ) + \\ell _ { h i n g e } ( \\tilde { \\pmb { X } } , \\tilde { \\pmb { Y } } ; \\tilde { \\pmb { \\theta } } ) \\ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            484,
+            714,
+            515
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\ell _ { h i n g e } ( X , Y ; \\pmb \\theta )$ denotes the SVM hinge loss of a data matrix $\\boldsymbol { X }$ , label vector $\\mathbf { Y }$ , and classifier hyperplane $\\pmb \\theta$ , and $\\Gamma$ is a hyperparameter which, as it increases, enforces more transfer from the source classifier. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "MMDT [15] The Max-margin Domain Transforms (MMDT) method from [15] jointly optimizes an SVM-like objective over a feature transformation matrix $A$ mapping target points to the source feature space and classifier parameters $\\pmb { \\theta }$ in the source feature space. In particular, MMDT minimizes the following loss function (assuming a binary classification task to simplify notation, and with $\\ell _ { h i n g e }$ defined as in PMT): ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/346101bc49ae7474678678a431bf5e466fea92ef01893d80cd01249f36e84562.jpg",
+        "text": "$$\n\\mathcal { L } _ { M M D T } ( \\pmb { \\theta } , A ) = \\frac { 1 } { 2 } \\| \\pmb { \\theta } \\| _ { 2 } ^ { 2 } + \\frac { 1 } { 2 } \\| A - I \\| _ { F } ^ { 2 } + C _ { s } \\ell _ { h i n g e } ( \\pmb { X } , \\pmb { Y } ; \\pmb { \\theta } ) + C _ { t } \\ell _ { h i n g e } ( A \\tilde { \\pmb { X } } , \\tilde { \\pmb { Y } } ; \\pmb { \\theta } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            647,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $C _ { s }$ and $C _ { t }$ are hyperparameters controlling the importance of correctly classifying the source and target points (respectively). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Evaluation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 Datasets ",
+        "text_level": 1,
+        "bbox": [
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+            763,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Office [20] dataset is a collection of images from three distinct domains: Amazon, DSLR, and Webcam. The 31 categories in the dataset consist of objects commonly encountered in office settings, such as keyboards, file cabinets, and laptops. Of these 31 categories, 16 overlap with the categories present in the 1000-category ImageNet classification task1. Thus, for our experiments, we limit ourselves to these 16 classes. In our experiments using Amazon as a source domain, we follow the standard training protocol for this dataset of using 20 source examples per category [20, 12], for a total of 320 images. ",
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+    {
+        "type": "text",
+        "text": "ImageNet [3] is the largest available dataset of image category labels. We use 1000 categories’ worth of data (1.2M images) to train the network, and use the 16 categories that overlap with Office (approximately 1200 examples per category or ${ \\approx } 2 0 \\mathrm { K }$ images total) as labeled source classifier data. ",
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+        "type": "text",
+        "text": "4.2 Experimental Setup & Baselines ",
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+        "text": "For our experiments, we use the fully trained deep CNN model described in Section 2, extracting feature representations from three different layers of the CNN. We then train a source classifier using these features on one of two source domains, and adapt to the target domain. ",
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+        "text": "The source domains we consider are either the Amazon domain, or the corresponding 16-category ImageNet subset where each category has many more examples. We focus on the Webcam domain as our target (test) domain, as Amazon-to-Webcam was shown to be the only challenging shift in [9] (the DSLR domain is much more similar to Webcam and did not require adaptation when using deep mid-level features). This combination exemplifies the shift from online web images to realworld images taken in typical office/home environments. Note that, regardless of the source domain chosen to learn the classifier, ImageNet data from all 1000 categories was used to train the network. ",
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+        "text": "In addition, for the supervised adaptation setting we assume access to only a single example per category from the target domain (Webcam). ",
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+        "type": "text",
+        "text": "Each method is then evaluated across 20 random train/test splits, and we report averages and standard errors for each setting. For each random train/test split we choose one example for training and 10 other examples for testing (so there is a balanced test set across categories). Therefore, each test split has 160 examples. The unsupervised adaptation methods operate in a transductive setting, so the target subspaces are learned from the unlabeled test data. ",
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+        "type": "text",
+        "text": "Non-adaptive Baselines In addition to the adaptation methods outlined in Section 3, we also evaluate using the following non-adaptive baselines. ",
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+        "type": "text",
+        "text": "• SVM (source only): A support vector machine trained only on source data. • SVM (target only): A support vector machine trained only on target data. • SVM (source and target): A support vector machine trained on both source and target data. To account for the large discrepancy between the number of training data points in the source and target domains, we weighted the data points such that the constraints from the source and target domains effectively contribute equally to the optimization problem. Specifically, each source data point receives a weight of $\\frac { n _ { t } } { n _ { s } + n _ { t } }$ , and each target data point receives a weight of $\\frac { n _ { s } } { n _ { s } + n _ { t } }$ , where $n _ { s } , n _ { t }$ denote the number of data points in the source and target, respectively. ",
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+        "type": "text",
+        "text": "Many of the adaptation methods we evaluate have hyperparameters that must be cross-validated for use in practice, so we set the parameters of the adaptation techniques as follows. ",
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+        "text": "First, the C value used for C-SVM in the classifier for all methods is set to $C = 1$ . Without any validation data we are not able to tune this parameter properly, so we choose to leave it as the default value. Since all methods we report require setting of this parameter, we feel that the relative comparisons between methods is sound even if the absolute numbers could be improved with a new setting for C. For Daume III and MMDT, which look at the source and target data simultaneously, ´ we use the same weighting scheme as we did for the source and target SVM. Late Fusion with the linear interpolation combination rule is reported across hyperparameter settings in Figure 1(a) to help understand how performance varies as we trade off emphasis between the learned classifiers from the source and target domains. Again, we do not have the validation data to tune this parameter so we report in the tables the performance averaged across parameter settings. The plot vs $\\alpha$ indicates that there is usually a best parameter setting that could be learned with more available data. For PMT, we choose $\\Gamma = 1 0 0 0$ , which corresponds to allowing a large amount of transfer from the source classifier to the target classifier. We do this because the source-only classifier is stronger than the target-only classifier (with ImageNet source). For the unsupervised methods GFK and SA, again we evaluated a variety of subspace dimensionalities and Figure 1(b) shows that the overall method performance does not vary significantly with the dimensionality choice. ",
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+        "table_body": "<table><tr><td>Adaptation Method</td><td>Training Data</td><td>DeCAF6</td><td>DeCAF7</td></tr><tr><td>SVM (source only)</td><td>Amazon</td><td>50.28 ± 1.8</td><td>54.08 ± 1.7</td></tr><tr><td>SVM (target only)</td><td>Webcam</td><td>62.28 ± 1.8</td><td>64.97 ± 1.8</td></tr><tr><td>GFK[12]</td><td>Amazon</td><td>53.13 ±1.1</td><td>53.39 ± 1.1</td></tr><tr><td>SA [10]</td><td>Amazon</td><td>51.74 ± 1.2</td><td>53.86 ± 1.0</td></tr><tr><td>SVM (source and target)</td><td>Amazon+Webcam</td><td>62.91 ±1.8</td><td>65.82 ± 1.4</td></tr><tr><td>Late Fusion (Max)</td><td>Amazon+Webcam</td><td>65.35 ± 1.7</td><td>58.42 ± 1.1</td></tr><tr><td>Late Fusion (Lin. Int. Avg)</td><td>Amazon+Webcam</td><td>63.23 ± 1.4</td><td>64.29 ± 1.3</td></tr><tr><td>Daumé III [7]</td><td>Amazon+Webcam</td><td>68.89 ± 1.9</td><td>72.09 ± 1.4</td></tr><tr><td>PMT[1]</td><td>Amazon+Webcam</td><td>64.84 ± 1.5</td><td>65.63 ± 1.8</td></tr><tr><td>MMDT[15]</td><td>Amazon+Webcam</td><td>65.47 ± 1.8</td><td>68.10 ± 1.5</td></tr><tr><td>Late Fusion (Lin. Int. Oracle)</td><td>Amazon+Webcam</td><td>71.1 ± 1.7</td><td>72.82 ±1.4</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: Amazon Webcam adaptation experiment. We show here multiclass accuracy on the target domain test set for both supervised and unsupervised adaptation experiments across the two fully connected layer features (similar to [9], but with one labeled target example). The best performing unsupervised adaptation algorithms are shown in blue and the best performing supervised adaptation algorithms are shown in red. ",
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+    {
+        "type": "text",
+        "text": "4.3 Effect of Source Domain Size ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Previous studies considered source domains from the Office dataset. In this section, we ask what happens when an orders-of-magnitute larger source dataset is used. ",
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+        "type": "text",
+        "text": "For completeness we begin by evaluating Amazon as a source domain. Preliminary results on this setting are reported in [9], here we extend the comparison here by presenting the results with more adaptation algorithms and more complete evaluation of hyperparameter settings. Table 1 presents multiclass accuracies for each algorithm using either layer 6 or 7 from the deep network, which corresponds to the output from each of the fully connected layers. ",
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+        "text": "An SVM trained using only Amazon data achieves $7 8 . 6 \\%$ in-domain accuracy (tested on the same domain) when using the ${ \\mathrm { D e C A F } } _ { 6 }$ feature and $8 0 . 2 \\%$ in-domain accuracy when using the ${ \\mathrm { D e C A F } } _ { 7 }$ feature. These numbers are significantly higher than the performance of the same classifier on Webcam test data, indicating that even with the DeCAF features, there is a still a domain shift between the Amazon and Webcam datasets. ",
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+    {
+        "type": "text",
+        "text": "Next, we consider an unsupervised adaptation setting where no labeled examples are available from the target dataset. In this scenario, we apply two state-of-the-art unsupervised adaptation methods, GFK [12] and SA [10]. Both of these methods make use of a subspace dimensionality hyperparameter. We show the results using a 100-dimensional subspace and leave the discussion of setting this parameter until Section 4.6. For this shift the adaptation algorithms increase performance when using the layer 6 feature, but offer no additional improvement when using the layer 7 feature. ",
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+        "type": "text",
+        "text": "We finally assume that a single example per category is available in the target domain. As the bottom rows of Table 1 show, supervised adaptation algorithms are able to provide significant improvement regardless of the feature space chosen, even in the one-shot scenario. For this experiment we noticed that using the second fully connected layer $( \\mathrm { D e C A F } _ { 7 }$ ) was a stronger overall feature in general. ",
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+    {
+        "type": "text",
+        "text": "4.4 Adapting with a Large Scale Source Domain ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We next address one of the main questions of this paper: Is there still a domain shift when using a large source dataset such as ImageNet? To begin to answer this question we follow the same experimental paradigm as the previous experiment, but use ImageNet as our source dataset. The results are shown in Table 2. ",
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+        "type": "text",
+        "text": "Again, we first verify that the source only SVM achieves higher performance when tested on indomain data than on Webcam data. Indeed, for the 16 overlapping labels, the source SVM produces $6 2 . 5 0 \\%$ accuracy on ImageNet data using ${ \\mathrm { D e C A F } } _ { 6 }$ features and $7 4 . 5 0 \\%$ accuracy when using ${ \\mathrm { D e C A F } } _ { 7 }$ features. Compare this to the $54 \\%$ and $59 \\%$ for Webcam evaluation and a dataset bias is still clearly evident. ",
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+        "table_caption": [
+            "Table 2: ImageNet Webcam adaptation experiment. Comparison of unsupervised and supervised adaptation algorithms on the ImageNet to Webcam domain shift. Results are computed using the outputs of each of the fully connected layers as features. The best supervised adaptation performance is indicated in red and the best unsupervised adaptation performance is highlighted in blue. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Adaptation Method</td><td>Training Data</td><td>DeCAF6</td><td>DeCAF7</td></tr><tr><td>SVM (source only)</td><td>ImageNet</td><td>53.51 ± 1.1</td><td>59.15 ± 1.1</td></tr><tr><td>SVM (target only)</td><td>Webcam</td><td>62.28 ± 1.8</td><td>64.97 ± 1.8</td></tr><tr><td>GFK [12]</td><td>ImageNet</td><td>65.16 ± 1.1</td><td>67.97 ± 1.4</td></tr><tr><td>SA [10]</td><td>ImageNet</td><td>59.30 ± 1.4</td><td>66.08 ± 1.4</td></tr><tr><td>SVM (source and target)</td><td>ImageNet+Webcam</td><td>56.68 ± 1.2</td><td>66.93 ± 1.3</td></tr><tr><td>Late Fusion (Max)</td><td>ImageNet+Webcam</td><td>59.59 ± 1.3</td><td>68.86 ± 1.2</td></tr><tr><td>Late Fusion (Lin. Int. Avg)</td><td>ImageNet+Webcam</td><td>60.64 ± 1.3</td><td>66.45 ± 1.1</td></tr><tr><td>Daumé III [7]</td><td>ImageNet+Webcam</td><td>59.21 ± 1.7</td><td>71.39 ± 1.5</td></tr><tr><td>PMT[1]</td><td>ImageNet+Webcam</td><td>66.30 ± 2.1</td><td>69.81 ± 1.8</td></tr><tr><td>MMDT[15]</td><td>ImageNet+Webcam</td><td>59.21 ± 1.3</td><td>67.75 ± 1.4</td></tr><tr><td>Late Fusion (Lin. Int. Oracle)</td><td>ImageNet+Webcam</td><td>71.65 ± 2.0</td><td>76.76 ± 1.3</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/fc0b69a785800dc75b28049f03d408a699d94ee282e7035c053a9fcc101fbd65.jpg",
+        "table_caption": [
+            "Table 3: ImageNet Webcam and Amazon Webcam adaptation experiments using $\\mathrm { D e C A F _ { 8 } }$ , the label activations of the CNN trained on the full ImageNet data. Again, we compare multiclass accuracy of various unsupervised and supervised adaptation methods. The best performing unsupervised adaptation algorithm is shown in blue and the best performing supervised adaptation algorithms are shown in red. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Adaptation Method</td><td>Training Data</td><td>Source=ImageNet</td><td>Source=Amazon</td></tr><tr><td>SVM (source only)</td><td>Source</td><td>66.23 ± 0.8</td><td>53.23 ± 1.6</td></tr><tr><td>SVM (target only)</td><td>Webcam</td><td>63.13 ± 1.9</td><td>63.13 ± 1.9</td></tr><tr><td>GFK[12]</td><td>Source</td><td>68.73 ±1.1</td><td>54.56 ± 1.2</td></tr><tr><td>SA[10]</td><td>Source</td><td>66.08 ± 1.1</td><td>55.98 ± 1.0</td></tr><tr><td> SVM (source and target)</td><td>Source+Webcam</td><td>75.13 ± 1.1</td><td>63.20 ± 1.7</td></tr><tr><td>Late Fusion (Max)</td><td>Source+Webcam</td><td>71.77 ± 1.4</td><td>62.25 ± 0.8</td></tr><tr><td>Late Fusion (LinInt Avg)</td><td>Source+Webcam</td><td>70.56 ± 1.2</td><td>64.56 ± 1.3</td></tr><tr><td>Daumé III [7]</td><td>Source+Webcam</td><td>77.15 ± 1.1</td><td>70.51 ± 1.7</td></tr><tr><td>PMT[1]</td><td>Source+Webcam</td><td>70.28 ± 1.8</td><td>66.77 ± 2.1</td></tr><tr><td>MMDT[15]</td><td>Source+Webcam</td><td>73.96 ± 1.2</td><td>66.23 ± 1.4</td></tr><tr><td>Late Fusion (Lin. Int. Oracle)</td><td>Source+Webcam</td><td>76.61 ± 1.5</td><td>71.49 ± 1.3</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Note that when using ImageNet as a source domain, overall performance of all algorithms improves. In addition, unsupervised adaptation approaches are more effective than for the smaller source domain experiment. ",
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+    {
+        "type": "text",
+        "text": "4.5 Adapting a Pre-trained Classifier to a New Label Set ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "$\\mathrm { D e C A F _ { 8 } }$ differs from the other DeCAF features in that it constitutes the 1000 activations corresponding to the 1000 labels in the ImageNet classification task. In the CNN proposed by [17], these activations are fed into a softmax unit to compute the label probabilities. We instead experiment with using the $\\mathrm { D e C A F _ { 8 } }$ activations directly as a feature representation, which is akin to training another classifier using the output of the 1000-way CNN classifier. ",
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+    {
+        "type": "text",
+        "text": "Table 3 shows results for various adaptation techniques using both ImageNet and Amazon as source domains. We use the same setup as before, but instead use $\\mathrm { D e C A F _ { 8 } }$ as the feature representation. ",
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+    {
+        "type": "text",
+        "text": "The ImageNet results are uniformly better with $\\mathrm { D e C A F _ { 8 } }$ than with ${ \\mathrm { D e C A F } } _ { 6 }$ or ${ \\mathrm { D e C A F } } _ { 7 }$ , likely due to the fact that $\\mathrm { D e C A F _ { 8 } }$ was explicitly trained on ImageNet data to effectively discriminate between ImageNet categories. Because it can more effectively classify images from the source domain, it is able to better adapt from the source domain to the target domain. ",
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+    {
+        "type": "text",
+        "text": "However, we see a negligible difference in performance for Amazon, with performance actually decreasing with respect to ${ \\mathrm { D e C A F } } _ { 7 }$ for certain adaptation methods. We believe this is because the final activation vector is too specific to the 1000-way ImageNet task, and that ${ \\mathrm { D e C A F } } _ { 7 }$ provides a more general representation that is better suited to the Amazon domain. This, in turn, results in improved adaptation. In general, however, the difference between the various DeCAF representations with Amazon as a source are small enough to be insignificant. ",
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+    {
+        "type": "text",
+        "text": "4.6 Analysis and Practical Considerations ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our adaptation experiments show that, despite its large size, even ImageNet is not large enough to cover all domains, and that traditional domain adaptation methods go a long way in increasing performance and mitigating the effects of this shift. Depending on the characteristics of the problem at hand, our results suggest different methods may be most suitable. ",
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+    {
+        "type": "text",
+        "text": "If no labels exist in the target domain, then there are unsupervised adaptation algorithms that are easy to use and fast to compute at adaptation time, yet still achieve increased performance over sourceonly methods. For this scenario, we experimented with two subspace alignment based methods that both require setting a parameter that indicates the dimensionality of the input subspaces. Figure 1(b) shows the effect that changing the subspace dimensionality has on the overall method performance. In general, we noticed that these methods were not particularly sensitive to this parameter so long as the dimensionality remains larger than the number of categories in our label set. Below this threshold, the subspace is less likely to capture all important discriminative information needed for classification. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the case where we have a large source dataset and a limited number of labeled target examples, it may be preferable to compute source classifier parameters in advance, then examine only the source parameters and the target data at adaptation time. Examples of these kinds of methods are Late Fusion and PMT. These methods are unaffected by the number of data points in the source domain at adaptation time, and can thus be applied quickly. In our experiments, we found that a properly tuned Late Fusion classifier with linear interpolation was the fastest and most effective approach. Figure 1(a) shows the performance of linear interpolation Late Fusion as we vary the hyperparameter $\\alpha$ . Although the method is sensitive to $\\alpha$ , we found that for both source domains, the basic strategy of setting $\\alpha$ around 0.8 provides a close approximation to optimal performance. This setting can be interpreted as trusting the target classifier more than the source, but not so much as to completely discount the information available from the source classifier. In each table we report both the performance of linear interpolation both averaged across hyper parameter settings $\\bar { \\alpha \\in [ 0 , 1 ] }$ as well as the performance of linear interpolation with the best possible setting of $\\alpha$ per experiment – this is denoted as “Oracle” performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "If there are no computational constraints and there are very few labels in the target domain, the best-performing method seems to be the “frustratingly easy” approach originally proposed by Daume III [7] and applied again for deep models in [5]. ´ ",
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+    {
+        "type": "text",
+        "text": "Finally, we found that feature representation can have a significant impact on adaptation performance. Our results show that ImageNet as source performs best with the $\\mathrm { D e C A F _ { 8 } }$ representation, whereas Amazon as source performs best with the ${ \\mathrm { D e C A F } } _ { 7 }$ representation. This, combined with our intuition, seems to indicate that for adaptation from source domains other than ImageNet, an intermediate representation other than $\\mathrm { D e C A F _ { 8 } }$ is more powerful for adaptation, whereas ImageNet classification works best with the full representation that was trained on it. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we presented the first evaluation of domain adaptation from a large-scale source dataset with deep features. We demonstrated that, although using ImageNet as a source domain generalizes better than other smaller source domains, there is still a domain shift when adapting to other visual domains. ",
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+        "image_caption": [
+            "Figure 1: Evaluation of hyperparameters for domain adaptation methods. (a) Analysis of the combination hyperparameter $\\alpha$ for Late Fusion with linear interpolation. (b) Analysis of the subspace dimensionality for the unsupervised adaptation algorithms "
+        ],
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+        "text": "Our experimental results show that deep adaptation methods can go a long way in mitigating the effects of this domain shift. Based on our results, we also provided a set of practical recommendations for choosing a feature representation and adaptation method accounting for constraints on runtime and accuracy. ",
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+        "text": "There are a number of interesting directions to take given our results. First we notice that though $\\mathrm { D e C A F _ { 8 } }$ is the strongest feature to use for learning a classifier on ImageNet data, ${ \\mathrm { D e C A F } } _ { 7 }$ is actually a better feature to use with the Amazon source domain and the Webcam target domain. This could lead to a hybrid approach where one uses different feature representations for the various domains and produces a combined adapted model. Another interesting direction that should be explored is to integrate the adaption algorithms into the deep models explicitly and even allow for feedback between the two stages. Current deep models although allow information flow between the final classifier and the representation learning architecture. We feel that the next step is to have a separate task specific adaptable layer that does not simply learn a new final layer, but instead learns a separate, but equivalent final layer, that is regularized by the final layer learned on the source dataset. ",
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+        "text": "This future work is a natural extension of the result we have shown in this paper: that pre-trained deep representations with large source domains can be effectively adapted to new target domains using only shallow, linear adaptation methods, and that in cases where the target data is limited, this approach is the best way to mitigate dataset bias. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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