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RPC-Bench

+# UNDERSTANDING GANS VIA GENERALIZATION ANALYSIS FOR DISCONNECTED SUPPORT
+Anonymous authors Paper under double-blind review
+# ABSTRACT
+This paper provides theoretical analysis of generative adversarial networks (GANs) to explain its advantages over other standard methods of learning probability measures. GANs learn a probability through observations, using the objective function with a generator and a discriminator. While many empirical results indicate that GANs can generate realistic samples, the reason for such successful performance remains unelucidated. This paper focuses the situation where the target probability measure satisfies the disconnected support property, which means a separate support of a probability, and relates it with the advantage of GANs. It is theoretically shown that, unlike other popular models, GANs do not suffer from the decrease of generalization performance caused by the disconnected support property. We rigorously quantify the generalization performance of GANs of a given architecture, and compare it with the performance of the other models. Based on the theory, we also provide a guideline for selecting deep network architecture for GANs. We demonstrate some numerical examples which support our results.
+# 1 INTRODUCTION
+Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) attract much attention as technology for learning a distribution and generating data. The purpose of GANs is to learn a probability measure from a given dataset and generate samples from the learned measure. It is often seen that samples generated by GANs can extract effectively features in the real world; it is difficult, for instance, to distinguish real images and generated images. By practical successes, a countless number of variations of GANs have been developed (Dziugaite et al., 2015; Arjovsky et al., 2017; Li et al., 2015; Nowozin et al., 2016; Gulrajani et al., 2017; Zhao et al., 2016), and applied to a wide range of tasks (Reed et al., 2016; Zhu et al., 2017; Gauthier, 2014).
+Understanding the remarkable performance of GANs is, however, still a challenging problem. There are active discussions on the role of generators in its learning scheme (Goodfellow, 2016; Arjovsky & Bottou, 2017; Arora et al., 2018; Creswell et al., 2018), and adversarial structures with discriminators are also a target of interest (Lotter et al., 2015; Zhang et al., 2018). A gaming structure between generators and discriminators is also regarded as a useful factor in the mechanism of GANs (Mescheder et al., 2017; Arora et al., 2017; Heusel et al., 2017). Generalization performance of GANs has been investigated in several studies (Liang, 2017; Liu et al., 2017; Tolstikhin et al., 2017). In spite of these studies, it is not yet clear why GANs can generate well-extracted data better than other standard methods.
+This paper introduces the disconnected support property, and explains an advantage of GANs in connection with this notion. The disconnected support property refers to a probability measures of which the support is divided into several disjoint sets, allowing non-differentiable density on the boundary. The property makes a probability measure be complex, hence it can be an obstacle for standard methods to learn the measure effectively. This property, however, is popularly seen in many real data, especially data with cluster structure, as demonstrated in Section 3.
+We investigate in detail the approximation and estimation ability of GANs and some other methods, and provide novel generalization analysis of probability measures with disconnected support. Firstly, we show that the other methods suffer worse generalization performance due to complex structures of disconnected supports (Proposition 1 and Lemma 2). Secondly, our generalization analysis reveals that GANs can learn the probability measure without loss of efficiency under the the disconnected supports (Theorem 1, Corollary 1 and ??). Additionally, we derive a guideline for choosing the number of layers or connections of the generator and discriminator from the generalization analysis. Numerical results support our theoretical findings.
+We remark that the disconnected support property is different from the low-dimensional supports studied in Arjovsky & Bottou (2017), where the support of a measure generated by neural networks is disjoint to the measure of observations. In contrast, this paper considers the case in which the support of the observation measure is divided into disjoint subsets. The problem of disconnected supports is complement to the low-dimensionality, hence these two problems can be investigated separately. In this paper, to simplify the discussion, we assume that the support of a probability measure is not low-dimensional.
+The contributions of this paper are summarized as follows:
+1. We show that GANs perform better than other standard methods of estimating probability measures when the measure satisfies the disconnected support property.
+2. We provide a new generalization error bound under a general formulation of GANs by analyzing an approximation error. The result is thus applicable to a wide range of variations of GANs.
+3. Based on the generalization bound, we provide a theoretical guideline for selecting architectures of generators and discriminators.
+All the proofs are given in Supplementary materials.
+# 2 PRELIMINARIES
+# 2.1 NOTATION
+We use notations $I : = [ 0 , 1 ]$ . A $j$ -th element of a vector $b$ is denoted by $b _ { j }$ , and $\begin{array} { r } { \| b \| _ { q } : = ( \sum _ { j } b _ { j } ^ { q } ) ^ { 1 / q } } \end{array}$ is the $q$ -norm $( q \in [ 0 , \infty ] )$ . $\mathrm { v e c } ( \cdot )$ is a vectorization operator for matrices. For $z \in \mathbb { N } , [ z ] : =$ $\{ 1 , 2 , \ldots , z \}$ is a set of positive integers no more than $z$ . For $\alpha \in \mathbb { R }$ , $\lfloor \alpha \rfloor$ denotes the largest integer which is not larger than ş $\alpha$ . For a domain $\Omega$ in a Euclidean space and a function $f : \Omega  \mathbb { R }$ , $\Vert f \Vert _ { L ^ { p } } : = ( \int _ { \Omega } | f ( t ) | ^ { p } d t ) ^ { 1 / p }$ denotes the $L ^ { p }$ norm for $p \in [ 0 , \infty ]$ . For $f : \Omega \to \mathbb { R } ^ { D }$ with a multidimensional output, $f _ { d }$ denotes a $d$ -th coordinate of $\boldsymbol { f } ( \boldsymbol { x } ) = ( f _ { 1 } ( \boldsymbol { x } ) , . . . , f _ { D } ( \boldsymbol { x } ) ) ^ { \top } ,$ . Let $H ^ { \beta } ( \Omega )$ be the Hölder space for $\beta > 0$ such as a set of $\beta$ -smooth functions $f : \Omega \to { \mathbb { R } }$ , namely, $f$ is $C ^ { \lfloor \beta \rfloor }$ -class and its $\lfloor \beta \rfloor$ -th derivative is $\beta - \lfloor \beta \rfloor$ -Hölder continuous. $\otimes$ denotes a tensor product, and $\bigcirc$ a composition of functions, namely, for functions $f$ and $f ^ { \prime }$ , $f \circ f ^ { \prime } = f ( f ^ { \prime } ( \cdot ) )$ . A Borel $\sigma$ -algebra of $\Omega$ is denoted as $\sigma ( \Omega )$ . For a measurable mapping $f : \Omega \to \Omega ^ { \prime }$ and $B ^ { \prime } \subset \Omega ^ { \prime }$ , a pre-image of $f$ is defined as $f ^ { - 1 } ( B ^ { \prime } ) : = \{ t \in \Omega \mid B ^ { \prime } \ni f ( t ) \}$ . Let $I _ { \Omega } : x \mapsto \{ 0 , 1 \}$ be an indicator function such that $\pmb { I } _ { \Omega } ( x ) = 1$ if $x \in \Omega$ , and $\pmb { I } _ { \Omega } ( x ) = 0$ otherwise.
+# 2.2 GENERAL FRAMEWORK OF GANS
+We provide a general formulation of a learning problem with generative adversarial networks $( G A N s )$ following Liu et al. (2017). In this paper, we consider a probability measure $P ^ { * }$ on a measurable space $( I ^ { \breve { D } } , \Sigma )$ with a dimensionality $D \in \mathbb { N }$ and $\Sigma : = \sigma ( I ^ { \hat { D } } )$ . Here, we set $D \geqslant 3$ . Suppose we have a set of $n$ observations ř ${ \mathcal { D } } : = \{ X _ { 1 } , . . . , X _ { n } \}$ which is independently and identically generated from $P ^ { * }$ . Let $\begin{array} { r } { P _ { n } : = \frac { 1 } { n } \sum _ { i \in [ n ] } \delta _ { X _ { i } } } \end{array}$ be an empirical measure where $\delta _ { x }$ is the Dirac measure at $x$ .
+The goal of generative networks is to estimate $P ^ { * }$ from $\mathcal { D }$ . To this end, we construct a probability measure by generators. Let $P _ { Z }$ be the uniform distribution on $( I ^ { D } , \Sigma )$ . For a measurable mapping $g : I ^ { D } \to { \bf \check { \cal I } } ^ { \check { D } }$ , we define $P _ { g }$ as the pushforward measure: i.e., $\begin{array} { r } { P _ { g } ( B ) = P _ { Z } ( g ^ { - 1 } ( B ) ) } \end{array}$ for $B \in \Sigma$ . We call $g$ as a generator and use $\mathcal { G }$ for a set of generators.
+GANs employ a learning scheme with a metric with discriminators (Goodfellow et al., 2014). Let $\mathcal { F } = \{ f : \mathbf { \bar { \chi } } _ { I } D ^ { \bullet }  \mathbb { R } \}$ be a a set of discriminators. This paper considers a general metric for GANs (Liu et al., 2017),
+$$
+d _ { \mathcal { F } } ( P , P ^ { \prime } ) : = \operatorname* { s u p } _ { f \in \mathcal { F } } \mathbb { E } _ { X \sim P } [ f ( X ) ] - \mathbb { E } _ { X \sim P ^ { \prime } } [ f ( X ) ] ,
+$$
+between probability measures $P$ and $P ^ { \prime }$ .
+For the learning process, we generate $m$ noise samples $\widetilde { Z } _ { 1 } , . . . , \widetilde { Z } _ { m }$ from $P _ { Z }$ and obtain generated samples as $\tilde { X } _ { j } : = g ( \tilde { Z } _ { j } )$ with $g \in { \mathcal { G } }$ for $j \in [ m ]$ . Let $\begin{array} { r } { P _ { g , m } : = \frac { 1 } { m } \sum _ { j \in [ m ] } \delta _ { \widetilde { X } _ { j } } } \end{array}$ denote the sampling measure. GANs construct an estimator $P _ { \hat { g } }$ for $P ^ { * }$ by learning $\widehat g$ with the following optimization problem
+$$
+{ \widehat { g } } \in \mathop { \mathrm { a r g m i n } } _ { g \in { \mathcal { G } } } d _ { { \mathcal { F } } } \left( P _ { n } , P _ { g , m } \right) .
+$$
+The metric (1) covers a wide variety of GANs by selecting $\mathcal { F }$ . Among others, the original GAN (Goodfellow et al., 2014) is realized if $\mathcal { F }$ contains a logarithm of density ratio; Wasserstein-GAN (Arjovsky et al., 2017), MMD-GAN (Dziugaite et al., 2015; Li et al., 2017) and Energy-Based GAN (Zhao et al., 2016) are given if $\mathcal { F }$ is the set of 1-Lipschitz functions, a reproducing kernel Hilbert space, and the bounded continuous functions, respectively. The $f$ -GAN (Nowozin et al., 2016) also belongs to this class.
+We assume that $\mathcal { F }$ is large enough to contain functions which can work as a discriminator, namely, we assume that the following holds:
+$$
+d _ { \mathcal { F } } ( P , P ^ { \prime } ) = 0 \Leftrightarrow P = P ^ { \prime } .
+$$
+A sufficient condition for (3) is investigated in Zhang et al. (2018).
+# 2.3 DEEP NEURAL NETWORKS FOR GENERATORS AND DISCRIMINATORS
+In the schemes of GANs, $\mathcal { F }$ and $\mathcal { G }$ are realized by deep neural networks (DNNs). For further discussion, we formulate the function class given by DNNs.
+Let $L \in \mathbb { N }$ be a number of layers in DNNs, and $D _ { \ell } ^ { \prime } \in \mathbb { N }$ be a dimensionality of variables in an $\ell \cdot$ -th layer for $\ell \in \left[ L + 1 \right]$ . Here, we set $D _ { L + 1 } ^ { \prime } = D$ for generators and $D _ { L + 1 } ^ { \prime } = \mathrm { \bar { 1 } }$ for discriminators. We introduce $A _ { \ell } \in \mathbb { R } ^ { D _ { \ell + 1 } ^ { \prime } \times D _ { \ell } ^ { \prime } }$ and $b _ { \ell } \in \mathbb { R } ^ { D _ { \ell } ^ { \prime } }$ as matrix and vector parametersof the $\ell$ -th layer. An architecture $\Theta$ of DNNs is defined as a set of $L$ pairs of $\left( A _ { \ell } , b _ { \ell } \right)$ as $\bar { \Theta : = ( ( A _ { 1 } , b _ { 1 } ) , . . . , ( A _ { L } , \bar { b } _ { L } ) ) }$ . We define notations for $\Theta$ as follow: $| \Theta | : = L$ as the number of layers, $\begin{array} { r } { \| \Theta \| _ { 0 } : = \sum _ { \ell \in [ L ] } \| \operatorname { v e c } ( A _ { \ell } ) \| _ { 0 } + \| b _ { \ell } \| _ { 0 } } \end{array}$ as the number of non-zero elements in $\Theta$ , and $\begin{array} { r } { \| \Theta \| _ { \infty } : = \operatorname* { m a x } \{ \operatorname* { m a x } _ { \ell \in [ L ] } \| \mathrm { v e c } ( \mathring { A } _ { \ell } ) \| _ { \infty } , \operatorname* { m a x } _ { \ell \in [ L ] } \| b _ { \ell } \| _ { \infty } \} } \end{array}$ be the scale of parameters in $\Theta$ . We employ the ReLU activation function $\eta : \mathbb { R } ^ { D ^ { \prime } } \to \mathbb { R } ^ { D ^ { \prime } }$ for each $D ^ { \prime } \in \mathbb { N }$ such as $\eta ( x ) = ( \operatorname* { m a x } \{ x _ { d } , 0 \} ) _ { d \in [ D ^ { \prime } ] }$ .
+We define functions of DNNs with an architecture $\Theta$ as $\xi [ \Theta ] : \mathbb { R } ^ { D ^ { \prime } }  \mathbb { R } ^ { D ^ { \prime \prime } }$ by
+$$
+\xi [ \Theta ] ( x ) = x ^ { ( L + 1 ) } , x ^ { ( 1 ) } : = x , x ^ { ( \ell + 1 ) } : = \eta ( A _ { \ell } x ^ { ( \ell ) } + b _ { \ell } ) , \mathrm { f o r } \ell \in [ L ] .
+$$
+The function class of DNNs is thus given by
+$$
+\Xi ( S , B , L ) : = \Big \{ \xi [ \Theta ] : I ^ { D } \to \mathbb { R } \ | \ \| \Theta \| _ { 0 } \leqslant S , \| \Theta \| _ { \infty } \leqslant B , | \Theta | \leqslant L \Big \} ,
+$$
+where $S \in \mathbb { N } , B > 0$ , and $L \in \mathbb { N }$ are hyper-parameters. Here, $S$ bounds the number of non-zero parameters of DNNs, namely, it controls the sparseness of DNNs. $B$ is a bound for scales of parameters.
+# 3 DISCONNECTED SUPPORT PROPERTY
+# 3.1 INTRODUCTION AND EXAMPLE
+It is often observed that data in real world data the support of its probability measure may not be connected but a union of disjoint subsets. This is typical if the data has cluster structures, as seen in many data sets for classification tasks. Moreover, the density function of the probability measure may not be smooth at a boundary of the support. Figures 1 (MNIST, (LeCun et al., 1998)) and 2 (Shelter Animal, Center) illustrate such examples in the real world. They are projected onto a 2-dimensional Euclidean space by t-SNE (Maaten & Hinton, 2008) so that they preserve the original distance structure among points. We can see that both of the data are concentrated on several disjoint subsets and there are a clear gap or empty regions between some of the subsets. This observation suggests that the disconnected property of probability measures should be addressed in discussing estimation of probability measures, while standard analysis does not consider this phenomenon. In fact, this paper will show that the disconnected supports property has an important role in showing an advantage of GANs over standard estimation methods.
+![](images/fe32edfd04a7a392e42a9668e55374015bbdbbcd473787ac6a8e24e86e9e385a.jpg)
+Figure 1: Plot of the MNIST data.
+![](images/9e1bdfe866576746ea80f727f2a4ff3e3c05341c6ba33fc8c5e775ffcb4bc9c3.jpg)
+Figure 2: Plot of the animal data.
+# 3.2 MATHEMATICAL FORMULATION OF DISCONNECTED SUPPORTS
+Here we make a rigorous definition of disconnected supports. The property of smoothness (i.e. differentiability) is involved, which is a key factor to analyze generalization performance in the fields of the statistics (Stone, 1982; Tsybakov, 2009); Stone (1982) shows, for instance, that smoothness and a dimension of data are sufficient to characterize an optimal convergence of generalization errors.
+We first prepare a family of subsets as a component in the disconnected supports:
+$\begin{array} { r } { S _ { \alpha , J } : = \left\{ S \subset I ^ { D } \ \right| } \end{array}$ A boundary of $S$ is $J$ combination of $\alpha$ -smooth hyper surfaces  .
+The supplementary material will provide a more rigorous definition.
+Now, we define the disconnected support property of probability measures as well as a probability measure with global support, i.e., with no the disconnected support property. Let $\operatorname { S u p p } ( P )$ be the support of $P$ ., i.e, ${ \dot { \operatorname { S u p p } } } ( P ) : = \{ x \in I ^ { D } \mid P ( V _ { x } ) > 0$ for all open neighborhood $V _ { x }$ of $x \}$ Hereafter, $M \geqslant 2$ is the number of disjoint components of a support.
+Definition 1. (Disconnected Supports / Global Support)
+Let $M \geqslant 2$ . A probability measure $P$ on $( I ^ { D } , \bar { \Sigma } )$ has $M$ disconnected supports, if there exist
+nonempty disjoint sets ${ \cal S } _ { 1 } , . . . , { \cal S } _ { M } \in { \cal S } _ { \alpha , J }$ such that
+$$
+\operatorname { S u p p } ( P ) = \bigcup _ { m \in [ M ] } S _ { m } .
+$$
+A probability measure $P$ on $( I ^ { D } , \Sigma )$ has a global support, if $\operatorname { S u p p } ( P ) = I ^ { D }$ .
+Figure 3 illustrates the disconnected support property.
+We next formulate a notion of smoothness for $P$ with disconnected supports. Let $\beta \geqslant 1$ be a parameter for a degree of smoothness of $P$ .
+Definition 2. (Local Smoothness)
+A probability measure $P$ with $M$ disconnected support is locally $\beta$ -smooth, if there exist $M$ pairs $( \widetilde { S } _ { m } , S _ { m } ) \in \mathrm { ~ \cal { S } ~ } _ { 2 \beta , J } \times S _ { 2 \beta , J }$ and $\beta + 1$ -smooth bijective measurable maps $\gamma _ { m } : \widetilde { S } _ { m }  S _ { m }$ as
+$$
+P ( B ) = P _ { Z } ( \gamma _ { m } ^ { - 1 } ( B ) ) , \forall B \in \sigma ( S _ { m } ) ,
+$$
+for $m \in [ M ]$
+This definition of local smoothness says that a probability measure $P$ with disconnected supports can be generated by sufficiently smooth mappings $\gamma _ { m }$ . It is used for considering a smooth density function of $P$ restricted on $S _ { m }$ .
+Lemma 1. (Locally Smooth Density Functions)
+If a probability measure $P$ with $M$ disconnected support is locally $\beta$ -smooth, then there exists a
+function $p _ { m } : S _ { m } \to \mathbb { R } _ { + }$ such that
+$$
+P ( B ) = \int _ { B } p _ { m } ( x ) d \lambda , B \in \sigma ( S _ { m } ) ,
+$$
+where $\lambda$ is the Lebesgue measure, and $p _ { m }$ is $\beta$ -smooth for all $m \in [ M ]$ .
+We call $p _ { m }$ as a local density function.
+Note that, since Ť $P$ with disconnected supports is not absolutely continuous to the Lebesgue measure on $\textstyle I ^ { D } \backslash \bigcup _ { m \in [ M ] } S _ { m }$ , an ordinary density function cannot be defined. Instead, a localized version of density functions for each $S _ { m }$ is introduced, which is guaranteed by the local smoothness.
+![](images/c3dfa9a97b096a21af5f6a1718c1eb7930b940550b40573c085a60e6e567de44.jpg)
+Figure 3: Illustration of a probability measure $P$ with a disconnected support. $\operatorname { S u p p } ( P )$ is a union of two disjoint sets $S _ { 1 }$ and $S _ { 2 }$ .
+![](images/cfd77cfd712870f7426ee318b45107ce716b4b22206e74bddb9e99ccc7ae3428.jpg)
+Figure 4: Illustration of a generator $g$ . To represent discontinuous $S _ { 1 }$ and $S _ { 2 }$ , $g$ should be discontinuous.
+# 3.3 DIFFICULTY WITH DISCONNECTED SUPPORTS
+As shown in this subsection, the generalization performance of many standard estimation methods is worsened with disconnected supports. We consider popular nonparametric methods, for which the generalization performance is well-studied in the asymptotics of the observation size $n$ . The considered methods are the kernel density estimator (KDE) (Nadaraya, 1964), the nonparametric Bayes (NB) by the Dirichlet mixtures of normal distributions (Ferguson, 1973), the series density estimator (SDE) (Efromovich et al., 2008; Efromovich, 2010) and the density estimator with Gaussian process (GP) (Leonard, 1978). Bounds of the generalization errors for these methods are already known (Tsybakov, 2009; Ghosal et al., 2007; van der Vaart $\&$ van Zanten, 2008), and they are optimal in the minimax sense. Here, their performance is evaluated with respect to a root of an expected squared loss with respect to the $L ^ { 2 }$ -norm, namely, $d _ { 2 } ( P , P ^ { \prime } ) : = \mathbb { E } [ \| p - p ^ { \prime } \| _ { L ^ { 2 } } ^ { 2 } ] ^ { 1 / 2 }$ where $p$ and $p ^ { \prime }$ are densities for $P$ and $P ^ { \prime }$ .
+We show the deterioration of their performance by the disconnected support property. Let $\hat { P }$ be an estimator for $P ^ { * }$ by KDE, NB, SDE, or GP. If $P ^ { * }$ has a global support and a $\beta$ -smooth density, the existing studies (Tsybakov, 2009; Ghosal et al., 2007; van der Vaart $\&$ van Zanten, 2008) show that
+$$
+d _ { 2 } ( P ^ { * } , \widehat { P } ) = O \left( n ^ { - \beta / ( 2 \beta + D ) } \right) .
+$$
+These bounds are sufficiently tight, since these bounds corresponds to an optimal rate (Stone, 1982), and the performance of the methods can be improved as the density for $P ^ { * }$ is smoother (larger $\beta$ ).
+On the other hand, we consider a case in which $P ^ { * }$ has the disconnected support property.
+Proposition 1. (Deterioration of other standard methods) There exists $P ^ { * }$ of the disconnected support property and locally $\beta$ -smooth such that
+$$
+d _ { 2 } ( P ^ { * } , \widehat { P } ) = { \cal O } \left( n ^ { - 1 / ( 2 + D ) } \right) .
+$$
+When $P ^ { * }$ has disconnected supports, the errors are worse than those for the global support, independent of $\beta$ . This worse generalization error can be understood by the non-smoothness or discontinuity of the density functions on the boundaries of the disconnected sets (see Figure 3).
+We next discuss other generative models for estimating a probability measure. To the best of our knowledge, other probabilistic generative methods (Koller et al., 2009) and the variational autoencoder (Kingma & Welling, 2013), their statistical generalization property is not well investigated. Here we provide a property of generators for probability measures with disconnected supports.
+Lemma 2. (Discontinuous Generators for Disconnected Supports)
+If $P ^ { * }$ has disconnected supports and $P ^ { * } = P _ { g ^ { * } }$ with a generator $g ^ { * }$ , then $g ^ { * }$ is not uniformly continuous.
+Lemma 2 states that a generator must be discontinuous to construct a probability measure with disconnected support sets. Intuitively, to make $P _ { g ^ { * } } ( B ) = 0$ for $B \in I ^ { D }$ with $\lambda ( B ) > 0$ , the slope of $g ^ { * }$ at $z \in I ^ { D }$ should be close to infinite for $z \in g ^ { * , - 1 } ( B )$ , hence $g ^ { * }$ cannot be uniformly continuous (see Figure 4). Because of the discontinuity, generative models with smooth functions, such as an adversarial generative model with kernel generators (Sinn & Rawat, 2018), cannot work well with disconnected supports.
+# 4 GENERALIZATION BY GANS
+We provide generalization analysis for GANs for probability measures with and without disconnected supports. For the purpose, we employ a metric $d _ { \mathcal { F } }$ with properly selected discriminators $\mathcal { F }$ and evaluate the generalization error $d _ { \mathcal { F } } ( P ^ { \ast } , P _ { \hat { g } } )$ with respect to an observation size $n$ and a sampling size $m$ . We assume $\mathcal { F }$ is realized by DNNs as $\mathcal { F } = \Xi ( S _ { f } , B _ { f } , L _ { f } ) \cap \widetilde { \mathcal { F } }$ with parameters $S _ { f } , B _ { f } , L _ { f }$ , where $\tilde { \mathcal { F } }$ is a specified functional class; for an example, $\tilde { \mathcal { F } }$ is 1-Lipschitz functions for WassersteinGAN. Here, we consider settings that all $f \in { \mathcal { F } }$ are $L _ { 1 }$ -Lipschitz continuous and $\| f \| _ { L ^ { \infty } } \leqslant B _ { F }$ with constants $L _ { 1 } , B _ { F } > 0 .$ Generators are also constructed by DNNs as $\mathcal { G } = \Xi ( S _ { g } , B _ { g } , L _ { g } )$ with parameters $S _ { g } , B _ { g } , L _ { g }$ .
+A standard line of discussing generalization, we should consider statistical errors and approximation errors. We define a measure of the complexity of $\mathcal { F }$
+$$
+\Upsilon _ { n } ( \mathcal { F } ) : = \operatorname* { i n f } _ { \eta > 0 } 4 \eta + 1 2 n ^ { - 1 / 2 } \int _ { \eta } ^ { c } \log \mathcal { N } ( ( L _ { 1 } + 1 ) ^ { - 1 } \epsilon , \mathcal { F } , \| \cdot \| _ { n } ) ^ { 1 / 2 } d \epsilon ,
+$$
+where $c > 0$ is a constant depends on $\mathcal { F }$ and $\mathcal { N } ( \epsilon , \tilde { \mathcal { F } } , \| \cdot \| )$ is a covering number of $\tilde { \mathcal { F } }$ with respect to an empirical norm $\| \cdot \|$ . We note that $\Upsilon _ { n } ( \mathcal { F } )$ bounds an expectation of the Rademacher complexity as
+$$
+\Upsilon _ { n } ( \mathcal { F } ) \geqslant \mathbb { E } \left[ \operatorname* { s u p } _ { f \in \mathcal { F } } \frac { 1 } { n } \left| \sum _ { i \in [ n ] } \tau _ { i } f ( X _ { i } ) \right| \right] ,
+$$
+where $\tau _ { i }$ is the i.i.d. Rademacher random variables; $\operatorname* { P r } ( \tau _ { i } = 1 ) = \operatorname* { P r } ( \tau _ { i } = 1 ) = 1 / 2$ , and the expectation is about $X _ { i }$ and $\tau _ { i }$ . Using the statistics and learning theory van der Vaart & Wellner (1996); Bartlett et al. (2005), we can apply a bound for $\Upsilon _ { n } ( \mathcal { F } )$ as
+$$
+\Upsilon _ { n } ( \mathcal { F } ) \leqslant C _ { \mathcal { F } } n ^ { - 1 / \kappa } ,
+$$
+with some constants $C _ { \mathcal { F } } > 0$ and $\kappa \geqslant 2$ .
+Regarding approximation errors, we need to consider approximation of a discontinuous function;
+since Lemma 2 shows that a discontinuous generator is necessary to represent disconnected supports.
+To approximate such generators, DNNs in GANs has an advantage.
+Lemma 3. (Approximation for Discontinuous $g$ by DNNs) Suppose $P ^ { * }$ has $M$ disconnected supports and locally $\beta$ -smooth, and also $P ^ { * } = P _ { g ^ { * } }$ holds with some $g ^ { * }$ . Then, for any $S _ { g }$ , there exist $\mathcal { G }$ , ${ \dot { g } } \in { \mathcal { G } }$ , and a constant $c _ { g } = c _ { g } ( B _ { g } , L _ { g } ) > 0$ such that
+$$
+\lVert \dot { \boldsymbol g } _ { d } - \boldsymbol g _ { d } ^ { * } \rVert _ { L ^ { 2 } } \leqslant c _ { g } M S _ { g } ^ { - \beta / D } , \forall d \in [ D ] .
+$$
+Furthermore, if $P ^ { * } = P _ { g ^ { * } }$ has a global support and it is $\beta$ -smooth, (4) holds with $M = 1$
+Lemma 3 shows that $\mathcal { G }$ for GANs can approximate $g ^ { * }$ for disconnected supports with the rate $\left( - \beta / D \right)$ by $S _ { g }$ , and the rate is same in the case of global support. This implies an advantage of GANs in comparison with the other standard methods (Proposition 1).
+Based on Lemma 3, we obtain the main theorem for generalization analysis.
+Theorem 1. (Generalization of GANs) Suppose that $P ^ { * }$ has $M$ disconnected supports and locally $\beta$ -smooth, and we have n observations and m samplings. Then, with $\mathcal { F }$ , an existing $\mathcal { G }$ , an estimator $P _ { \hat { g } }$ by (2), and finite constants $c _ { 1 } =$ $c _ { 1 } ( L _ { f } , B _ { f } , L _ { g } , B _ { g } ) , c _ { 2 } , c _ { 3 } = c _ { 3 } ( L _ { f } , B _ { f } ) > 0$ , the following inequality holds with high probability,
+$$
+d _ { \mathcal { F } } \big ( P ^ { * } , P _ { \widehat { g } } \big ) \leqslant \underbrace { \Upsilon _ { m } ( \widetilde { \mathcal { F } } ) + c _ { 1 } \frac { \sqrt { S _ { g } } + \sqrt { S _ { f } } } { \sqrt { m } } } _ { = : I } + \underbrace { c _ { 2 } M D S _ { g } ^ { - \beta / D } } _ { = : I I } + \underbrace { \Upsilon _ { n } ( \widetilde { \mathcal { F } } ) + c _ { 3 } \sqrt { \frac { S _ { f } } { n } } } _ { = : I I I } .
+$$
+Furthermore, $i f P ^ { * }$ has a global support and it is $\beta$ -smooth, (5) holds with $M = 1$
+Each of the terms $I , I I$ and $I I I$ has the following role: $I$ bounds an error by the $m$ samplings, $I I$ bounds an error from approximation by $\mathcal { G }$ , and $I I I$ bounds an error by $n$ observations.
+Proof Outline: By the definition of $P _ { \hat { g } }$ in (2) and standard calculation, we obtain the inequality
+$$
+d _ { \mathcal { F } } ( P ^ { * } , P _ { \hat { g } } ) \leqslant \underbrace { 2 \operatorname* { s u p } _ { g \in \mathcal { G } } \operatorname* { s u p } _ { f \in \mathcal { F } } \big | \mathbb { E } _ { P _ { g , m } } [ f ( X ) ] - \mathbb { E } _ { P _ { g } } [ f ( X ) ] \big | } _ { = : i } + \underbrace { \operatorname* { i n f } _ { g \in \mathcal { G } } d _ { \mathcal { F } } ( P _ { g } , P ^ { * } ) } _ { = : i i } + \underbrace { 2 d _ { \mathcal { F } } ( P _ { n } , P _ { 0 } ) } _ { = : i i i } .
+$$
+To obtain $i \leqslant I$ and $i i i \leqslant I I I$ , we apply an empirical process technique (van der Vaart & Wellner, 1996), especially convergence of integral probability measures (Sriperumbudur et al., 2012) and the entropy control technique (Lemma 4 and 5 in the supplementary material). To show $i i \leqslant I I$ , we employ recent results on approximation ability of DNNs (Yarotsky, 2017; Petersen & Voigtlaender, 2017; Imaizumi & Fukumizu, 2018) and obtain an approximation bound for $d _ { \mathcal { F } } ( P _ { g } , P ^ { * } )$ (Lemma 3). Combining these results, we obtain the statement of Theorem 1. □
+Theorem 1 provides two trade-off relations with respect to $S _ { g }$ and $S _ { f }$ . The generator class $\mathcal { G }$ controls uncertainty by sampling and the approximation error, while $S _ { f }$ controls uncertainty of observations and discrimination. For balancing the trade-offs, we select the number of parameters (connections of DNNs) with some constants $c _ { g } , c _ { f } > 0$ as
+$$
+S _ { g } = c _ { g } m ^ { D / ( 2 \beta + D ) } , ~ \mathrm { a n d } ~ S _ { f } = c _ { f } n ^ { ( \kappa - 2 ) / \kappa } ,
+$$
+for optimizing the bound (5). We then obtain the following corollary.
+Corollary 1. (Convergence Rate of GANs)
+Make the same assumptions as Theorem $^ { l }$ , and set $S _ { f }$ and $S _ { g }$ as in (6). Then, with high probability
+converging to 1, we obtain
+$$
+d _ { \mathcal { F } } ( P ^ { \ast } , P _ { \hat { g } } ) = O \left( n ^ { - 1 / \kappa } \right) + O \left( m ^ { - 1 / \kappa } + m ^ { - \beta / ( 2 \beta + D ) } \right) .
+$$
+A selection of $\tilde { \mathcal { F } }$ determines the first term in (7), since \` $\kappa$ depends on ˘ $\tilde { \mathcal { F } }$ . For an example, when $\mathcal { F }$ is a set of 1-Lipschitz functions, the first term is $O \left( n ^ { - 1 / \left( 2 + 2 D \right) } \right)$ (Sriperumbudur et al., 2012).
+Remark 1. (Heterogeneous Smoothness)
+Corollary 1 can be extended when $P ^ { * }$ has different smoothness for each $m \in [ M ]$ , i.e., $P ^ { * }$ is locally $\beta _ { m }$ -smooth on a set $S _ { m }$ . In this case, we can easily extend our analysis in Theorem 1 and Corollary 1, and obtain the following convergence rate.
+$$
+d _ { \mathcal { F } } ( P ^ { \ast } , P _ { \hat { g } } ) = O \left( n ^ { - 1 / \kappa } \right) + O \left( m ^ { - 1 / \kappa } + m ^ { - \widetilde { \beta } / ( 2 \widetilde { \beta } + D ) } \right) ,
+$$
+where $\widetilde { \beta } : = \operatorname* { m i n } _ { m \in \left[ M \right] } \beta _ { m }$
+# 5 DISCUSSION
+We emphasize the theoretical results show that GANs do not suffer from the effect of disconnected supports. A larger $\beta$ improves performance of GANs even with disconnected supports, as shown in Theorem 1 and Corollary 1. This phenomenon is different from the result of the other methods discussed in Proposition 1. Hence, we can state that GANs have advantages over the other methods which are deteriorated by the disconnected property (Section 3.3). In other words, when data are generated from a probability measure with disconnected supports and sufficiently smooth in each of the sets, only GANs can estimate the measure effectively and the other methdos cannot. This advantage of GANs comes from the approximation power for discontinuous generators shown in Lemma 3.
+The results (5) and (7) provide interpretation about performance of GANs. About convergence with $n$ the complexity of $\tilde { \mathcal { F } }$ and $S _ { f }$ control a trade-off between convergence and a power of discrimination. While smaller $S _ { f }$ reduce the errors in terms of $d _ { \mathcal { F } }$ , too small $S _ { f }$ can lose the power of discrimination to satisfy (1). Hence, setting $S _ { f }$ as in (6) can keep the discrimination power and does not worsen the overall rate of convergence $O ( n ^ { - 1 / \kappa } )$ . About convergence with $m$ , $S _ { g }$ controls the trade-off between the bias and variance of the estimator. An optimal way to select $S _ { g }$ is provided in (6) which depends on $\beta$ and $D$ , and it is more important when $\kappa$ is small (e.g. $\kappa = 2$ as MMD-GAN). Based on the interpretation and the selection rule (6), our study can provide a guideline for a design of the architecture of DNNs.
+# 5.1 RELATED WORKS
+Compared with studies for understanding GANs (Dziugaite et al., 2015; Liang, 2017; Liu et al., 2017; Zhang et al., 2018; Biau et al., 2018; Arora et al., 2017), we show that GANs can avoid influences of disconnected supports, unlike the other methods, hence it is a source of the advantage of GANs. This paper is the first work to focus on the disconnected support property, while several discussions (Goodfellow, 2016; Liang, 2017; Creswell et al., 2018) focus on models and metrics of the learning scheme of GANs.
+It is important to compare our result with other studies for generalization analysis. Although some existing studies (Dziugaite et al., 2015; Liang, 2017; Liu et al., 2017; Zhang et al., 2018; Biau et al., 2018; Arora et al., 2017) provide generalization analysis, they do not analyze an approximation effect, namely, they evaluate $\begin{array} { r } { d _ { \mathcal { F } } ( P ^ { * } , P _ { \hat { g } } ) - \operatorname* { i n f } _ { g \in \mathcal { G } } ( P ^ { * } , P _ { g } ) } \end{array}$ . Since we analyze the term ${ \operatorname* { i n f } } _ { g \in { \mathcal { G } } } ( P ^ { * } , P _ { g } )$ , we can provide a more general bound and discuss the effect of disconnected support.
+As we mentioned in the introduction, this paper is not along with studies for low-dimensional supports (Arjovsky & Bottou, 2017). We also note that an optimization aspect and gaming aspect of GANs are out of concerns of this paper. We focus on the statistical aspects of GANs such as sample complexity.
+# 6 NUMERICAL EXPERIMENTS
+We compare the numerical performance of GANs and the other methods with toy data with. We generate synthetic data from the following two settings: (A) Gaussian distribution restricted on a compact set (global support), and (B) a probability measure with two disconnected supports (density function is the black solid line in Figure 6). We generate $n = 5 0 0$ , 1000, ..., 5000 observations and estimate the true probability measures with Wasserstein GAN, MMD-GAN, KDE (the Gaussian kernel and the Epanechnikov kernel), SDE (Fourier basis), and NB (Dirichlet process prior). Hyperparameters for these methods are selected by cross-validation. For GANs, we set $m = n$ . We use $d _ { \mathcal { F } }$ to evaluate errors by GANs, and a root of the expected squared errors with the $L ^ { 2 }$ -norm for the other methods. The plots are the mean of 30 replications.
+Figure 5 shows generalization errors by the methods. With the disconnected support case (B), we plot the estimated density in Figure 6. The black line shows the true density, the dashed line is by estimated densities of the other methods, and bars are histograms by GANs. The results by Wasserstein-GAN and MMD-GAN are almost same, so we omit the result by Wasserstein-GAN.
+In Figure 5, we see that in the case of global support (A), the error of GANs and the other methods are comparable. In contrast, in the case of disconnected supports (B), the other standard methods show worse generalization and only GANs keep the high performance. From Figure 6, we can see that GANs can reveal the disconnected supports, while some of the other methods fail to fit. KDE with the Gaussian kernel represents the disconnected support by employing a small bandwidth. However, the small bandwidth yields a too sharp density, tending to worsen the generalization performance.
+![](images/b7802bb8ea485d4b37d7f38150a4da78b2af66548ec317a150627253f99cdfd2.jpg)
+![](images/fdfdd8b80275cb2384c0f8e41eeca8edf4984999a69ba9efe06b8bc5ead17296.jpg)
+Figure 5: Generalization errors.
+Figure 6: Estimated density functions with the case (B).
+# 7 CONCLUSION
+We investigate a generalization performance of GANs with a situation such that a support of real probability measures is divided into several sets. We find that GANs do not suffer from the division of supports, while some of the other nonparametric methods loss their efficiency by the division. Since real data are often distributed on such divided supports, the finding in this paper is related to the question of why GANs perform well with real datasets.
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+# Supplementary Materials for “Understanding GANs via disconnected Support Detection”
+We introduce a new notation $P f : = \mathbb { E } _ { X \sim P } [ f ( X ) ]$ with a probability measure $P$ and a function $f$ . For a set $\Omega$ with equipped distance $d$ , let $\mathcal { N } ( \epsilon , \Omega , d )$ be a covering number which is a minimum number of $\epsilon$ -balls to cover $\Omega$ .
+# A SOME ADDITIONAL INFORMATION
+# A Rigorous Definition of $S _ { \alpha , J }$
+We consider a set represented by a combination of multiple horizon functions, which has been used in Petersen & Voigtlaender (2017). Given $\alpha$ -smooth function $h \in \check { H ^ { \alpha } } ( I ^ { D - 1 } )$ with $\alpha \geqslant 1$ , a horizon function $\Psi _ { h } : I ^ { D } \overset { \sim } {  } \{ 0 , 1 \}$ is defined for some $d \in [ D ]$ as
+$$
+\Psi _ { h } = \Psi ( x _ { 1 } , \ldots , x _ { d - 1 } , x _ { d } \pm h ( x _ { 1 } , \ldots , x _ { d - 1 } , x _ { d + 1 } , \ldots , x _ { D } ) , x _ { d + 1 } , . . . , x _ { D } ) ,
+$$
+where $\Psi$ is the Heaviside function; $\Psi ( x ) = I _ { \{ x \in I ^ { D } | x _ { d } \geqslant 0 \} }$ . We define a set by the intersection of $J$ horizon functions $\Psi _ { h _ { 1 } } , . . . , \Psi _ { h _ { J } }$ ; namely the family of sets is defined by
+$$
+S _ { \alpha , J } : = \left\{ S \subset [ 0 , 1 ] ^ { D } \mid { \cal I } _ { S } = \Psi _ { h _ { 1 } } \otimes \cdot \cdot \cdot \otimes \Psi _ { h _ { J } } \right\} .
+$$
+Intuitively, $h$ is regarded as an $\alpha$ -smooth curved surface in $I ^ { D }$ , and $\Psi _ { h }$ describes a set which is one side of the surface. Also, $S \in S _ { \alpha , J }$ is a set which is a intersection of $J$ sets by $\Psi _ { h _ { 1 } } , . . . , \Psi _ { h _ { J } }$ .
+# A Support of Probability Measures
+Let $N _ { x }$ denote an open neighborhood of $x \in I ^ { D }$ , and For a probability measure $P$ , a support of $P$ is defined as
+$$
+\mathrm { S u p p } ( P ) : = \bigg \{ x \in I ^ { D } \ | \ P ( N _ { x } ) > 0 , \forall N _ { x } \in \Sigma \bigg \} .
+$$
+# B PROOFS
+# B.1 PROOF OF LEMMA 1
+Fix $m \in [ M ]$ and a corresponding $\widetilde { S } _ { m } , S _ { m }$ and $g _ { m }$ . For any $B \in \sigma ( S _ { m } )$ , the definition of $\gamma _ { m }$ yields
+$$
+P _ { X } ( B ) = P _ { Z } ( \gamma _ { m } ^ { - 1 } ( B ) ) = \int _ { \gamma _ { m } ^ { - 1 } ( B ) } p _ { Z } ( z ) d z ,
+$$
+where $p _ { Z }$ is a density function of a uniform measure $P _ { Z }$ . By changing variables $x = \gamma _ { m } ( z )$ , we have
+$$
+\int _ { \gamma _ { m } ^ { - 1 } ( B ) } p _ { Z } ( z ) d z = \int _ { B } p _ { Z } ( \gamma _ { m } ^ { - 1 } ( x ) ) J _ { \gamma _ { m } } ( x ) d x ,
+$$
+where $J _ { \gamma _ { m } } ( x ) = | \operatorname* { d e t } \nabla g _ { m } ^ { - 1 } ( x ) |$ . Using $p _ { Z } ( z ) = 1$ for all $z \in I ^ { D }$ , we obtain the following form of $P _ { X } ( B )$ using a function $p _ { m } : \widetilde { S } _ { m }  S _ { m }$ as
+$$
+P _ { X } ( B ) = \int _ { B } J _ { g _ { m } } ( x ) d x = : \int _ { B } p _ { m } ( x ) d x ,
+$$
+and $p _ { m }$ is $\beta$ -smooth since $\gamma _ { m }$ is $\beta + 1$ -smooth and bijective.
+# B.2 PROOF OF LEMMA 2
+Firstly, we show the first points. Suppose that $g$ is a continuous mapping. By the generalized intermediate value theorem (Theorem 24.3 in Munkres (2000)), we know that $g ( I ^ { D } )$ is connected since $I ^ { D }$ is a connected set. Thus, a support of $P _ { g }$ is connected. However, $P _ { g }$ has a disconnected support, thus there is a contradiction. □
+In this proof, $a \lesssim b$ denotes that $b$ is larger than $a$ up to a finite constant. $a = b$ denotes that $a \lesssim b$ and $a \gtrsim b$ hold.
+By the definition of $\{ g _ { m } \} _ { m \in [ M ] }$ for the measure with local smoothness, we consider an explicit form of $g _ { m }$ . By Stein (2016), we can extend $g _ { m } : \widetilde { S } _ { m }  S _ { m }$ to $\tilde { g } _ { m } : I ^ { D } \to S _ { m }$ since boundaries of $\widetilde { S } _ { m }$ are Lipschitz continuous. Then, we provide the following formulation
+$$
+\widetilde { \boldsymbol { g } } _ { m } ( \boldsymbol { x } ) = ( \gamma _ { m , 1 } ( \boldsymbol { x } ) , . . . , \gamma _ { m , D } ( \boldsymbol { x } ) ) ^ { \top } ,
+$$
+where $\gamma _ { m , d } \in H ^ { \beta } ( I ^ { D } )$ . Then, we obtain the form of $g ^ { * }$ as
+$$
+g ^ { * } = \sum _ { m \in [ M ] } \widetilde { g } _ { m } \otimes \pmb { I } _ { \widetilde { S } _ { m } } .
+$$
+Also, by the definition of $\mathcal { S } _ { 2 \beta , J }$ which contains $\widetilde { S } _ { m }$ , we obtain the form
+$$
+{ \cal I } _ { { \widetilde { \cal S } } _ { m } } = \bigotimes _ { j \in [ J ] } \psi _ { h _ { m , j } } ,
+$$
+with existing $\psi _ { h _ { m , j } }$ . Then, we have
+$$
+g ^ { * } = \sum _ { m \in [ M ] } \widetilde { g } _ { m } \bigotimes _ { j \in [ J ] } \psi _ { h _ { m , j } } .
+$$
+Preliminarily, we apply sub-neural networks from Yarotsky (2017) and Petersen & Voigtlaenderř (2017). Let $\zeta [ \Theta _ { + } ]$ be a network for summation such that $\begin{array} { r } { { \bf \Pi } \dot { \zeta } [ \Theta _ { + } ] ( x _ { 1 } , . . . , x _ { D ^ { \prime } } ) = \sum _ { d \in [ D ^ { \prime } ] } x _ { d } } \end{array}$ , and $\zeta [ \Theta _ { \times } ]$ be a network for approximate multiplication such as $| \zeta [ \Theta _ { \times } ] ( x _ { 1 } , . . . , x _ { D ^ { \prime } } ) - \prod _ { d \in [ D ^ { \prime } ] } x _ { d } | < \epsilon$ with some $\epsilon > 0$ for all $x , x ^ { \prime } \in I$ (Proposition 3 in Yarotsky (2017) and Lemma 1 in Imaizumi $\&$ Fukumizu (2018)).
+We consider approximation $\begin{array} { r } { \sum _ { m \in [ M ] } \gamma _ { m , d } \bigotimes _ { j \in [ J ] } \psi _ { h _ { m , j } } } \end{array}$ for each $d \in \mathsf { \Gamma } [ D ]$ . Let $\zeta [ \Theta _ { \gamma , d , m } ]$ and $\zeta [ \Theta _ { h , m , j } ]$ for $d \in [ D ] , m \in [ M ]$ and $j \in [ J ]$ , and we will specify the networks later. Also, let $\zeta \bar { [ \Theta _ { S , m } ] } ^ { - } = \zeta [ \Theta _ { \times } ] ( \bar { \zeta } [ \bar { \Theta } _ { h , m , 1 } ] \bar { ( \cdot ) } , \cdot . . . , \zeta [ \Theta _ { h , m , 1 } ] \bar { ( \cdot ) } )$ .
+We consider a neural network
+$$
+\zeta [ \Theta _ { d } ] = \zeta [ \Theta _ { + } ] ( \zeta [ \Theta _ { \times } ] ( \zeta [ \Theta _ { \gamma , d , 1 } ] ( \cdot ) , \zeta [ \Theta _ { S , 1 } ] ( \cdot ) ) , . . . , \zeta [ \Theta _ { \times } ] ( \zeta [ \Theta _ { \gamma , d , M } ] ( \cdot ) , \zeta [ \Theta _ { S , M } ] ( \cdot ) ) ) .
+$$
+Then, an approximation error is evaluated as
+$$
+\begin{array} { r l } & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ &  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \end{array}
+$$
+where the last inequality follows the Hölder’s inequality.
+About $T _ { 1 , m }$ , there exists a corresponding $\zeta [ \Theta _ { \gamma , d , m } ]$ such that
+$$
+\begin{array} { r } { \| \gamma _ { m , d } - \zeta [ \Theta _ { \gamma , d , m } ] \| _ { L ^ { 2 } } \lesssim \| \Theta _ { \gamma , d , m } \| _ { 1 } ^ { - ( \beta + 1 ) / D } , } \end{array}
+$$
+by following Theorem A.8 in Petersen & Voigtlaender (2017). Also, since $\gamma _ { m , d }$ is bounded by its smoothness and compact support, we have $\| \gamma _ { m , d } \| _ { L ^ { \infty } } < \infty$ hence
+$$
+T _ { 1 , m } \lesssim \| \Theta _ { \gamma , d , m } \| _ { 1 } ^ { - ( \beta + 1 ) / D } .
+$$
+For $T _ { 2 , m }$ , we specify $\zeta [ \Theta _ { h , m , j } ]$ as Theorem 3.1 in Petersen & Voigtlaender (2017). Then, we evaluate the following as
+$$
+\begin{array} { r l } & { \| \gamma _ { m , d } - \zeta [ \Theta _ { \gamma , d , m } ] \| _ { L ^ { 2 } } } \\ & { \leqslant \| \displaystyle \bigoplus _ { j \in [ J ] } \psi _ { h _ { m , j } } - \zeta [ \Theta _ { \star } ] ( \zeta [ \Theta _ { h , m , 1 } ] ( \cdot ) , \ldots , \zeta [ \Theta _ { h , m , 1 } ] ( \cdot ) ) \| _ { L ^ { 2 } } } \\ & { \leqslant \| \displaystyle \bigoplus _ { j \in [ J ] } \psi _ { h _ { m , j } } - \zeta [ \Theta _ { h , m , j } ] \| _ { L ^ { 2 } } + \| \displaystyle \bigotimes _ { j \in [ J ] } \psi _ { h _ { m , j } } - \zeta [ \Theta _ { \star } ] ( \zeta [ \Theta _ { h , m , 1 } ] ( \cdot ) , \ldots , \zeta [ \Theta _ { h , m , 1 } ] ( \cdot ) ) \| _ { L ^ { 2 } } } \\ & { \leqslant \displaystyle \sum _ { j \in [ J ] } \prod _ { j \in [ J ] } \prod _ { l = \delta , j } ( | \psi _ { h _ { m , j } } , | \| _ { L ^ { 2 } } \forall [ \Theta _ { h , m , j } ] \| _ { L ^ { 2 } } ) \| \psi _ { h _ { m , j } } - \zeta [ \Theta _ { h , m , j } ] \| _ { L ^ { 2 } } + \epsilon _ { \times } } \\ & { \leqslant \displaystyle \sum _ { j \in [ J ] } \| \psi _ { h _ { m , j } } - \zeta [ \Theta _ { h , m , j } ] \| _ { L ^ { 2 } } + \epsilon _ { \times } . } \end{array}
+$$
+Here, the last inequality follows the boundedness of $\psi _ { h _ { m , j ^ { \prime } } }$ and $\zeta [ \Theta _ { h , m , j } ]$ by Theorem 3.1 in Petersen & Voigtlaender (2017). Also, Theorem 3.1 in Petersen $\&$ Voigtlaender (2017) provides an existence of $[ \Theta _ { h , m , j }$ such that
+$$
+\begin{array} { r } { \| \psi _ { h _ { m , j } } - \zeta [ \Theta _ { h , m , j } ] \| _ { L ^ { 2 } } \leqslant \| \Theta _ { h , m , j } \| _ { 1 } ^ { - \beta / ( D - 1 ) } . } \end{array}
+$$
+We apply the boundedness of $\zeta [ \Theta _ { h , m , j } ]$ , we have
+$$
+T _ { 2 , m } \lesssim \sum _ { j \in [ J ] } \| \Theta _ { h , m , j } \| _ { 1 } ^ { - \beta / ( D - 1 ) } + \epsilon _ { \times } .
+$$
+Combining the bounds for $T _ { 1 , m }$ and $T _ { 2 , m }$ , we bound
+$$
+\begin{array} { r l } & { \left\| \displaystyle \sum _ { m \in [ M ] } \gamma _ { m , d } \bigotimes \psi _ { h _ { m , j } } - \zeta [ \Theta _ { d } ] \right\| _ { L ^ { 2 } } } \\ & { \leqslant \displaystyle \sum _ { m \in [ M ] } \| \Theta _ { \gamma , d , m } \| _ { 1 } ^ { - ( \beta + 1 ) / D } + \displaystyle \sum _ { m \in [ M ] } \sum _ { j \in [ J ] } \| \Theta _ { h , m , j } \| _ { 1 } ^ { - \beta / ( D - 1 ) } + ( M + 1 ) \epsilon _ { \times } . } \end{array}
+$$
+Here, we consider a parameter $\begin{array} { r } { S = \sum _ { d \in [ D ] } \| \Theta _ { d } \| _ { 1 } } \end{array}$ such that $S = \| \Theta _ { \gamma , d , m } \| _ { 1 } \asymp \| \Theta _ { h , m , j } \| _ { 1 } \asymp \| \Theta _ { \times } \| _ { 1 }$ Also, following Yarotsky (2017) and Petersen & Voigtlaender (2017), a proper selection of $L = | \Theta _ { \times } |$ and $B = \| \Theta _ { \times } \| _ { \infty }$ provides $\epsilon _ { \times } \lesssim \| \Theta _ { \times } \| _ { 0 } ^ { - \beta / D }$ }´β{D0 . Then, we have
+$$
+\left. \sum _ { m \in [ M ] } \gamma _ { m , d } \bigotimes _ { j \in [ J ] } \psi _ { h _ { m , j } } - \zeta [ \Theta _ { d } ] \right. _ { L ^ { 2 } } \lesssim M J S ^ { - \beta / D } .
+$$
+Since $J$ is finite, we obtain the result.
+# B.4 PROOF OF THEOREM 1
+By the definition of $\widehat g$ in (2), the following inequality holds
+$$
+d _ { \mathcal { F } } ( P _ { n } , P _ { \hat { g } , m } ) = \operatorname* { s u p } _ { f \in \mathcal { F } } ( P _ { n } f - P _ { \hat { g } , m } f ) \leqslant d _ { \mathcal { F } } ( P _ { n } , P _ { g , m } ) = \operatorname* { s u p } _ { f \in \mathcal { F } } ( P _ { n } f - P _ { g , m } f ) ,
+$$
+for arbitrary $g \in { \mathcal { G } }$ .
+We consider a bound for $d _ { \mathcal { F } } ( P ^ { \ast } , P _ { \hat { g } } )$ as
+$$
+\begin{array} { r l } & { d _ { \mathcal { F } } \big ( P ^ { * } , P _ { \hat { g } } \big ) = \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \big ( P ^ { * } f - P _ { \hat { g } } f \big ) } \\ & { \quad \quad = \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \big ( P ^ { * } f - P _ { n } f + P _ { n } f - P _ { \hat { g } , m } f + P _ { \hat { g } , m } f - P _ { \hat { g } } f \big ) } \\ & { \quad \quad \leqslant \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \big ( P ^ { * } f - P _ { n } f + P _ { \hat { g } , m } f - P _ { \hat { g } } f \big ) + \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \big ( P _ { n } f - P _ { \hat { g } , m } f \big ) , } \end{array}
+$$
+where the inequality follows (9) with an existing ${ \dot { g } } \in { \mathcal { G } }$ . We will provide a detailed construction of $g ^ { * }$ . We continue the bound as
+$$
+\begin{array} { r l } & { d _ { \mathcal { F } } ( P ^ { * } , P _ { \widehat { g } } ) } \\ & { \leqslant \underset { f \in \mathcal { F } } { \operatorname* { s u p } } ( P ^ { * } f - P _ { n } f + P _ { \widehat { g } , m } f - P _ { \widehat { g } } f ) + \underset { f \in \mathcal { F } } { \operatorname* { s u p } } ( P _ { n } f - P ^ { * } f + P ^ { * } f - P _ { \widehat { g } } f + P _ { \widehat { g } } f - P _ { \widehat { g } , m } f ) } \\ & { \leqslant 2 \underset { g \in \mathcal { G } } { \operatorname* { s u p } } \underset { f \in \mathcal { F } } { \operatorname* { s u p } } | P _ { g , m } f - P _ { g } f | + \underset { f \in \mathcal { F } } { \operatorname* { s u p } } ( P ^ { * } f - P _ { \widehat { g } } f ) + 2 \underset { f \in \mathcal { F } } { \operatorname* { s u p } } | P _ { n } f - P ^ { * } f | } \\ & { = : i + i i + i i i . } \end{array}
+$$
+Here, $i$ denotes an effect from $m$ samplings, $\romannumeral 2$ denotes an approximation error, and iii denotes an uncertainty with the $n$ observations.
+To evaluate $i$ and $i i i$ , we provide the following lemma. This result follows a standard technique of the empirical process theory and we provide its outline for a sake of completeness.
+Lemma 4. Let $\mathcal { H }$ be a some set of measurable functions and $X _ { 1 } , . . . , X _ { n } \sim P$ be i.i.d. n observations. Suppose that $\mathbb { E } _ { P } [ h ^ { 2 } ( X ) ] \leqslant \sigma ^ { 2 }$ and $\| h \| _ { L ^ { \infty } } < C _ { h }$ hold with finite parameters $\sigma ^ { 2 } > 0$ and $C _ { h } > 0$ . Then, there exists a constant $C _ { \theta }$ and we obtain
+$$
+\begin{array} { r l r } {  { \operatorname* { s u p } _ { h \in \mathcal { H } } | \frac { 1 } { n } \sum _ { i \in [ n ] } h ( X _ { i } ) - \mathbb { E } _ { P } [ h ( X ) ] | } } \\ & { } & { \leqslant \operatorname* { i n f } _ { \eta > 0 } \{ 4 \eta + 1 2 \int _ { \eta } ^ { C _ { h } } \sqrt { \frac { \log \mathcal { N } ( \epsilon , \mathcal { H } , \| \cdot \| _ { L ^ { \infty } } ) } { n } } d \epsilon + \sqrt { \frac { 2 \tau \sigma ^ { 2 } + 4 C _ { \theta } } { n } } + \frac { \tau C _ { h } } { n } ( \frac { 2 } { 3 } + C _ { \theta } ) \} } \end{array}
+$$
+with probability at least $1 - 2 \exp ( - \tau )$ for all $\tau > 0$ .
+Proof. At the beginning, we bound an expectation of the empirical process (Steinwart & Christmann, 2008; Bartlett et al., 2005; Massart, 2000; Sriperumbudur et al., 2012). Afterward, we evaluate a concentration of the empirical process around the expectation.
+By applying the symmetrization and concentration techniques (Proposition 7.10 in Steinwart & Christmann (2008)), we obtain
+$$
+\mathbb { E } _ { P ^ { \otimes n } } \left[ \operatorname* { s u p } _ { h \in \mathcal { H } } \left| \frac { 1 } { n } \sum _ { i \in [ n ] } h ( X _ { i } ) - \mathbb { E } [ h ( X ) ] \right| \right] \leqslant 2 \mathbb { E } _ { P ^ { \otimes n } \otimes \nu ^ { \otimes n } } \left[ \operatorname* { s u p } _ { h \in \mathcal { H } } \left| \frac { 1 } { n } \sum _ { i \in [ n ] } u _ { i } h ( X _ { i } ) \right| \right] ,
+$$
+where $u _ { i } \sim \nu$ is the Rademacher variable which takes 0 or 1 with probability 0.5. Combining this bound with the Taralgand’s inequality (Theorem A.9.1 in Steinwart $\&$ Christmann (2008)), we obtain the following inequality
+$$
+\begin{array} { r l } & { \displaystyle \operatorname* { s u p } _ { h \in \mathcal { H } } \left| \frac { 1 } { n } \sum _ { i \in [ n ] } h ( X _ { i } ) - \mathbb { E } _ { P } [ h ( X ) ] \right| } \\ & { \leqslant ( 1 + \theta ) \mathbb { E } _ { P ^ { \otimes n } \otimes \nu ^ { \otimes n } } \left[ \displaystyle \operatorname* { s u p } _ { h \in \mathcal { H } } \left| \frac { 1 } { n } \sum _ { i \in [ n ] } u _ { i } h ( X _ { i } ) \right| \right] + \sqrt { \frac { 2 \tau \sigma ^ { 2 } } { n } } + \frac { \tau C _ { h } } { n } \left( \frac { 2 } { 3 } + \displaystyle \frac { 1 } { \theta } \right) , } \end{array}
+$$
+with probability at least $1 - \exp ( - \tau )$ for all $\tau > 0$ and $\theta > 0$ .
+About the term with the Rademacher variable, we also apply a similar strategy (Lemma A.4 in Bartlett et al. (2005)), then obtain
+$$
+\mathbb { E } _ { P \hat { \otimes } n } \otimes _ { \mathcal { V } } \otimes n ^ { \prime } \left[ \operatorname* { s u p } _ { h \in \mathcal { H } } \left| \frac { 1 } { n } \sum _ { i \in [ n ] } u _ { i } h ( X _ { i } ) \right| \right] \leqslant \frac { 1 } { 1 - \theta ^ { \prime } } \mathbb { E } _ { \nu } \otimes n \left[ \operatorname* { s u p } _ { h \in \mathcal { H } } \left| \frac { 1 } { n } \sum _ { i \in [ n ] } u _ { i } h ( X _ { i } ) \right| \right] + \frac { \tau ^ { \prime } C _ { h } } { n \theta ^ { \prime } ( 1 - \theta ^ { \prime } ) } ,
+$$
+with probability at least $1 - \exp ( - \tau ^ { \prime } )$ for all $\tau ^ { \prime } > 0$ and $\theta ^ { \prime } > 0$ .
+Let $\| { \bf \nabla } \cdot { \bf \nabla } \| _ { n }$ be an empirical norm as ˇ ˇı $\begin{array} { r l r } { \| f \| _ { n } ^ { 2 } } & { { } = } & { n ^ { - 1 } \sum _ { i \in [ n ] } f ( X _ { i } ) ^ { 2 } } \end{array}$ . About the term $\begin{array} { r } { \mathbb { E } _ { \nu \otimes n } \left[ \operatorname* { s u p } _ { h \in \mathcal { H } } \left| \frac { 1 } { n } \sum _ { i \in [ n ] } u _ { i } h ( X _ { i } ) \right| \right] } \end{array}$ , we apply the chaining technique and obtain
+$$
+\begin{array} { r } { \mathbb { E } _ { \boldsymbol \nu \otimes \boldsymbol n } \left[ \displaystyle \operatorname* { s u p } _ { h \in \mathcal { H } } \left| \frac { 1 } { n } \sum _ { i \in [ n ] } u _ { i } h ( X _ { i } ) \right| \right] \leqslant \displaystyle \operatorname* { i n f } _ { \boldsymbol \theta ^ { \prime \prime } > 0 } \left\{ 4 \boldsymbol \theta ^ { \prime \prime } + 1 2 \int _ { \boldsymbol \theta ^ { \prime \prime } } ^ { \tilde { C } _ { h } } \sqrt { \frac { \log \mathcal { N } ( \epsilon , \mathcal { H } , \| \cdot \| _ { n } ) } { n } } d \epsilon \right\} } \\ { \leqslant \displaystyle \operatorname* { i n f } _ { \boldsymbol \theta ^ { \prime \prime } > 0 } \left\{ 4 \boldsymbol \theta ^ { \prime \prime } + 1 2 \int _ { \boldsymbol \theta ^ { \prime \prime } } ^ { C _ { h } } \sqrt { \frac { \log \mathcal { N } ( \epsilon , \mathcal { H } , \| \cdot \| _ { L ^ { \infty } } ) } { n } } d \epsilon \right\} , } \end{array}
+$$
+where the last inequality follows a bound for an empirical norm and the boundedness of $\mathcal { H }$ .
+Combining (10), (11) and (12) and changing variables, we obtain the result.
+To bound $I$ with $\widetilde { X } _ { 1 } , . . . , \widetilde { X } _ { m } \sim P _ { g }$ , we consider the following value
+$$
+\begin{array} { r l } & { \underset { g \in \mathcal { G } } { \operatorname* { s u p } } \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \left| \frac { 1 } { m } \sum _ { i \in [ m ] } f ( \widetilde { X } _ { i } ) - \mathbb { E } _ { P _ { g } } [ h ( X ) ] \right| = \underset { g \in \mathcal { G } } { \operatorname* { s u p } } \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \left| \frac { 1 } { m } \sum _ { i \in [ m ] } f \circ g ( Z _ { i } ) - \mathbb { E } _ { P _ { \mathcal { Z } } } [ f \circ g ( X ) ] \right| } \\ & { \qquad = : \underset { h \in \mathcal { H } } { \operatorname* { s u p } } \left| \frac { 1 } { m } \sum _ { i \in [ m ] } h ( Z _ { i } ) - \mathbb { E } _ { P _ { \mathcal { Z } } } [ h ( X ) ] \right| , } \end{array}
+$$
+where we define $\mathcal { H } = \left\{ h = f \circ g \vert f \in \mathcal { F } , g \in \mathcal { G } \right\}$ . To apply Lemma 4, we investigate a covering number $\mathcal { N } ( \epsilon , \mathcal { H } , \| \cdot \| _ { L ^ { \infty } } )$ .
+Lemma 5. Assume that $f \in { \mathcal { F } }$ is $L _ { 1 }$ -Lipschitz continuous. We obtain
+$$
+\log \mathcal { N } ( \epsilon , \mathcal { H } , \| \cdot \| _ { L ^ { \infty } } ) \leqslant \log \mathcal { N } ( ( 1 + L _ { 1 } ) ^ { - 1 } \epsilon , \mathcal { G } , \| \cdot \| _ { L ^ { \infty } } ) + \log \mathcal { N } ( ( 1 + L _ { 1 } ) ^ { - 1 } \epsilon , \mathcal { F } , \| \cdot \| _ { L ^ { \infty } } )
+$$
+Proof. Fix $\epsilon > 0$ . Let $G \subset { \mathcal { G } }$ and $F \subset { \mathcal { F } }$ be covering sets as a set of centers of $\epsilon$ -balls for the covering $\mathcal { G }$ and $\mathcal { F }$ . Obviously, $| G | = \mathcal { N } ( \epsilon , \mathcal { G } , \| \cdot \| _ { L ^ { \infty } } )$ and $| F | = \mathcal { N } ( \epsilon , \mathcal { F } , \| \cdot \| _ { L ^ { \infty } } )$ . We define a subset
+$$
+H : = \left\{ h = f \circ g \vert g \in G , h \in F \right\} \subset \mathcal { H } ,
+$$
+and we known $| H | = | G | \times | F |$ .
+For any $h \in \mathcal H$ , there exist $g \in { \mathcal { G } }$ and $f \in { \mathcal { F } }$ , then $f = f \circ g$ holds. Also, by the definition of covering sets, there exist $f ^ { \prime } \in F$ and $g ^ { \prime } \in G$ such that $\| f - f ^ { \prime } \| _ { L ^ { \infty } } \leqslant \epsilon$ and $\| g - g ^ { \prime } \| _ { L ^ { \infty } } \leqslant \epsilon$ . Let $h ^ { \prime } = f ^ { \prime } \circ g ^ { \prime }$ , and we measure the distance
+$$
+\begin{array} { r l } & { \| h - h ^ { \prime } \| _ { L ^ { \infty } } \leqslant \| f \circ g - f ^ { \prime } \circ g ^ { \prime } \| _ { L ^ { \infty } } } \\ & { \qquad = \| f \circ g - f \circ g ^ { \prime } + f \circ g ^ { \prime } - f ^ { \prime } \circ g ^ { \prime } \| _ { L ^ { \infty } } } \\ & { \qquad \leqslant \| f \circ g - f \circ g ^ { \prime } \| _ { L ^ { \infty } } + \| f \circ g ^ { \prime } - f ^ { \prime } \circ g ^ { \prime } \| _ { L ^ { \infty } } } \\ & { \qquad \leqslant L _ { 1 } \| g - g ^ { \prime } \| _ { L ^ { \infty } } + \| f - f ^ { \prime } \| _ { L ^ { \infty } } } \\ & { \qquad \leqslant ( L _ { 1 } + 1 ) \epsilon . } \end{array}
+$$
+Here, the third inequality follows the Lipschitz property of $f \in { \mathcal { F } }$ . Here, we know that $\mathcal { H }$ is covered by $( L _ { 1 } + 1 ) \epsilon$ -balls with the center $H$ . Since $| H | = | G | \times | F |$ , the result holds. □
+Now, we have the following entropy bound
+$$
+\begin{array} { r l } & { \log \mathcal { N } ( \epsilon , \mathcal { H } , \| \cdot \| _ { L ^ { \infty } } ) } \\ & { \leqslant \log { \mathcal { N } ( ( 1 + L _ { 1 } ) ^ { - 1 } \epsilon , \mathcal { G } , \| \cdot \| _ { L ^ { \infty } } ) } + \log \mathcal { N } ( ( 1 + L _ { 1 } ) ^ { - 1 } \epsilon , \mathcal { F } , \| \cdot \| _ { L ^ { \infty } } ) } \\ & { \leqslant ( S _ { g } + 1 ) \log ( 2 ( L _ { 1 } + 1 ) \epsilon ^ { - 1 } ( L _ { g } + 1 ) D _ { g } B _ { g } ) } \\ & { \quad + \operatorname* { m i n } \Biggl \{ ( S _ { f } + 1 ) \log ( 2 ( L _ { 1 } + 1 ) \epsilon ^ { - 1 } ( L _ { f } + 1 ) D _ { f } B _ { f } ) , C _ { \kappa } ( 1 + L _ { 1 } ) \epsilon ^ { - \kappa } \Biggr \} } \end{array}
+$$
+with $\begin{array} { r } { D _ { g } : = \prod _ { \ell \in [ L _ { g } + 1 ] } ( D _ { \ell } + 1 ) } \end{array}$ and $\begin{array} { r } { D _ { f } : = \prod _ { \ell \in [ L _ { f } + 1 ] } ( D _ { \ell } + 1 ) } \end{array}$ . Let $\Gamma _ { f } : = 2 ( L _ { f } + 1 ) D _ { f } B _ { f }$ , and $N _ { \epsilon } ( \widetilde { \mathcal { F } } ) : = \log \mathcal { N } ( \epsilon , \widetilde { \mathcal { F } } , \| \cdot \| _ { n } )$ for brevity. Here, we apply Lemma 8 in Schmidt-Hieber (2017) for the entropy bound for $\mathcal { G }$ and $\mathcal { F }$ . Using the entropy bound and Lemma 4, we obtain
+$$
+\begin{array} { r l r } {  { 2 \operatorname* { s u p } _ { j \in \mathcal { T } } \big | P _ { n } f - P ^ { * } f \big | } } \\ & { \leqslant 4 \eta + \frac { 1 2 } { n ^ { 1 / 2 } } \int _ { \eta } ^ { \infty } \operatorname* { m i n } \Big \{ N _ { c } ( \widetilde { \mathcal { F } } ) , ( S _ { f } + 1 ) \log ( \Gamma _ { f } \epsilon ^ { - 1 } ) \Big \} ^ { 1 / 2 } d \epsilon }  \\ & { } & { + \frac { ( 2 \tau \sigma ^ { 2 } + C _ { \theta } ) ^ { 1 / 2 } } { n ^ { 1 / 2 } } + \frac { \tau C _ { h } ( 2 / 3 + C _ { \theta } ) } { n } } \\ & { \leqslant 4 \eta + \frac { 1 2 } { n ^ { 1 / 2 } } \int _ { \eta } ^ { \infty } N _ { c } ( \widetilde { \mathcal { F } } ) ^ { 1 / 2 } d \epsilon + \frac { 1 2 } { n ^ { 1 / 2 } } ( S _ { f } + 1 ) ^ { 1 / 2 } ( \log ^ { 1 / 2 } \Gamma _ { f } + C _ { \widetilde { \mathcal { F } } } \log ^ { 1 / 2 } C _ { \widetilde { \mathcal { F } } } - \eta \log ^ { 1 / 2 } \eta ) } \\ & { } & { + \frac { ( 2 7 \sigma ^ { 2 } + C _ { \theta } ) ^ { 1 / 2 } } { n ^ { 1 / 2 } } + \frac { \tau C _ { h } ( 2 / 3 + C _ { \theta } ) } { n } } \\ & { \leqslant \Upsilon _ { n } ( \widetilde { \mathcal { F } } ) + \frac { 1 } { n ^ { 1 / 2 } } ( ( S _ { f } + 1 ) ^ { 1 / 2 } A _ { 1 } + A _ { 2 } ) + \frac { A _ { 3 } } { n } , \qquad \mathrm { ~ o ~ } } \end{array}
+$$
+with some $\eta > 0$ , $A _ { 1 } = 1 2 \log ^ { 1 / 2 } \Gamma _ { f } , A _ { 2 } = C _ { \tilde { \pi } } \log ^ { 1 / 2 } C _ { \tilde { \pi } } + ( 2 \tau \sigma ^ { 2 } + C _ { \theta } ) ^ { 1 / 2 }$ , and $A _ { 3 } = \tau C _ { h } ( 2 / 3 +$ $C _ { \theta }$ q. Also, we set $\begin{array} { r } { \Upsilon _ { n } ( \widetilde { \mathcal { F } } ) = 4 \eta + \frac { 1 2 } { n ^ { 1 / 2 } } \int _ { \eta } ^ { C _ { \widetilde { \mathcal { F } } } } N _ { ( L _ { 1 } + 1 ) ^ { - 1 } \epsilon } ( \widetilde { \mathcal { F } } ) ^ { 1 / 2 } d \epsilon } \end{array}$ . Then, we have
+$$
+i i i \leqslant \Upsilon _ { n } ( \widetilde { \mathcal { F } } ) + \frac { 1 } { n ^ { 1 / 2 } } \left( ( S _ { f } + 1 ) ^ { 1 / 2 } A _ { 1 } + A _ { 2 } \right) + \frac { A _ { 3 } } { n } .
+$$
+About $I$ , we define $\Gamma _ { g } : = 2 ( L _ { f } + 1 ) D _ { g } B _ { g }$ and obtain a similar bound as
+$$
+\begin{array} { r l r } & { } & { i \leqslant \Upsilon _ { m } ( \widetilde { \mathcal { F } } ) + \displaystyle \frac { 1 } { m ^ { 1 / 2 } } \left( ( S _ { g } + 1 ) ^ { 1 / 2 } A _ { 1 } ^ { \prime } + A _ { 2 } ^ { \prime } \right) + \displaystyle \frac { A _ { 3 } ^ { \prime } } { m } , } \\ & { } & { A _ { 1 } ^ { \prime } = \smash { 1 2 ( \log ^ { 1 / 2 } \Gamma _ { f } + \log ^ { 1 / 2 } \Gamma _ { g } ) } , A _ { 2 } = C _ { \widetilde { \mathcal { F } } } \log ^ { 1 / 2 } C _ { \widetilde { \mathcal { F } } } + C _ { \mathcal { G } } \log ^ { 1 / 2 } C _ { \mathcal { G } } + 2 ( 2 \tau \sigma ^ { 2 } + C _ { \theta } ) ^ { 1 / 2 } , } \end{array}
+$$
+where and $A _ { 3 } \stackrel { - } { = } 2 \tau C _ { h } ( 2 / 3 + \check { C } _ { \theta } )$ .
+About $\romannumeral 2$ , we evaluate the error from approximation by constructing a specific deep neural network for generators. We apply Lemma 3 and let $\dot { \boldsymbol g } = ( \dot { g } _ { 1 } , . . . , \dot { g } _ { D } )$ be a generator specified in Lemma 3.
+$$
+\begin{array} { r l } & { \mathrm { i } i = P _ { g } \ast f - P _ { g } f } \\ & { = \displaystyle \int \big ( f o { g ^ { \ast } } - f o \bar { g } i \big ) d P _ { Z } } \\ & { \lesssim L \displaystyle \int \big | \displaystyle { g ^ { \ast } } - \bar { g } \big | | d P _ { Z } } \\ & { \lesssim L _ { 1 } \left( \displaystyle \sum _ { i \neq [ D ] } \left\lceil g _ { i } ^ { \ast } ( x ) - \bar { g } _ { d } ( x ) \right\rceil ^ { 2 } d P _ { Z } ( x ) \right) ^ { 1 / 2 } } \\ & { - L _ { 1 } \left( \displaystyle \sum _ { i \neq [ D ] } \| g _ { i } ^ { \ast } - \bar { g } _ { d } \| _ { L ^ { 2 } } ^ { 2 } \right) ^ { 1 / 2 } } \\ & { \lesssim \epsilon _ { g , L } \displaystyle L _ { 1 } D M _ { S } e ^ { \frac { 1 } { g } / D } , } \end{array}
+$$
+which follows $L _ { 1 }$ -Lipschitz continuity of $f$ , the Jensen’s inequality, the Cauchy-Schwartz inequality, compactness of the support $I ^ { D }$ , and uniformity of $P _ { Z }$ . Then, we have
+$$
+\begin{array} { r } { i i \leqslant c _ { 2 } M D S _ { g } ^ { - \beta / D } . } \end{array}
+$$
+Combining the result, we obtain the result of Theorem 1.
+# B.5 PROOF OF PROPOSITION 1
+When $P$ are globally smooth, we obtain a $\beta$ -smooth density function on $I ^ { D }$ by its definition. Due to the smoothness, the studies for nonparametric statistics (Nadaraya, 1964; Ghosal et al., 2007; Efromovich, 2010; van der Vaart $\&$ van Zanten, 2008; Tsybakov, 2009) guarantees that the methods (KDE,NB,SDE, and GP) obtain the rate $O ( n ^ { - \beta / ( 2 \beta + D ) } )$ with respect to the roof of $L ^ { 2 }$ norm.
+When $P$ have disconnected supports and locally smooth, we consider a following specific $P$ . Fix $M = 2$ . Let us define supports as $S _ { 1 } = \widetilde { S } _ { 1 } = \{ x \in I ^ { D } \mid x _ { 1 } \leqslant 0 . 5 \} \subset I ^ { D }$ and $S _ { 2 } = \bar { S } _ { 2 } = \{ x \in I ^ { D } \mid $ $x _ { 1 } \leqslant 0 . 5 \} \subset I ^ { D }$ . Also, $g _ { 1 } ( z ) = z$ and $g _ { 2 } : \widetilde { S } _ { 2 }  S _ { 2 }$ as
+$$
+g _ { 2 } ( z ) = ( g _ { 2 , 1 } ( z _ { 1 } ) , . . . , g _ { 2 , D } ( z _ { D } ) ) ^ { \top } ,
+$$
+where $g _ { 2 , 1 } ( z _ { 1 } ) = 0 . 6 + c ( z _ { 1 } - 0 . 5 ) ^ { 1 / 3 }$ and $g _ { 2 , d } ( z _ { d } ) = z _ { d }$ for $d \in [ D ] \backslash \{ 1 \}$ with a constant $c$ . Then, by the proof of Lemma 1, $p _ { 2 } ( x )$ on $S _ { 2 }$ is a quadratic function with respect to $z _ { 1 }$ is 1-times differentiable but not twice-differentiable at the boundary $\{ x \in I ^ { D } \mid x _ { 1 } = 0 . 6 \dot  \}$ . Hence, the studies (Nadaraya, 1964; Ghosal et al., 2007; Efromovich, 2010; van der Vaart & van Zanten, 2008; Tsybakov, 2009) provides that the generalization error of the methods is bounded by $O ( n ^ { - \beta / ( 2 \beta + D ) } )$ with $\beta = 1$ .

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+# DEEP UNSUPERVISED LEARNING THROUGH SPATIAL CONTRASTING
+Elad Hoffer
+Technion - Israel Institute of Technology
+Haifa, Israel
+ehoffer@tx.technion.ac.il
+Itay Hubara
+Technion - Israel Institute of Technology
+Haifa, Israel
+itayh@tx.technion.ac.il
+Nir Ailon
+Technion - Israel Institute of Technology
+Haifa, Israel
+nailon@cs.technion.ac.il
+# ABSTRACT
+Convolutional networks have marked their place over the last few years as the best performing model for various visual tasks. They are, however, most suited for supervised learning from large amounts of labeled data. Previous attempts have been made to use unlabeled data to improve model performance by applying unsupervised techniques. These attempts require different architectures and training methods. In this work we present a novel approach for unsupervised training of Convolutional networks that is based on contrasting between spatial regions within images. This criterion can be employed within conventional neural networks and optimized using standard techniques such as SGD and backpropagation, thus complementing supervised methods.
+# 1 INTRODUCTION
+For the past few years convolutional networks (ConvNets, CNNs) LeCun et al. (1998) have proven themselves as a successful model for vision related tasks Krizhevsky et al. (2012) Mnih et al. (2015) Pinheiro et al. (2015) Razavian et al. (2014). A convolutional network is composed of multiple convolutional and pooling layers, followed by a fully-connected affine transformations. As with other neural network models, each layer is typically followed by a non-linearity transformation such as a rectified-linear unit (ReLU).
+A convolutional layer is applied by cross correlating an image with a trainable weight filter. This stems from the assumption of stationarity in natural images, which means that parameters learned for one local region in an image can be shared for other regions and images.
+Deep learning models, including convolutional networks, are usually trained in a supervised manner, requiring large amounts of labeled data (ranging between thousands to millions of examples per-class for classification tasks) in almost all modern applications. These models are optimized using a variant of stochastic-gradient-descent (SGD) over batches of images sampled from the whole training dataset and their ground truth-labels. Gradient estimation for each one of the optimized parameters is done by back propagating the objective error from the final layer towards the input. This is commonly known as ”backpropagation” Rumelhart et al..
+In early works, unsupervised training was used as a part of pre-training procedure to obtain an effective initial state of the model. The network was later fine-tuned in a supervised manner as displayed by Hinton (2007). Such unsupervised pre-training procedures were later abandoned, since they provided no apparent benefit over other initialization heuristics in more careful fully supervised training regimes. This led to the de-facto almost exclusive usage of neural networks in supervised environments.
+In this work we will present a novel unsupervised learning criterion for convolutional network based on comparison of features extracted from regions within images. Our experiments indicate that by using this criterion to pre-train networks we can improve their performance and achieve state-ofthe-art results.
+# 2 PREVIOUS WORKS
+Using unsupervised methods to improve performance have been the holy grail of deep learning for the last couple of years and vast research efforts have been focused on that. We hereby give a short overview of the most popular and recent methods that tried to tackle this problem.
+AutoEncoders and reconstruction loss These are probably the most popular models for unsupervised learning using neural networks, and ConvNets in particular. Autoencoders are NNs which aim to transform inputs into outputs with the least possible amount of distortion. An Autoencoder is constructed using an encoder $G ( x ; w _ { 1 } )$ that maps an input to a hidden compressed representation, followed by a decoder $F ( y ; w _ { 2 } )$ , that maps the representation back into the input space. Mathematically, this can be written in the following general form:
+$$
+\hat { x } = F ( G ( x ; w _ { 1 } ) ; w _ { 2 } )
+$$
+The underlying encoder and decoder contain a set of trainable parameters that can be tied together and optimized for a predefined criterion. The encoder and decoder can have different architectures, including fully-connected neural networks, ConvNets and others. The criterion used for training is the reconstruction loss, usually the mean squared error (MSE) between the original input and its reconstruction Zeiler et al. (2010)
+$$
+m i n \lVert x - { \hat { x } } \rVert ^ { 2 }
+$$
+This allows an efficient training procedure using the aforementioned backpropagation and SGD techniques. Over the years autoencoders gained fundamental role in unsupervised learning and many modification to the classic architecture were made. $\mathrm { N g }$ (2011) regularized the latent representation to be sparse, Vincent et al. (2008) substituted the input with a noisy version thereof, requiring the model to denoise while reconstructing. Kingma et al. (2014) obtained very promising results with variational autoencoders (VAE). A variational autoencoder model inherits typical autoencoder architecture, but makes strong assumptions concerning the distribution of latent variables. They use variational approach for latent representation learning, which results in an additional loss component which required a new training algorithm called Stochastic Gradient Variational Bayes (SGVB). VAE assumes that the data is generated by a directed graphical model $p ( x | z )$ and require the encoder to learn an approximation $q _ { w _ { 1 } } ( z | x )$ to the posterior distribution $p _ { w _ { 2 } } ( z | x )$ where $w _ { 1 }$ and $w _ { 2 }$ denote the parameters of the encoder and decoder. The objective of the variational autoencoder in that case has the following form:
+$$
+\mathcal { L } ( w _ { 1 } , w _ { 2 } , x ) = - D _ { K L } \big ( q _ { w _ { 1 } } ( z | x ) | | p _ { w _ { 2 } } ( z ) \big ) + \mathbb { E } _ { q _ { w _ { 1 } } ( z | x ) } \big ( \log p _ { w _ { 2 } } ( x | z ) \big )
+$$
+Recently, a stacked set of denoising autoencoders architectures showed promising results in both semi-supervised and unsupervised tasks. A stacked what-where autoencoder by Zhao et al. (2015) computes a set of complementary variables that enable reconstruction whenever a layer implements a many-to-one mapping. Ladder networks by Rasmus et al. (2015) - use lateral connections and layer-wise cost functions to allow the higher levels of an autoencoder to focus on invariant abstract features.
+Exemplar Networks: The unsupervised method introduced byDosovitskiy et al. (2014) takes a different approach to this task and trains the network to discriminate between a set of pseudo-classes. Each pseudo-class is formed by applying multiple transformations to a randomly sampled image patch. The number of pseudo-classes can be as big as the size of the input samples. This criterion ensures that different input samples would be distinguished while providing robustness to the applied transformations. In this work we will explore an alternative method with a similar motivation.
+Context prediction Another method for unsupervised learning by context was introduced by Doersch et al. (2015). This method uses an auxiliary criterion of predicting the location of an image patch given another from the same image. This is done by classification to 1 of 9 possible locations. Although the work of Doersch et al. (2015) and ours both use patches from an image to perform unsupervised learning, the methods are quite different. Whereas the former used a classification criterion over the spatial location of each patch within a single image, our work is concerned with comparing patches from several images to each other. We claim that this encourages discriminability between images (which we feel to be important aspect of feature learning), and was not an explicit goal in previous work.
+Adversarial Generative Models: This a recently introduced model that can be used in an unsupervised fashion Goodfellow et al. (2014). Adversarial Generative Models uses a set of networks, one trained to discriminate between data sampled from the true underlying distribution (e.g., a set of images), and a separate generative network trained to be an adversary trying to confuse the first network. By propagating the gradient through the paired networks, the model learns to generate samples that are distributed similarly to the source data. As shown by Radford et al. (2015),this model can create useful latent representations for subsequent classification tasks.
+Sampling Methods: Methods for training models to discriminate between a very large number of classes often use a noise contrasting criterion. In these methods, roughly speaking, the posterior probability $P ( t | y _ { t } )$ of the ground-truth target $t$ given the model output on an input sampled from the true distribution $y _ { t } = F ( x )$ is maximized, while the probability $P ( t | y _ { n } )$ given a noise measurement $y = F ( n )$ is minimized. This was successfully used in a language domain to learn unsupervised representation of words. The most noteworthy case is the word2vec model introduced by Mikolov et al. (2013). When using this setting in language applications, a natural contrasting noise is a smooth approximation of the Unigram distribution. A suitable contrasting distribution is less obvious when data points are sampled from a high dimensional continuous space, such as the case of image patches.
+# 2.1 PROBLEMS WITH CURRENT APPROACHES
+Only recently the potential of ConvNets in an unsupervised environment began to bear fruit, still we believe it is not fully uncovered.
+The majority of unsupervised optimization criteria currently used are based on variations of reconstruction losses. One limitation of this fact is that a pixel level reconstruction is non-compliant with the idea of a discriminative objective, which is expected to be agnostic to low level information in the input. In addition, it is evident that MSE is not best suited as a measurement to compare images, for example, viewing the possibly large square-error between an image and a single pixel shifted copy of it. Another problem with recent approaches such as Rasmus et al. (2015); Zeiler et al. (2010) is their need to extensively modify the original convolutional network model. This leads to a gap between unsupervised method and the state-of-the-art, supervised, models for classification - which can hurt future attempt to reconcile them in a unified framework, as well as efficiently leverage unlabeled data with otherwise supervised regimes.
+# 3 LEARNING BY COMPARISONS
+The most common way to train NN is by defining a loss function between the target values and the network output. Learning by comparison approaches the supervised task from a different angle. The main idea is to use distance comparisons between samples to learn useful representations. For example, we consider relative and qualitative examples of the form $X _ { 1 }$ is closer to $X _ { 2 }$ than $X _ { 1 }$ is to $X _ { 3 }$ . Using a comparative measure with neural network to learn embedding space was introduced in the “Siamese network” framework by Bromley et al. (1993) and later used in the works of Chopra et al. (2005). One use for this methods is when the number of classes is too large or expected to vary over time, as in the case of face verification, where a face contained in an image has to compared against another image of a face. This problem was recently tackled by Schroff et al. (2015) for training a convolutional network model on triplets of examples. There, one image served as an anchor $x$ , and an additional pair of images served as a positive example $x _ { + }$ (containing an instance of the face of the same person) together with a negative example $x _ { - }$ , containing a face of a different person. The training objective was on the embedded distance of the input faces, where the distance between the anchor and positive example is adjusted to be smaller by at least some constant $\alpha$ from the negative distance. More precisely, the loss function used in this case was defined as
+$$
+L ( x , x _ { + } , x _ { - } ) = \operatorname* { m a x } \left\{ \| F ( x ) - F ( x _ { + } ) \| _ { 2 } - \| F ( x ) - F ( x _ { - } ) \| _ { 2 } + \alpha , 0 \right\}
+$$
+where $F ( x )$ is the embedding (the output of a convolutional neural network), and $\alpha$ is a predefined margin constant. Another similar model used by Hoffer & Ailon (2015) with triplets comparisons for classification, where examples from the same class were trained to have a lower embedded distance than that of two images from distinct classes. This work introduced a concept of a distance ratio loss, where the defined measure amounted to:
+$$
+L ( x , x _ { + } , x _ { - } ) = \frac { e ^ { - \| F ( x ) - F ( x _ { + } ) \| _ { 2 } } } { e ^ { - \| F ( x ) - F ( x _ { + } ) \| _ { 2 } } + e ^ { - \| F ( x ) - F ( x _ { - } ) \| _ { 2 } } }
+$$
+This loss has a flavor of a probability of a biased coin flip. By ‘pushing’ this probability to zero, we express the objective that pairs of samples coming from distinct classes should be less similar to each other, compared to pairs of samples coming from the same class. It was shown empirical by Balntas et al. (2016) to provide better feature embeddings than the margin based distance loss 1
+# 4 OUR CONTRIBUTION: SPATIAL CONTRASTING
+One implicit assumption in convolutional networks, is that features are gradually learned hierarchically, each level in the hierarchy corresponding to a layer in the network. Each spatial location within a layer corresponds to a region in the original image. It is empirically observed that deeper layers tend to contain more ‘abstract’ information from the image. Intuitively, features describing different regions within the same image are likely to be semantically similar (e.g. different parts of an animal), and indeed the corresponding deep representations tend to be similar. Conversely, regions from two probably unrelated images (say, two images chosen at random) tend to be far from each other in the deep representation. This logic is commonly used in modern deep networks such as Szegedy et al. (2015) Lin et al. (2013) He et al. (2015), where a global average pooling is used to aggregate spatial features in the final layer used for classification.
+Our suggestion is that this property, often observed as a side effect of supervised applications, can be used as a desired objective when learning deep representations in an unsupervised task. Later, the resulting representation can be used, as typically done, as a starting point or a supervised learning task. We call this idea which we formalize below Spatial contrasting. The spatial contrasting criterion is similar to noise contrasting estimation Gutmann & Hyvarinen (2010) Mnih & Kavukcuoglu ¨ (2013), in trying to train a model by maximizing the expected probability on desired inputs, while minimizing it on contrasting sampled measurements.
+# 4.1 FORMULATION
+We will concern ourselves with samples of images patches $\tilde { x } ^ { ( m ) }$ taken from an image $x$ . Our convolutional network model, denoted by $F ( x )$ , extracts spatial features $f$ so that $f ^ { ( m ) } = F ( \tilde { x } ^ { ( m ) } )$ for an image patch $\tilde { x } ^ { ( m ) }$ . We will also define $P ( f _ { 1 } | f _ { 2 } )$ as the probability for two features $f _ { 1 } , f _ { 2 }$ to occur together in the same image.
+We wish to optimize our model such that for two features representing patches taken from the same image x˜(1)i , x˜ $\tilde { x } _ { i } ^ { ( 1 ) } , \bar { x _ { i } ^ { ( 2 ) } } \in x _ { i }$ for which $f _ { i } ^ { ( 1 ) } = F ( \tilde { x } _ { i } ^ { ( 1 ) } )$ and $f _ { i } ^ { ( 2 ) } \stackrel { - } { = } F ( \tilde { x } _ { i } ^ { ( 2 ) } )$ , $P ( f _ { i } ^ { ( 1 ) } | f _ { i } ^ { ( 2 ) } )$ will be maxi
+This means that features from a patch taken from a specific image can effectively predict, under our model, features extracted from other patches in the same image. Conversely, we want our model to minimize $P ( f _ { i } | f _ { j } )$ for $i , j$ being two patches taken from distinct images. Following the logic presented before, we will need to sample contrasting patch $\tilde { x } _ { j } ^ { ( 1 ) }$ from a different image $x _ { j }$ such that $P ( f _ { i } ^ { ( 1 ) } | f _ { i } ^ { ( 2 ) } ) > P ( f _ { j } ^ { ( 1 ) } | f _ { i } ^ { ( 2 ) } )$ , where $\underset { . } { f _ { j } ^ { ( 1 ) } } = F ( \tilde { x } _ { j } ^ { ( 1 ) } )$ . In order to obtain contrasting samples, we use regions from two random images in the training set. We will use a distance ratio, described earlier in Eq. (2) for the supervised case, to represent the probability two feature vectors were taken from the same image. The resulting training loss for a pair of images will be defined as
+$$
+L _ { S C } ( x _ { 1 } , x _ { 2 } ) = - \log \frac { e ^ { - \| f _ { 1 } ^ { ( 1 ) } - f _ { 1 } ^ { ( 2 ) } \| _ { 2 } } } { e ^ { - \| f _ { 1 } ^ { ( 1 ) } - f _ { 1 } ^ { ( 2 ) } \| _ { 2 } } + e ^ { - \| f _ { 1 } ^ { ( 1 ) } - f _ { 2 } ^ { ( 1 ) } \| _ { 2 } } }
+$$
+Effectively minimizing a log-probability under the SoftMax measure. This formulation is portrayed in figure 4.1. Since we sample our contrasting sample from the same underlying distribution, we can evaluate this loss considering the image patch as both patch compared (anchor) and contrast symmetrically. The final loss will be the average between these estimations:
+$$
+\widehat { L } _ { S C } ( x _ { 1 } , x _ { 2 } ) = \frac { 1 } { 2 } \left[ L _ { S C } ( x _ { 1 } , x _ { 2 } ) + L _ { S C } ( x _ { 2 } , x _ { 1 } ) \right]
+$$
+![](images/f412cdc401c9a392a006634319c43ab46c8b2578a7cd2274d9536d24baa26d3b.jpg)
+Figure 1: Spatial contrasting depiction.
+# 4.2 METHOD
+Convolutional network are usually trained using SGD over mini-batch of samples, therefore we can extract patches and contrasting patches without changing the network architecture. Each image serves as both anchor and positive patches, for which the corresponding features should be closer, as well as contrasting samples for other images in that batch. For a batch of $N$ images, two samples from each image are taken, and $N ^ { 2 }$ different distance comparisons are made. The final loss is defined as the average distance ratio for all images in the batch:
+$$
+\overline { { L } } _ { S C } ( \{ x \} _ { i = 1 } ^ { N } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } L _ { S C } ( x _ { i } , \{ x \} _ { j \neq i } ) = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \frac { e ^ { - \| f _ { i } ^ { ( 1 ) } - f _ { i } ^ { ( 2 ) } \| _ { 2 } } } { \sum _ { j = 1 } ^ { N } e ^ { - \| f _ { i } ^ { ( 1 ) } - f _ { j } ^ { ( 2 ) } \| _ { 2 } } }
+$$
+Since the criterion is differentiable with respect to its inputs, it is fully compliant with standard methods for training convolutional network and specifically using backpropagation and gradient descent. Furthermore, SC can be applied to any layer in the network hierarchy. In fact, SC can be used at multiple layers within the same convolutional network. The spatial properties of the features means that we can sample directly from feature space $\tilde { f } ^ { ( m ) } \in f$ instead of from the original image. Therefore SC has a simple implementation which doesn’t require substation amount of computation. The complete algorithm for batch training is described in Algorithm (1). Similar to the batch normalization (BN) layer Ioffe & Szegedy (2015), a recent usage for batch statistics in neural networks, SC also uses the batch statistics. While BN normalize the input based on the batch statistics, SC sample from it. This can be viewed as a simple sampling from the space of possible features describing a patch of image.
+Algorithm 1 Calculation the spatial contrasting loss
+Require: $X = \{ x \} _ { i = 1 } ^ { N }$ # Training on batches of images
+# Get the spatial features for the whole batch of images
+# Size: $N \times W _ { f } \times H _ { f } \times C$
+$\{ f \} _ { i = 1 } ^ { N }  \mathsf { C o n v N e t } ( X )$
+$$
+\begin{array} { r } { d _ { i } \gets - \log \frac { \bar { e } ^ { - D i s t ( i , i ) } } { \sum _ { k = 1 } ^ { N } e ^ { - D i s t ( i , k ) } } } \end{array}
+$$
+# Spatial contrasting loss is the mean of distance ratios return $\textstyle { \frac { 1 } { N } } \sum _ { i = 1 } ^ { N } d _ { i }$ g
+# 5 EXPERIMENTS
+In this section we report empirical results showing that using SC loss as an unsupervised pretraining procedure can improve state-of-the-art performance on subsequent classification. We experimented with MNIST, CIFAR-10 and STL10 datasets. We used modified versions of well studied networks such as those of Lin et al. (2013) and Rasmus et al. (2015). A detailed description of our architecture can be found in 4.
+In each one of the experiments, we used the spatial contrasting criterion to train the network on the unlabeled images. In each usage of SC criterion, patch features were sampled from the preceding layer in uniform. We note that spatial size of sampled patches ranged between datasets, where on STL10 and Cifar10 it covered about $3 0 \%$ of the image, MNIST required the use of larger patches covering almost the entire image.Training was done by using SGD with an initial learning rate of 0.1 that was decreased by a factor of 10 whenever the measured loss stopped decreasing. After convergence, we used the trained model as an initialization for a supervised training on the complete labeled dataset. The supervised training was done following the same regime, only starting with a lower initial learning rate of 0.01. We used mild data augmentations, such as small translations and horizontal mirroring.
+The datasets we used are:
+• STL10 (Coates et al. (2011)). This dataset consists of $1 0 0 , 0 0 0 9 6 \times 9 6$ colored, unlabeled images, together with another set of 5, 000 labeled training images and 8, 000 test images . The label space consists of 10 object classes. • Cifar10 (Krizhevsky & Hinton (2009)). The well known CIFAR-10 is an image classification benchmark dataset containing 50, 000 training images and 10, 000 test images. The image sizes $3 2 \times 3 2$ pixels, with color. The classes are airplanes, automobiles, birds, cats, deer, dogs, frogs, horses, ships and trucks.
+Table 1: State of the art results on STL-10 dataset
+<table><tr><td>Model</td><td>STL-10 test accuracy</td></tr><tr><td>Zero-bias Convnets - Paine et al. (2014)</td><td>70.2%</td></tr><tr><td>Triplet network -Hoffer &amp; Ailon (2015)</td><td>70.7%</td></tr><tr><td>Exemplar Convnets - Dosovitskiy et al. (2014)</td><td>72.8%</td></tr><tr><td>Target Coding - Yang et al. (2015)</td><td>73.15%</td></tr><tr><td>Stacked what-where AE - Zhao et al. (2015)</td><td>74.33%</td></tr><tr><td>Spatial contrasting initialization (this work)</td><td>81.34% ± 0.1</td></tr><tr><td>The same model without initialization</td><td>72.6%±0.1</td></tr></table>
+• MNIST (LeCun et al. (1998)). The MNIST database of handwritten digits is one of the most studied dataset benchmark for image classification. The dataset contains 60,000 examples of handwritten digits from 0 to 9 for training and 10,000 additional examples for testing. Each sample is a $2 8 \times 2 8$ pixel gray level image.
+All experiments were conducted using the Torch7 framework by Collobert et al. (2011). Code reproducing these results will by available at https://github.com/eladhoffer/ SpatialContrasting.
+# 5.1 RESULTS ON STL10
+Since STL10 dataset is comprised of mostly unlabeled data, it is most suitable to highlight the benefits of the spatial contrasting criterion. The initial training was unsupervised, as described earlier, using the entire set of 105, 000 samples (union of the original unlabeled set and labeled training set). The representation outputted by the training, was used to initialize supervised training on the 5, 000 labeled images. Evaluation was done on a separate test set of 8, 000 samples. Comparing with state of the art results, we see an improvement of $7 \%$ in test accuracy over the best model by Zhao et al. (2015), setting the SC as best model at $8 1 . 3 \%$ test classification accuracy (see Table (1)). We note that the results of Dosovitskiy et al. (2014) are achieved with no fine-tuning over labeled examples, which may be unfair to this work. We also compare with the same network, but without SC initialization, which achieves a lower classification of $7 2 . 6 \%$ . This is an indication that indeed SC managed to leverage unlabeled examples to provide a better initialization point for the supervised model.
+# 5.2 RESULTS ON CIFAR10
+For Cifar10 dataset, we use the same setting as Coates & $\mathrm { N g } \left( 2 0 1 2 \right)$ and Hui (2013) to test a model’s ability to learn from unlabeled images. Here, only 4, 000 samples out of 50, 000 are used with their label annotation, and the rest of the samples can be used only in an unsupervised manner. The final test accuracy is measured on the entire 10, 000 test set.
+In our experiments, we trained our model using SC criterion on the entire dataset, and then used only 400 labeled samples per class (for a total of 4000) in a supervised regime over the initialized network. The results are compared with previous efforts in Table (2). Using the SC criterion allowed an improvement of $6 . 8 \%$ over a non-initialized model, and achieved a final test accuracy of $7 9 . 2 \%$ . This is a competitive result with current state-of-the-art models.
+# 5.3 RESULTS ON MNIST
+The MNIST dataset is very different in nature from the Cifar10 and STL10 datasets, we experimented earlier. The biggest difference, relevant to this work, is that spatial regions sampled from MNIST images usually provide very little, or no information. Thus, SC is much less suited for MNIST dataset, and was conjured to have little benefit. We still, however, experimented with initializing a model with SC criterion and continuing with a fully-supervised regime over all labeled examples. We found again that this provided benefit over training the same network without preinitialization, improving results from ${ \bar { 0 } } . 6 3 \%$ to $0 . 3 4 \%$ error on test set. As mentioned previously, the effective compared patches of MNIST covered almost the entire image area. This can be attributed to the fact that MNIST requires global features to differentiate between digits. The results, compared with previous attempts are included in Table (3).
+Table 2: State of the art results on Cifar10 dataset with only 4000 labeled samples
+<table><tr><td>Model</td><td>Cifar10 (400 per class) test accuracy</td></tr><tr><td>Convolutional K-means Network - Coates &amp; Ng (2012)</td><td>70.7%</td></tr><tr><td>View-Invariant K-means -Hui (2013)</td><td>72.6%</td></tr><tr><td>DCGAN - Radford et al. (2015)</td><td>73.8%</td></tr><tr><td>Exemplar Convnets - Dosovitskiy et al. (2014)</td><td>76.6%</td></tr><tr><td>Ladder networks - Rasmus et al. (2015)</td><td>79.6%</td></tr><tr><td>Conv-CatGan Springenberg (2016)</td><td>80.42% (± 0.58)</td></tr><tr><td>ImprovedGan Salimans et al. (2016)</td><td>81.37% (± 2.32)</td></tr><tr><td>Spatial contrasting initialization (this work)</td><td>79.2%(±0.3)</td></tr><tr><td>The same model without initialization</td><td>72.4%(±0.1)</td></tr></table>
+Table 3: results on MNIST dataset
+<table><tr><td>Model</td><td>MNIST test error</td></tr><tr><td>Stacked what-where AE - Zhao et al. (2015)</td><td>0.71%</td></tr><tr><td>Triplet network - Hoffer &amp; Ailon (2015)</td><td>0.56%</td></tr><tr><td>Jarrett et al. (2009)</td><td>0.53%</td></tr><tr><td>Ladder networks - Rasmus et al. (2015)</td><td>0.36%</td></tr><tr><td>DropConnect - Wan et al. (2013)</td><td>0.21%</td></tr><tr><td>Spatial contrasting initialization (this work)</td><td>0.34%± 0.02</td></tr><tr><td>The same model without initialization</td><td>0.63%± 0.02</td></tr></table>
+# 6 CONCLUSIONS AND FUTURE WORK
+In this work we presented spatial contrasting - a novel unsupervised criterion for training convolutional networks on unlabeled data. Its is based on comparison between spatial features sampled from a number of images. We’ve shown empirically that using spatial contrasting as a pretraining technique to initialize a ConvNet, can improve its performance on a subsequent supervised training. In cases where a lot of unlabeled data is available, such as the STL10 dataset, this translates to state-of-the-art classification accuracy in the final model.
+Since the spatial contrasting loss is a differentiable estimation that can be computed within a network parallel to supervised losses, in future work we plan to embed it as a semi-supervised model. This usage will allow to create models that can leverage both labeled an unlabeled data, and can be compared to similar semi-supervised models such as the ladder network Rasmus et al. (2015). It is is also apparent that contrasting can occur in dimensions other than the spatial, the most straightforward is the temporal dimension. This suggests that similar training procedure can be applied on segments of sequences to learn useful representation without explicit supervision.
+# REFERENCES
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+Kevin Jarrett, Koray Kavukcuoglu, Marc’Aurelio Ranzato, and Yann LeCun. What is the best multistage architecture for object recognition? In Computer Vision, 2009 IEEE 12th International Conference on, pp. 2146–2153. IEEE, 2009.
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Computer Science Department, University of Toronto, Tech. Rep, 2009.
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+Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
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+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pp. 3111–3119, 2013.
+Andriy Mnih and Koray Kavukcuoglu. Learning word embeddings efficiently with noise-contrastive estimation. In Advances in Neural Information Processing Systems, pp. 2265–2273, 2013.
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+Andrew Ng. Sparse autoencoder. 2011.
+Tom Le Paine, Pooya Khorrami, Wei Han, and Thomas S Huang. An analysis of unsupervised pre-training in light of recent advances. arXiv preprint arXiv:1412.6597, 2014.
+Pedro O Pinheiro, Ronan Collobert, and Piotr Dollar. Learning to segment object candidates. In Advances in Neural Information Processing Systems, pp. 1981–1989, 2015.
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+Matthew D Zeiler, Dilip Krishnan, Graham W Taylor, and Rob Fergus. Deconvolutional networks. In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, pp. 2528–2535. IEEE, 2010.
+Junbo Zhao, Michael Mathieu, Ross Goroshin, and Yann Lecun. Stacked what-where auto-encoders. arXiv preprint arXiv:1506.02351, 2015.
+# 7 APPENDIX
+Table 4: Convolutional models used, based on Lin et al. (2013), Rasmus et al. (2015)
+<table><tr><td rowspan=1 colspan=3>Model</td></tr><tr><td rowspan=1 colspan=1>STL10</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>MNIST</td></tr><tr><td rowspan=1 colspan=1>Input: 96×96 RGB</td><td rowspan=1 colspan=1>Input:32×32RGB</td><td rowspan=1 colspan=1>Input:28 × 28 monochrome</td></tr><tr><td rowspan=1 colspan=1>5 ×5 conv. 64 BN ReLU</td><td rowspan=1 colspan=1>3 × 3 conv. 96 BN LeakyReLU</td><td rowspan=6 colspan=1>5 ×5 conv.32 ReLU2 × 2 max-pooling,stride 2 BN3 × 3 conv. 64 BN ReLU3 × 3 conv. 64 BN ReLU2 × 2 max-pooling,stride 2 BN</td></tr><tr><td rowspan=1 colspan=1>1 ×1 conv.160 BN ReLU</td><td rowspan=5 colspan=1>3 ×3 conv. 96 BNLeakyReLU3 × 3 conv. 96 BN LeakyReLU2 × 2 max-pooling,stride 2 BN3 × 3 conv.192 BNLeakyReLU3 × 3 conv. 192 BN LeakyReLU3 × 3 conv.192 BN LeakyReLU2 × 2 max-pooling,stride 2 BN</td></tr><tr><td rowspan=1 colspan=1>1 ×1 conv. 96 BN ReLU</td></tr><tr><td rowspan=1 colspan=1>3 × 3 max-pooling, stride 2</td></tr><tr><td rowspan=1 colspan=1>5 × 5 conv. 192 BN ReLU</td></tr><tr><td rowspan=1 colspan=1>1 ×1 conv.192 BN ReLU1 ×1 conv. 192 BN ReLU3 × 3 max-pooling,stride 23 × 3 conv. 192 BN ReLU1 ×1 conv.192 BN ReLU1 ×1 conv.192 BN ReLU</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Spatial contrasting criterion</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>3×3 conv.256 ReLU3 × 3 max-pooling,stride 2dropout, p = 0.53 × 3 conv.128 ReLUdropout, p = 0.5fully-connected 10</td><td rowspan=1 colspan=1>3×3 conv.192BNLeakyReLU1 ×1 conv.192 BNLeakyReLU1 ×1 conv.10 BNLeakyReLUglobal average pooling</td><td rowspan=1 colspan=1>3×3 conv.128 BN ReLU1 × 1 conv.10 BN ReLUglobal average pooling</td></tr></table>
+10-way softmax
+![](images/dbd928541e70043deb704a91118b985563d8e3c93a5cbdcb28b93b6bd8b098bc.jpg)
+Figure 2: First layer convolutional filters after spatial-contrasting training

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+        "text": "DEEP UNSUPERVISED LEARNING THROUGH SPATIAL CONTRASTING ",
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+        "text": "Elad Hoffer   \nTechnion - Israel Institute of Technology   \nHaifa, Israel   \nehoffer@tx.technion.ac.il   \nItay Hubara   \nTechnion - Israel Institute of Technology   \nHaifa, Israel   \nitayh@tx.technion.ac.il   \nNir Ailon   \nTechnion - Israel Institute of Technology   \nHaifa, Israel   \nnailon@cs.technion.ac.il ",
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+        "text": "ABSTRACT ",
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+        "text": "Convolutional networks have marked their place over the last few years as the best performing model for various visual tasks. They are, however, most suited for supervised learning from large amounts of labeled data. Previous attempts have been made to use unlabeled data to improve model performance by applying unsupervised techniques. These attempts require different architectures and training methods. In this work we present a novel approach for unsupervised training of Convolutional networks that is based on contrasting between spatial regions within images. This criterion can be employed within conventional neural networks and optimized using standard techniques such as SGD and backpropagation, thus complementing supervised methods. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "For the past few years convolutional networks (ConvNets, CNNs) LeCun et al. (1998) have proven themselves as a successful model for vision related tasks Krizhevsky et al. (2012) Mnih et al. (2015) Pinheiro et al. (2015) Razavian et al. (2014). A convolutional network is composed of multiple convolutional and pooling layers, followed by a fully-connected affine transformations. As with other neural network models, each layer is typically followed by a non-linearity transformation such as a rectified-linear unit (ReLU). ",
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+        "text": "A convolutional layer is applied by cross correlating an image with a trainable weight filter. This stems from the assumption of stationarity in natural images, which means that parameters learned for one local region in an image can be shared for other regions and images. ",
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+        "text": "Deep learning models, including convolutional networks, are usually trained in a supervised manner, requiring large amounts of labeled data (ranging between thousands to millions of examples per-class for classification tasks) in almost all modern applications. These models are optimized using a variant of stochastic-gradient-descent (SGD) over batches of images sampled from the whole training dataset and their ground truth-labels. Gradient estimation for each one of the optimized parameters is done by back propagating the objective error from the final layer towards the input. This is commonly known as ”backpropagation” Rumelhart et al.. ",
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+        "text": "In early works, unsupervised training was used as a part of pre-training procedure to obtain an effective initial state of the model. The network was later fine-tuned in a supervised manner as displayed by Hinton (2007). Such unsupervised pre-training procedures were later abandoned, since they provided no apparent benefit over other initialization heuristics in more careful fully supervised training regimes. This led to the de-facto almost exclusive usage of neural networks in supervised environments. ",
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+        "text": "In this work we will present a novel unsupervised learning criterion for convolutional network based on comparison of features extracted from regions within images. Our experiments indicate that by using this criterion to pre-train networks we can improve their performance and achieve state-ofthe-art results. ",
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+        "text": "2 PREVIOUS WORKS ",
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+        "text": "Using unsupervised methods to improve performance have been the holy grail of deep learning for the last couple of years and vast research efforts have been focused on that. We hereby give a short overview of the most popular and recent methods that tried to tackle this problem. ",
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+        "text": "AutoEncoders and reconstruction loss These are probably the most popular models for unsupervised learning using neural networks, and ConvNets in particular. Autoencoders are NNs which aim to transform inputs into outputs with the least possible amount of distortion. An Autoencoder is constructed using an encoder $G ( x ; w _ { 1 } )$ that maps an input to a hidden compressed representation, followed by a decoder $F ( y ; w _ { 2 } )$ , that maps the representation back into the input space. Mathematically, this can be written in the following general form: ",
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+        "text": "$$\n\\hat { x } = F ( G ( x ; w _ { 1 } ) ; w _ { 2 } )\n$$",
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+        "text": "The underlying encoder and decoder contain a set of trainable parameters that can be tied together and optimized for a predefined criterion. The encoder and decoder can have different architectures, including fully-connected neural networks, ConvNets and others. The criterion used for training is the reconstruction loss, usually the mean squared error (MSE) between the original input and its reconstruction Zeiler et al. (2010) ",
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+        "text": "$$\nm i n \\lVert x - { \\hat { x } } \\rVert ^ { 2 }\n$$",
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+        "text": "This allows an efficient training procedure using the aforementioned backpropagation and SGD techniques. Over the years autoencoders gained fundamental role in unsupervised learning and many modification to the classic architecture were made. $\\mathrm { N g }$ (2011) regularized the latent representation to be sparse, Vincent et al. (2008) substituted the input with a noisy version thereof, requiring the model to denoise while reconstructing. Kingma et al. (2014) obtained very promising results with variational autoencoders (VAE). A variational autoencoder model inherits typical autoencoder architecture, but makes strong assumptions concerning the distribution of latent variables. They use variational approach for latent representation learning, which results in an additional loss component which required a new training algorithm called Stochastic Gradient Variational Bayes (SGVB). VAE assumes that the data is generated by a directed graphical model $p ( x | z )$ and require the encoder to learn an approximation $q _ { w _ { 1 } } ( z | x )$ to the posterior distribution $p _ { w _ { 2 } } ( z | x )$ where $w _ { 1 }$ and $w _ { 2 }$ denote the parameters of the encoder and decoder. The objective of the variational autoencoder in that case has the following form: ",
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/b80a08937376cb8f40f66af938aa10f2a90d8332197f760430b1611dab07074e.jpg",
+        "text": "$$\n\\mathcal { L } ( w _ { 1 } , w _ { 2 } , x ) = - D _ { K L } \\big ( q _ { w _ { 1 } } ( z | x ) | | p _ { w _ { 2 } } ( z ) \\big ) + \\mathbb { E } _ { q _ { w _ { 1 } } ( z | x ) } \\big ( \\log p _ { w _ { 2 } } ( x | z ) \\big )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Recently, a stacked set of denoising autoencoders architectures showed promising results in both semi-supervised and unsupervised tasks. A stacked what-where autoencoder by Zhao et al. (2015) computes a set of complementary variables that enable reconstruction whenever a layer implements a many-to-one mapping. Ladder networks by Rasmus et al. (2015) - use lateral connections and layer-wise cost functions to allow the higher levels of an autoencoder to focus on invariant abstract features. ",
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+    {
+        "type": "text",
+        "text": "Exemplar Networks: The unsupervised method introduced byDosovitskiy et al. (2014) takes a different approach to this task and trains the network to discriminate between a set of pseudo-classes. Each pseudo-class is formed by applying multiple transformations to a randomly sampled image patch. The number of pseudo-classes can be as big as the size of the input samples. This criterion ensures that different input samples would be distinguished while providing robustness to the applied transformations. In this work we will explore an alternative method with a similar motivation. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Context prediction Another method for unsupervised learning by context was introduced by Doersch et al. (2015). This method uses an auxiliary criterion of predicting the location of an image patch given another from the same image. This is done by classification to 1 of 9 possible locations. Although the work of Doersch et al. (2015) and ours both use patches from an image to perform unsupervised learning, the methods are quite different. Whereas the former used a classification criterion over the spatial location of each patch within a single image, our work is concerned with comparing patches from several images to each other. We claim that this encourages discriminability between images (which we feel to be important aspect of feature learning), and was not an explicit goal in previous work. ",
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+    },
+    {
+        "type": "text",
+        "text": "Adversarial Generative Models: This a recently introduced model that can be used in an unsupervised fashion Goodfellow et al. (2014). Adversarial Generative Models uses a set of networks, one trained to discriminate between data sampled from the true underlying distribution (e.g., a set of images), and a separate generative network trained to be an adversary trying to confuse the first network. By propagating the gradient through the paired networks, the model learns to generate samples that are distributed similarly to the source data. As shown by Radford et al. (2015),this model can create useful latent representations for subsequent classification tasks. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Sampling Methods: Methods for training models to discriminate between a very large number of classes often use a noise contrasting criterion. In these methods, roughly speaking, the posterior probability $P ( t | y _ { t } )$ of the ground-truth target $t$ given the model output on an input sampled from the true distribution $y _ { t } = F ( x )$ is maximized, while the probability $P ( t | y _ { n } )$ given a noise measurement $y = F ( n )$ is minimized. This was successfully used in a language domain to learn unsupervised representation of words. The most noteworthy case is the word2vec model introduced by Mikolov et al. (2013). When using this setting in language applications, a natural contrasting noise is a smooth approximation of the Unigram distribution. A suitable contrasting distribution is less obvious when data points are sampled from a high dimensional continuous space, such as the case of image patches. ",
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+    {
+        "type": "text",
+        "text": "2.1 PROBLEMS WITH CURRENT APPROACHES",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Only recently the potential of ConvNets in an unsupervised environment began to bear fruit, still we believe it is not fully uncovered. ",
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+    {
+        "type": "text",
+        "text": "The majority of unsupervised optimization criteria currently used are based on variations of reconstruction losses. One limitation of this fact is that a pixel level reconstruction is non-compliant with the idea of a discriminative objective, which is expected to be agnostic to low level information in the input. In addition, it is evident that MSE is not best suited as a measurement to compare images, for example, viewing the possibly large square-error between an image and a single pixel shifted copy of it. Another problem with recent approaches such as Rasmus et al. (2015); Zeiler et al. (2010) is their need to extensively modify the original convolutional network model. This leads to a gap between unsupervised method and the state-of-the-art, supervised, models for classification - which can hurt future attempt to reconcile them in a unified framework, as well as efficiently leverage unlabeled data with otherwise supervised regimes. ",
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+    {
+        "type": "text",
+        "text": "3 LEARNING BY COMPARISONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The most common way to train NN is by defining a loss function between the target values and the network output. Learning by comparison approaches the supervised task from a different angle. The main idea is to use distance comparisons between samples to learn useful representations. For example, we consider relative and qualitative examples of the form $X _ { 1 }$ is closer to $X _ { 2 }$ than $X _ { 1 }$ is to $X _ { 3 }$ . Using a comparative measure with neural network to learn embedding space was introduced in the “Siamese network” framework by Bromley et al. (1993) and later used in the works of Chopra et al. (2005). One use for this methods is when the number of classes is too large or expected to vary over time, as in the case of face verification, where a face contained in an image has to compared against another image of a face. This problem was recently tackled by Schroff et al. (2015) for training a convolutional network model on triplets of examples. There, one image served as an anchor $x$ , and an additional pair of images served as a positive example $x _ { + }$ (containing an instance of the face of the same person) together with a negative example $x _ { - }$ , containing a face of a different person. The training objective was on the embedded distance of the input faces, where the distance between the anchor and positive example is adjusted to be smaller by at least some constant $\\alpha$ from the negative distance. More precisely, the loss function used in this case was defined as ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/484d212cac854adcb5189435d1af769a6d494575cce095b409510330c3aaa3c2.jpg",
+        "text": "$$\nL ( x , x _ { + } , x _ { - } ) = \\operatorname* { m a x } \\left\\{ \\| F ( x ) - F ( x _ { + } ) \\| _ { 2 } - \\| F ( x ) - F ( x _ { - } ) \\| _ { 2 } + \\alpha , 0 \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $F ( x )$ is the embedding (the output of a convolutional neural network), and $\\alpha$ is a predefined margin constant. Another similar model used by Hoffer & Ailon (2015) with triplets comparisons for classification, where examples from the same class were trained to have a lower embedded distance than that of two images from distinct classes. This work introduced a concept of a distance ratio loss, where the defined measure amounted to: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/4fb638bda29994a614957769f108d611ae9a93ad6b9a11972de0d8d2dfad9e99.jpg",
+        "text": "$$\nL ( x , x _ { + } , x _ { - } ) = \\frac { e ^ { - \\| F ( x ) - F ( x _ { + } ) \\| _ { 2 } } } { e ^ { - \\| F ( x ) - F ( x _ { + } ) \\| _ { 2 } } + e ^ { - \\| F ( x ) - F ( x _ { - } ) \\| _ { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This loss has a flavor of a probability of a biased coin flip. By ‘pushing’ this probability to zero, we express the objective that pairs of samples coming from distinct classes should be less similar to each other, compared to pairs of samples coming from the same class. It was shown empirical by Balntas et al. (2016) to provide better feature embeddings than the margin based distance loss 1 ",
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+    {
+        "type": "text",
+        "text": "4 OUR CONTRIBUTION: SPATIAL CONTRASTING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "One implicit assumption in convolutional networks, is that features are gradually learned hierarchically, each level in the hierarchy corresponding to a layer in the network. Each spatial location within a layer corresponds to a region in the original image. It is empirically observed that deeper layers tend to contain more ‘abstract’ information from the image. Intuitively, features describing different regions within the same image are likely to be semantically similar (e.g. different parts of an animal), and indeed the corresponding deep representations tend to be similar. Conversely, regions from two probably unrelated images (say, two images chosen at random) tend to be far from each other in the deep representation. This logic is commonly used in modern deep networks such as Szegedy et al. (2015) Lin et al. (2013) He et al. (2015), where a global average pooling is used to aggregate spatial features in the final layer used for classification. ",
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+        "type": "text",
+        "text": "Our suggestion is that this property, often observed as a side effect of supervised applications, can be used as a desired objective when learning deep representations in an unsupervised task. Later, the resulting representation can be used, as typically done, as a starting point or a supervised learning task. We call this idea which we formalize below Spatial contrasting. The spatial contrasting criterion is similar to noise contrasting estimation Gutmann & Hyvarinen (2010) Mnih & Kavukcuoglu ¨ (2013), in trying to train a model by maximizing the expected probability on desired inputs, while minimizing it on contrasting sampled measurements. ",
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+    {
+        "type": "text",
+        "text": "4.1 FORMULATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We will concern ourselves with samples of images patches $\\tilde { x } ^ { ( m ) }$ taken from an image $x$ . Our convolutional network model, denoted by $F ( x )$ , extracts spatial features $f$ so that $f ^ { ( m ) } = F ( \\tilde { x } ^ { ( m ) } )$ for an image patch $\\tilde { x } ^ { ( m ) }$ . We will also define $P ( f _ { 1 } | f _ { 2 } )$ as the probability for two features $f _ { 1 } , f _ { 2 }$ to occur together in the same image. ",
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+        "type": "text",
+        "text": "We wish to optimize our model such that for two features representing patches taken from the same image x˜(1)i , x˜ $\\tilde { x } _ { i } ^ { ( 1 ) } , \\bar { x _ { i } ^ { ( 2 ) } } \\in x _ { i }$ for which $f _ { i } ^ { ( 1 ) } = F ( \\tilde { x } _ { i } ^ { ( 1 ) } )$ and $f _ { i } ^ { ( 2 ) } \\stackrel { - } { = } F ( \\tilde { x } _ { i } ^ { ( 2 ) } )$ , $P ( f _ { i } ^ { ( 1 ) } | f _ { i } ^ { ( 2 ) } )$ will be maxi",
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+    {
+        "type": "text",
+        "text": "This means that features from a patch taken from a specific image can effectively predict, under our model, features extracted from other patches in the same image. Conversely, we want our model to minimize $P ( f _ { i } | f _ { j } )$ for $i , j$ being two patches taken from distinct images. Following the logic presented before, we will need to sample contrasting patch $\\tilde { x } _ { j } ^ { ( 1 ) }$ from a different image $x _ { j }$ such that $P ( f _ { i } ^ { ( 1 ) } | f _ { i } ^ { ( 2 ) } ) > P ( f _ { j } ^ { ( 1 ) } | f _ { i } ^ { ( 2 ) } )$ , where $\\underset { . } { f _ { j } ^ { ( 1 ) } } = F ( \\tilde { x } _ { j } ^ { ( 1 ) } )$ . In order to obtain contrasting samples, we use regions from two random images in the training set. We will use a distance ratio, described earlier in Eq. (2) for the supervised case, to represent the probability two feature vectors were taken from the same image. The resulting training loss for a pair of images will be defined as ",
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+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8f724b214737dc3deeda6105635fda1e52f720cb451698c586fb2eb8bbf58739.jpg",
+        "text": "$$\nL _ { S C } ( x _ { 1 } , x _ { 2 } ) = - \\log \\frac { e ^ { - \\| f _ { 1 } ^ { ( 1 ) } - f _ { 1 } ^ { ( 2 ) } \\| _ { 2 } } } { e ^ { - \\| f _ { 1 } ^ { ( 1 ) } - f _ { 1 } ^ { ( 2 ) } \\| _ { 2 } } + e ^ { - \\| f _ { 1 } ^ { ( 1 ) } - f _ { 2 } ^ { ( 1 ) } \\| _ { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Effectively minimizing a log-probability under the SoftMax measure. This formulation is portrayed in figure 4.1. Since we sample our contrasting sample from the same underlying distribution, we can evaluate this loss considering the image patch as both patch compared (anchor) and contrast symmetrically. The final loss will be the average between these estimations: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/48d1ad05b47f0bfd56d89fe5cdf1208bf9a6684314e8cf2034f8e0e959f35323.jpg",
+        "text": "$$\n\\widehat { L } _ { S C } ( x _ { 1 } , x _ { 2 } ) = \\frac { 1 } { 2 } \\left[ L _ { S C } ( x _ { 1 } , x _ { 2 } ) + L _ { S C } ( x _ { 2 } , x _ { 1 } ) \\right]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "image",
+        "img_path": "images/f412cdc401c9a392a006634319c43ab46c8b2578a7cd2274d9536d24baa26d3b.jpg",
+        "image_caption": [
+            "Figure 1: Spatial contrasting depiction. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.2 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Convolutional network are usually trained using SGD over mini-batch of samples, therefore we can extract patches and contrasting patches without changing the network architecture. Each image serves as both anchor and positive patches, for which the corresponding features should be closer, as well as contrasting samples for other images in that batch. For a batch of $N$ images, two samples from each image are taken, and $N ^ { 2 }$ different distance comparisons are made. The final loss is defined as the average distance ratio for all images in the batch: ",
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+    {
+        "type": "equation",
+        "img_path": "images/8b403568fa24822998826ddcf40e5c6cf8dbd89796e01189a57ab00cf49c6a19.jpg",
+        "text": "$$\n\\overline { { L } } _ { S C } ( \\{ x \\} _ { i = 1 } ^ { N } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } L _ { S C } ( x _ { i } , \\{ x \\} _ { j \\neq i } ) = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\frac { e ^ { - \\| f _ { i } ^ { ( 1 ) } - f _ { i } ^ { ( 2 ) } \\| _ { 2 } } } { \\sum _ { j = 1 } ^ { N } e ^ { - \\| f _ { i } ^ { ( 1 ) } - f _ { j } ^ { ( 2 ) } \\| _ { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since the criterion is differentiable with respect to its inputs, it is fully compliant with standard methods for training convolutional network and specifically using backpropagation and gradient descent. Furthermore, SC can be applied to any layer in the network hierarchy. In fact, SC can be used at multiple layers within the same convolutional network. The spatial properties of the features means that we can sample directly from feature space $\\tilde { f } ^ { ( m ) } \\in f$ instead of from the original image. Therefore SC has a simple implementation which doesn’t require substation amount of computation. The complete algorithm for batch training is described in Algorithm (1). Similar to the batch normalization (BN) layer Ioffe & Szegedy (2015), a recent usage for batch statistics in neural networks, SC also uses the batch statistics. While BN normalize the input based on the batch statistics, SC sample from it. This can be viewed as a simple sampling from the space of possible features describing a patch of image. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Calculation the spatial contrasting loss ",
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+    },
+    {
+        "type": "text",
+        "text": "Require: $X = \\{ x \\} _ { i = 1 } ^ { N }$ # Training on batches of images ",
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+    },
+    {
+        "type": "text",
+        "text": "# Get the spatial features for the whole batch of images   \n# Size: $N \\times W _ { f } \\times H _ { f } \\times C$   \n$\\{ f \\} _ { i = 1 } ^ { N }  \\mathsf { C o n v N e t } ( X )$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f8048d850b865de754aae5170c2abf7c3592ec395d8fff3b67e4bf25bb0663a.jpg",
+        "text": "$$\n\\begin{array} { r } { d _ { i } \\gets - \\log \\frac { \\bar { e } ^ { - D i s t ( i , i ) } } { \\sum _ { k = 1 } ^ { N } e ^ { - D i s t ( i , k ) } } } \\end{array}\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "# Spatial contrasting loss is the mean of distance ratios return $\\textstyle { \\frac { 1 } { N } } \\sum _ { i = 1 } ^ { N } d _ { i }$ g ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we report empirical results showing that using SC loss as an unsupervised pretraining procedure can improve state-of-the-art performance on subsequent classification. We experimented with MNIST, CIFAR-10 and STL10 datasets. We used modified versions of well studied networks such as those of Lin et al. (2013) and Rasmus et al. (2015). A detailed description of our architecture can be found in 4. ",
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+    {
+        "type": "text",
+        "text": "In each one of the experiments, we used the spatial contrasting criterion to train the network on the unlabeled images. In each usage of SC criterion, patch features were sampled from the preceding layer in uniform. We note that spatial size of sampled patches ranged between datasets, where on STL10 and Cifar10 it covered about $3 0 \\%$ of the image, MNIST required the use of larger patches covering almost the entire image.Training was done by using SGD with an initial learning rate of 0.1 that was decreased by a factor of 10 whenever the measured loss stopped decreasing. After convergence, we used the trained model as an initialization for a supervised training on the complete labeled dataset. The supervised training was done following the same regime, only starting with a lower initial learning rate of 0.01. We used mild data augmentations, such as small translations and horizontal mirroring. ",
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+    {
+        "type": "text",
+        "text": "The datasets we used are: ",
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+    {
+        "type": "text",
+        "text": "• STL10 (Coates et al. (2011)). This dataset consists of $1 0 0 , 0 0 0 9 6 \\times 9 6$ colored, unlabeled images, together with another set of 5, 000 labeled training images and 8, 000 test images . The label space consists of 10 object classes. • Cifar10 (Krizhevsky & Hinton (2009)). The well known CIFAR-10 is an image classification benchmark dataset containing 50, 000 training images and 10, 000 test images. The image sizes $3 2 \\times 3 2$ pixels, with color. The classes are airplanes, automobiles, birds, cats, deer, dogs, frogs, horses, ships and trucks. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f70e680c4da009ba6c944376177710a0a64eecc57507eb76161f1f476fa15955.jpg",
+        "table_caption": [
+            "Table 1: State of the art results on STL-10 dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>STL-10 test accuracy</td></tr><tr><td>Zero-bias Convnets - Paine et al. (2014)</td><td>70.2%</td></tr><tr><td>Triplet network -Hoffer &amp; Ailon (2015)</td><td>70.7%</td></tr><tr><td>Exemplar Convnets - Dosovitskiy et al. (2014)</td><td>72.8%</td></tr><tr><td>Target Coding - Yang et al. (2015)</td><td>73.15%</td></tr><tr><td>Stacked what-where AE - Zhao et al. (2015)</td><td>74.33%</td></tr><tr><td>Spatial contrasting initialization (this work)</td><td>81.34% ± 0.1</td></tr><tr><td>The same model without initialization</td><td>72.6%±0.1</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• MNIST (LeCun et al. (1998)). The MNIST database of handwritten digits is one of the most studied dataset benchmark for image classification. The dataset contains 60,000 examples of handwritten digits from 0 to 9 for training and 10,000 additional examples for testing. Each sample is a $2 8 \\times 2 8$ pixel gray level image. ",
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+    },
+    {
+        "type": "text",
+        "text": "All experiments were conducted using the Torch7 framework by Collobert et al. (2011). Code reproducing these results will by available at https://github.com/eladhoffer/ SpatialContrasting. ",
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+    {
+        "type": "text",
+        "text": "5.1 RESULTS ON STL10 ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since STL10 dataset is comprised of mostly unlabeled data, it is most suitable to highlight the benefits of the spatial contrasting criterion. The initial training was unsupervised, as described earlier, using the entire set of 105, 000 samples (union of the original unlabeled set and labeled training set). The representation outputted by the training, was used to initialize supervised training on the 5, 000 labeled images. Evaluation was done on a separate test set of 8, 000 samples. Comparing with state of the art results, we see an improvement of $7 \\%$ in test accuracy over the best model by Zhao et al. (2015), setting the SC as best model at $8 1 . 3 \\%$ test classification accuracy (see Table (1)). We note that the results of Dosovitskiy et al. (2014) are achieved with no fine-tuning over labeled examples, which may be unfair to this work. We also compare with the same network, but without SC initialization, which achieves a lower classification of $7 2 . 6 \\%$ . This is an indication that indeed SC managed to leverage unlabeled examples to provide a better initialization point for the supervised model. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 RESULTS ON CIFAR10 ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For Cifar10 dataset, we use the same setting as Coates & $\\mathrm { N g } \\left( 2 0 1 2 \\right)$ and Hui (2013) to test a model’s ability to learn from unlabeled images. Here, only 4, 000 samples out of 50, 000 are used with their label annotation, and the rest of the samples can be used only in an unsupervised manner. The final test accuracy is measured on the entire 10, 000 test set. ",
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+    {
+        "type": "text",
+        "text": "In our experiments, we trained our model using SC criterion on the entire dataset, and then used only 400 labeled samples per class (for a total of 4000) in a supervised regime over the initialized network. The results are compared with previous efforts in Table (2). Using the SC criterion allowed an improvement of $6 . 8 \\%$ over a non-initialized model, and achieved a final test accuracy of $7 9 . 2 \\%$ . This is a competitive result with current state-of-the-art models. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 RESULTS ON MNIST ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The MNIST dataset is very different in nature from the Cifar10 and STL10 datasets, we experimented earlier. The biggest difference, relevant to this work, is that spatial regions sampled from MNIST images usually provide very little, or no information. Thus, SC is much less suited for MNIST dataset, and was conjured to have little benefit. We still, however, experimented with initializing a model with SC criterion and continuing with a fully-supervised regime over all labeled examples. We found again that this provided benefit over training the same network without preinitialization, improving results from ${ \\bar { 0 } } . 6 3 \\%$ to $0 . 3 4 \\%$ error on test set. As mentioned previously, the effective compared patches of MNIST covered almost the entire image area. This can be attributed to the fact that MNIST requires global features to differentiate between digits. The results, compared with previous attempts are included in Table (3). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c88b927fef791956d5991cf312d4b4a2383f9011ccdd7db2a08eb502a090bb5d.jpg",
+        "table_caption": [
+            "Table 2: State of the art results on Cifar10 dataset with only 4000 labeled samples "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Cifar10 (400 per class) test accuracy</td></tr><tr><td>Convolutional K-means Network - Coates &amp; Ng (2012)</td><td>70.7%</td></tr><tr><td>View-Invariant K-means -Hui (2013)</td><td>72.6%</td></tr><tr><td>DCGAN - Radford et al. (2015)</td><td>73.8%</td></tr><tr><td>Exemplar Convnets - Dosovitskiy et al. (2014)</td><td>76.6%</td></tr><tr><td>Ladder networks - Rasmus et al. (2015)</td><td>79.6%</td></tr><tr><td>Conv-CatGan Springenberg (2016)</td><td>80.42% (± 0.58)</td></tr><tr><td>ImprovedGan Salimans et al. (2016)</td><td>81.37% (± 2.32)</td></tr><tr><td>Spatial contrasting initialization (this work)</td><td>79.2%(±0.3)</td></tr><tr><td>The same model without initialization</td><td>72.4%(±0.1)</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/e1e70e8fd494548a2eb23c22fe6d4cb4355a7dd6bbd7414322563dce6d766cd4.jpg",
+        "table_caption": [
+            "Table 3: results on MNIST dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>MNIST test error</td></tr><tr><td>Stacked what-where AE - Zhao et al. (2015)</td><td>0.71%</td></tr><tr><td>Triplet network - Hoffer &amp; Ailon (2015)</td><td>0.56%</td></tr><tr><td>Jarrett et al. (2009)</td><td>0.53%</td></tr><tr><td>Ladder networks - Rasmus et al. (2015)</td><td>0.36%</td></tr><tr><td>DropConnect - Wan et al. (2013)</td><td>0.21%</td></tr><tr><td>Spatial contrasting initialization (this work)</td><td>0.34%± 0.02</td></tr><tr><td>The same model without initialization</td><td>0.63%± 0.02</td></tr></table>",
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+    },
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work we presented spatial contrasting - a novel unsupervised criterion for training convolutional networks on unlabeled data. Its is based on comparison between spatial features sampled from a number of images. We’ve shown empirically that using spatial contrasting as a pretraining technique to initialize a ConvNet, can improve its performance on a subsequent supervised training. In cases where a lot of unlabeled data is available, such as the STL10 dataset, this translates to state-of-the-art classification accuracy in the final model. ",
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+    {
+        "type": "text",
+        "text": "Since the spatial contrasting loss is a differentiable estimation that can be computed within a network parallel to supervised losses, in future work we plan to embed it as a semi-supervised model. This usage will allow to create models that can leverage both labeled an unlabeled data, and can be compared to similar semi-supervised models such as the ladder network Rasmus et al. (2015). It is is also apparent that contrasting can occur in dimensions other than the spatial, the most straightforward is the temporal dimension. This suggests that similar training procedure can be applied on segments of sequences to learn useful representation without explicit supervision. ",
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+        "text": "Matthew D Zeiler, Dilip Krishnan, Graham W Taylor, and Rob Fergus. Deconvolutional networks. In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, pp. 2528–2535. IEEE, 2010. ",
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+    {
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+        "text": "Junbo Zhao, Michael Mathieu, Ross Goroshin, and Yann Lecun. Stacked what-where auto-encoders. arXiv preprint arXiv:1506.02351, 2015. ",
+        "bbox": [
+            171,
+            856,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/efc31b8705bda0bdbc12ca3710873dff63e9832b26979b917469fbce29b15310.jpg",
+        "table_caption": [
+            "Table 4: Convolutional models used, based on Lin et al. (2013), Rasmus et al. (2015) ",
+            ""
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=3>Model</td></tr><tr><td rowspan=1 colspan=1>STL10</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>MNIST</td></tr><tr><td rowspan=1 colspan=1>Input: 96×96 RGB</td><td rowspan=1 colspan=1>Input:32×32RGB</td><td rowspan=1 colspan=1>Input:28 × 28 monochrome</td></tr><tr><td rowspan=1 colspan=1>5 ×5 conv. 64 BN ReLU</td><td rowspan=1 colspan=1>3 × 3 conv. 96 BN LeakyReLU</td><td rowspan=6 colspan=1>5 ×5 conv.32 ReLU2 × 2 max-pooling,stride 2 BN3 × 3 conv. 64 BN ReLU3 × 3 conv. 64 BN ReLU2 × 2 max-pooling,stride 2 BN</td></tr><tr><td rowspan=1 colspan=1>1 ×1 conv.160 BN ReLU</td><td rowspan=5 colspan=1>3 ×3 conv. 96 BNLeakyReLU3 × 3 conv. 96 BN LeakyReLU2 × 2 max-pooling,stride 2 BN3 × 3 conv.192 BNLeakyReLU3 × 3 conv. 192 BN LeakyReLU3 × 3 conv.192 BN LeakyReLU2 × 2 max-pooling,stride 2 BN</td></tr><tr><td rowspan=1 colspan=1>1 ×1 conv. 96 BN ReLU</td></tr><tr><td rowspan=1 colspan=1>3 × 3 max-pooling, stride 2</td></tr><tr><td rowspan=1 colspan=1>5 × 5 conv. 192 BN ReLU</td></tr><tr><td rowspan=1 colspan=1>1 ×1 conv.192 BN ReLU1 ×1 conv. 192 BN ReLU3 × 3 max-pooling,stride 23 × 3 conv. 192 BN ReLU1 ×1 conv.192 BN ReLU1 ×1 conv.192 BN ReLU</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Spatial contrasting criterion</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>3×3 conv.256 ReLU3 × 3 max-pooling,stride 2dropout, p = 0.53 × 3 conv.128 ReLUdropout, p = 0.5fully-connected 10</td><td rowspan=1 colspan=1>3×3 conv.192BNLeakyReLU1 ×1 conv.192 BNLeakyReLU1 ×1 conv.10 BNLeakyReLUglobal average pooling</td><td rowspan=1 colspan=1>3×3 conv.128 BN ReLU1 × 1 conv.10 BN ReLUglobal average pooling</td></tr></table>",
+        "bbox": [
+            194,
+            171,
+            803,
+            438
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "10-way softmax ",
+        "bbox": [
+            452,
+            436,
+            547,
+            448
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/dbd928541e70043deb704a91118b985563d8e3c93a5cbdcb28b93b6bd8b098bc.jpg",
+        "image_caption": [
+            "Figure 2: First layer convolutional filters after spatial-contrasting training "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            366,
+            477,
+            627,
+            588
+        ],
+        "page_idx": 10
+    }
+]

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+# A GENERATIVE MODEL FOR MOLECULAR DISTANCE GEOMETRY
+Anonymous authors Paper under double-blind review
+# ABSTRACT
+Computing equilibrium states for many-body systems, such as molecules, is a long-standing challenge. In the absence of methods for generating statistically independent samples, great computational effort is invested in simulating these systems using, for example, Markov chain Monte Carlo. We present a probabilistic model that generates such samples for molecules from their graph representations. Our model learns a low-dimensional manifold that preserves the geometry of local atomic neighborhoods through a principled learning representation that is based on Euclidean distance geometry. In a new benchmark for molecular conformation generation, we show experimentally that our generative model achieves state-ofthe-art accuracy. Finally, we show how to use our model as a proposal distribution in an importance sampling scheme to compute molecular properties.
+# 1 INTRODUCTION
+Over the last few years, many highly-effective deep learning methods generating small molecules with desired properties (e.g., novel drugs) have emerged (Gomez-Bombarelli et al., 2018; Segler ´ et al., 2018; Dai et al., 2018; Jin et al., 2018; Bradshaw et al., 2019a; Liu et al., 2018; You et al., 2018; Bradshaw et al., 2019b). These methods operate using graph representations of molecules in which nodes and edges represent atoms and bonds, respectively. A representation that is closer to the physical system is one in which a molecule is described by its geometry or conformation. A conformation $\mathbf { x }$ of a molecule is defined by a set of atoms $\{ ( \boldsymbol { \epsilon } _ { i } , \mathbf { r } _ { i } ) \bar  \} _ { i = 1 } ^ { N _ { v } }$ , where $N _ { v }$ is the number of atoms in the molecule, $\epsilon _ { i } \in \{ \mathrm { H } , \mathrm { C } , \mathrm { O } , \dots \}$ is the chemical element of the atom $i$ , and $\mathbf { r } _ { i } \in \mathbb { R } ^ { 3 }$ is its position in Cartesian coordinates. Importantly, the relative positions of the atoms are restricted by the bonds in the molecule and the angles between them. Due to thermal fluctuations resulting in stretching of and rotations around bonds, there exist infinitely many conformations of a molecule. A molecule’s graph representation and a set of its conformations are shown in Fig. 1. Under a wide range of conditions, the probability $p ( \mathbf { x } )$ of a conformation $\mathbf { x }$ , is governed by the Boltzmann distribution and is proportional to $\exp \{ - E ( \mathbf { x } ) / k _ { B } T \}$ , where $E ( \mathbf { x } ) \in \mathbb { R }$ is the conformation’s energy, $k _ { B }$ is the Boltzmann constant, and $T$ is the temperature.
+To compute a molecular property for a molecule, one must sample from $p ( \mathbf { x } )$ . The main approach is to start with one conformation and make small changes to it over time, e.g., by using Markov chain Monte Carlo (MCMC) or molecular dynamics (MD). These methods can be used to accurately sample equilibrium states of molecules, but they become computationally expensive for larger ones (Shim & MacKerell, 2011; Ballard et al., 2015; De Vivo et al., 2016). Other heuristic approaches exist in which distances between atoms are set to fixed idealized values (Havel, 2002; Blaney & Dixon, 2007). Several methods based on statistical learning have also recently been developed to tackle the issue of conformation generation. However, they are mainly geared towards studying proteins and their folding dynamics (AlQuraishi, 2019). Some of these models are not targeting a distribution over conformations but the most stable folded configuration (Evans et al., 2018; Ingraham et al., 2019), while others are not transferable between different molecules (Lemke & Peter, 2019; Noe´ et al., 2019).
+This work includes the following key contributions:
+• We introduce a novel probabilistic model for learning conformational distributions of molecules with graph neural networks.
+![](images/434d845d7c8ccab7aba98bd6f5f40a3ddc2c0a355bf36f4ceacd18ef6cbb7a0b.jpg)
+Figure 1: Standard graph representation of a molecule (left) with a set of possible conformations $\{ { \bf { x } } _ { i } \}$ (right). Hydrogen (H), carbon (C), and oxygen (O) atoms are colored white, gray, and red, respectively. Conformations feature the same atom types and bonds but the atoms are arranged differently in space. These differences arise from rotations around and stretching of bonds in the molecule.
+• We create a new, challenging benchmark for conformation generation, which is made publicly available. To the best of our knowledge, this is the first benchmark of this kind.
+• By combining a conditional variational autoencoder (CVAE) with an Euclidean distance geometry (EDG) algorithm we present a state-of-the-art approach for generating one-shot samples of molecular conformations for unseen molecules that is independent of their size and shape.
+• We develop a rigorous experimental approach for evaluating and comparing the accuracy of conformation generation methods based on the mean maximum deviation distance metric.
+• We show how this generative model can be used as a proposal distribution in an importance sampling (IS) scheme to estimate molecular properties.
+# 2 METHOD
+Our goal is to build a statistical model that generates molecular conformations in a one-shot fashion from a molecule’s graph representation. First, we describe how a molecule’s conformation can be represented by a set of pairwise distances between atoms and why this presentation is advantageous over one in Cartesian coordinates (Section 2.1). Second, we present a generative model in Section 2.2 that will generate sets of atomic distances for a given molecular graph. Third, we explain in Section 2.3 how a set of predicted distances can be transformed into a molecular conformation and why this transformation is necessary. Finally, we detail in Section 2.4 how our generative model can be used as a proposal distribution in an IS scheme to estimate molecular properties.
+# 2.1 EXTENDED MOLECULAR GRAPHS AND DISTANCE GEOMETRY
+In this study, a molecule is represented by an undirected graph which is defined as a tuple $\mathcal { G } =$ $( V , E )$ . $V \stackrel { \cdot } { = } \{ v _ { i } \} _ { i = 1 } ^ { N _ { v } }$ is the set of nodes representing atoms, where each $v _ { i } \in \mathbb { R } ^ { F _ { v } }$ holds atomic attributes (e.g., the element type $\epsilon _ { i }$ ). $E = \{ ( e _ { k } , r _ { k } , s _ { k } ) \} _ { k = 1 } ^ { N _ { e } }$ is the set of edges, where each $e _ { k } \in \mathbb { R } ^ { F _ { e } }$ holds an edge’s attributes (e.g., the bond type), and $r _ { k }$ and $s _ { k }$ are the nodes an edge is connecting. Here, $E$ represents the molecular bonds (and the auxiliary edges which are explained below) in the molecule.
+We assume that, givof atomic distances r graph , where $\mathcal { G }$ ent one of its conformations  is the Euclidean distance b $\mathbf { x }$ by a setween the $\mathbf { d } = \{ d _ { k } \} _ { k = 1 } ^ { N _ { e } }$ $d _ { k } = | \mathbf { r } _ { r _ { k } } - \mathbf { r } _ { s _ { k } } |$
+$r _ { k }$ $s _ { k }$
+$( E _ { \mathrm { b o n d } } )$ alone would not suffice to describe a conformation, we expand the traditional graph representation of a molecule by adding auxiliary edges. Auxiliary edges between atoms that are second neighbors in the original graph fix angles between atoms, and those between third neighbors fix dihedral angles (denoted $E _ { \mathrm { a n g l e } }$ and $E _ { \mathrm { d i h e d r a l } }$ , respectively). In this work, $E _ { \mathrm { a n g l e } }$ consists of edges between all second neighbors in the original graph. Edges between third neighbors are added according to a heuristic (see Appendix A.1). From now on we are always referring to this extended molecular graph when talking about molecular graphs. In Fig. 2, the process of extending the molecular graph and the extraction of $\mathbf { d }$ from $\mathbf { x }$ and $\mathcal { G }$ are illustrated.
+![](images/08c1b3e3b35a33713718a18e75ad2f38d7311ffb7d495eaa5cf2fd437d3fe2a8.jpg)
+Figure 2: A) The structural formula of a molecule is converted to an extended molecular graph $\mathcal { G }$ consisting of nodes representing atoms (circles, e.g., $v _ { 1 }$ ) and edges representing molecular bonds (solid lines, e.g., $e _ { 1 } \in E _ { \mathrm { b o n d } } )$ ) and auxiliary edges (dotted lines, e.g., $e _ { 2 } \in E _ { \mathrm { a n g l e } }$ and $e _ { 3 } \in E _ { \mathrm { d i h e d r a l } } )$ . B) The distances $\mathbf { d }$ are extracted from a conformation $\mathbf { x }$ based on the edges $E$ . C) Graphical model of the variational autoencoder: generative model $p _ { \theta } ( \mathbf { d } | \mathbf { z } , \mathcal { G } ) p _ { \theta } ( \mathbf { z } | \mathcal { G } )$ (solid lines) and variational approximation $q _ { \phi } ( \mathbf { z } | \mathbf { d } , \mathcal { G } )$ (dashed lines).
+A key advantage of a representation in terms of distances is its invariance to rotation and translation; by contrast, Cartesian coordinates depend on the (arbitrary) choice of origin, for example. In addition, it reflects pair-wise physical interactions and their generally local nature. Auxiliary edges can be placed between higher-order neighbors depending on how far the physical interactions dominating the potential energy of the system reach.
+samples We have a set of $\{ \mathbf { x } _ { l , j } \} _ { j = 1 } ^ { S _ { l } }$ $N _ { \mathcal { G } }$ l=1from the ground-truth distribution resulting in molecular graphs $\{ \mathcal { G } _ { l } \} _ { l = 1 } ^ { N _ { g } }$ . Further, for each $S _ { l }$ $\mathcal { G } _ { l }$ , we have sets of distances $S _ { l }$ conformational $\{ \mathbf { d } _ { l , j } \} _ { j = 1 } ^ { S _ { l } }$ With this data, we will train a generative model which we detail in the following section.
+# 2.2 GENERATIVE MODEL
+We employ a CVAE (Kingma & Welling, 2014; Pagnoni et al., 2018) to model the distribution over distances d given a molecular graph $\mathcal { G }$ . A CVAE first encodes $\mathcal { G }$ together with d into a latent space $\mathbf { z } ~ \in ~ \mathbb { R } ^ { k N _ { v } }$ , where $k \in \mathbb { N } ^ { + }$ , with an encoder $q _ { \phi } ( \mathbf { z } | \mathbf { d } , \mathcal { G } )$ . Subsequently, the decoder $p _ { \boldsymbol { \theta } } ( \mathbf { d } | \mathbf { z } , \mathcal { G } )$ decodes $\mathbf { z }$ back into a set of distances. A graphical model is shown in Fig. 2 C).
+A conformation has, in general, $3 N _ { v } - 6$ spatial degrees of freedom (dofs): one dof per spacial dimension per atom minus three translational and three rotational dofs. Therefore, the latent space should be proportional to the number of atoms in the molecule. In addition, the latent space should be smaller than $3 N _ { v }$ as it is the role of the encoder to project the conformation into a lower-dimensional space. As a result, we set $k = 1$ to avoid overfitting.
+Here, $q _ { \phi } ( \mathbf { z } | \mathbf { d } , \mathcal { G } )$ and $p _ { \boldsymbol { \theta } } ( \mathbf { d } | \mathbf { z } , \mathcal { G } )$ are Gaussian distributions, the mean and variance of which are modeled by two artificial neural networks. At the center of this model are message-passing neural networks (MPNNs) (Gilmer et al., 2017) with multi-head attention (Velickovi ˇ c et al., 2018). In short, ´ an MPNN is a convolutional neural network that allows end-to-end learning of prediction pipelines whose inputs are graphs of arbitrary size and shape. In a convolution, neighboring nodes exchange so-called messages between neighbors to update their attributes. Edges update their attributes with the features of the nodes they are connecting. The MPNN is a well-studied technique that achieves state-of-the-art performance in representation learning for molecules (Kipf & Welling, 2017; Duvenaud et al., 2015; Kearnes et al., 2016; Schutt et al., 2017b; Gilmer et al., 2017; Kusner et al., 2017; ¨ Bradshaw et al., 2019a).
+In the following, we describe the details of the model.2 In Fig. 3, an illustration of the model is shown. In the encoder $q _ { \phi } ( \mathbf { z } | \mathbf { d } , \mathcal { G } )$ , each $d _ { k }$ is concatenated with the respective edge feature $e _ { k }$ to give $e _ { k } ^ { \prime } \in \mathbb { R } ^ { F _ { e } + 1 }$ . Then, each $v _ { i }$ and each $\boldsymbol { e } _ { k } ^ { \prime }$ are passed to $F _ { \mathrm { e n c , } v }$ and $F _ { \mathrm { e n c } , e }$ (two multilayer perceptrons,
+![](images/5f0738038473b071de697991b53166ab0b2ba48a790fd8a700a36244723f5ab6.jpg)
+Figure 3: The molecular graph $\mathcal { G }$ together with the distances $\mathbf { d }$ are passed through the model consisting of an encoder $q _ { \phi } ( \mathbf { z } | \mathbf { d } , \mathcal { G } )$ and a decoder $p _ { \boldsymbol { \theta } } ( \mathbf { d } | \mathbf { z } , \mathcal { G } )$ . See the main text for details.
+MLPs), respectively, to give $\mathcal { G } _ { \mathrm { e n c } } ^ { ( 0 ) }$ , where $\mathcal { G } _ { \mathrm { e n c } } ^ { ( t ) } = ( \{ v _ { i , \mathrm { e n c } } ^ { ( t ) } \} _ { i = 1 } ^ { N _ { v } } , \{ ( e _ { k , \mathrm { e n c } } ^ { ( t ) } , r _ { k } , s _ { k } ) \} _ { k = 1 } ^ { N _ { e } } ) , v _ { i } ^ { ( }$ $v _ { i , \mathrm { e n c } } ^ { ( t ) } \in \mathbb { R } ^ { L _ { v } }$ and $e _ { k , \mathrm { e n c } } ^ { ( t ) } \in \mathbb { R } ^ { L _ { \mathrm { c } } }$ . Then, $T$ MPNNs of depth 1, $\{ \mathbf { M P } _ { \mathrm { e n c } } ^ { ( t ) } \} _ { t = 1 } ^ { T }$ , are consecutively applied to obtain $\mathcal { G } _ { \mathrm { e n c } } ^ { ( T ) }$ . Finally, the read-out function $R _ { \mathrm { e n c } }$ (an MLP) takes each $v _ { \mathrm { i , e n c } } ^ { ( T ) }$ to predict the mean $\mu _ { z _ { i } } \in \mathbb { R }$ and the variance $\sigma _ { z _ { i } } ^ { 2 } \in \mathbb { R }$ of the Gaussian distribution for $z _ { i }$ . The so-called reparametrization trick is employed to draw a sample for $z _ { i }$ . In summary,
+$$
+\begin{array} { r l } { v _ { i , \mathrm { e n c } } ^ { ( 0 ) } = F _ { \mathrm { e n c } , v } ( v _ { i } ) , } & { e _ { k , \mathrm { e n c } } ^ { ( 0 ) } = F _ { \mathrm { e n c } , e } ( e _ { i } ^ { \prime } ) , } \\ { \mathcal { G } _ { \mathrm { e n c } } ^ { ( 1 ) } = \mathsf { M P } _ { \mathrm { e n c } } ^ { ( 0 ) } ( \mathcal { G } _ { \mathrm { e n c } } ^ { ( 0 ) } ) , } & { \mathcal { G } _ { \mathrm { e n c } } ^ { ( t + 1 ) } = \mathsf { M P } _ { \mathrm { e n c } } ^ { ( t ) } ( \mathcal { G } _ { \mathrm { e n c } } ^ { ( t ) } ) , \quad \mathcal { G } _ { \mathrm { e n c } } ^ { ( T ) } = \mathsf { M P } _ { \mathrm { e n c } } ^ { ( T - 1 ) } ( \mathcal { G } _ { \mathrm { e n c } } ^ { ( T - 1 ) } ) , } \\ { \mu _ { z _ { i } } , \sigma _ { z _ { i } } ^ { 2 } = R _ { \mathrm { e n c } } ( v _ { i , \mathrm { e n c } } ^ { ( T ) } ) . } \end{array}
+$$
+decode. Each $p _ { \boldsymbol { \theta } } ( \mathbf { d } | \mathbf { z } , \mathcal { G } )$ ach are $z _ { i }$ is concassed to d wand respective node feature (two MLPs), respective $v _ { i }$ to give  to give $v _ { i } ^ { \prime } \in$
+where $\mathbb { R } ^ { F _ { v } + 1 }$ $\mathcal { G } _ { \mathrm { d e c } } ^ { ( t ) } = ( \{ v _ { i , \mathrm { d e c } } ^ { ( t ) } \} _ { i = 1 } ^ { N _ { v } } , \{ ( e _ { k , \mathrm { d e c } } ^ { ( t ) } , r _ { k } , s _ { k } ) \} _ { k = 1 } ^ { N _ { e } } )$ $\boldsymbol { v } _ { i } ^ { \prime }$ $e _ { k }$ $F _ { \mathrm { d e c } , v }$ , $v _ { i , \mathrm { d e c } } ^ { ( t ) } \in \mathbb { R } ^ { L _ { v } }$ $F _ { \mathrm { d e c } , e }$ , and $e _ { k , \mathrm { d e c } } ^ { ( t ) } \in \mathbb { R } ^ { L _ { \mathrm { e } } }$ . Then, dec MPNNs
+of depth 1, {MP(t)dec}Tt= , are consecutively applied to obtain $\mathcal { G } _ { \mathrm { d e c } } ^ { ( T ) }$ . Finally, the read-out function
+(an MLP) takes ch $e _ { \mathbf { k } , \mathrm { d e c } } ^ { ( T ) }$ to predict the mean $\mu _ { d _ { k } } \in \mathbb { R }$ dec  and the variance $\sigma _ { d _ { k } } ^ { 2 } \in \mathbb { R }$ decof the Gaussian $d _ { k }$ . In summary,
+$$
+\begin{array} { r l } & { v _ { i , \mathrm { d e c } } ^ { ( 0 ) } = F _ { \mathrm { d e c } , v } ( v _ { i } ^ { \prime } ) , \quad e _ { k , \mathrm { d e c } } ^ { ( 0 ) } = F _ { \mathrm { d e c } , e } ( e _ { i } ) , } \\ & { \mathcal { G } _ { \mathrm { d e c } } ^ { ( 1 ) } = \mathbf { M } \mathbf { P } _ { \mathrm { d e c } } ^ { ( 0 ) } ( \mathcal { G } _ { \mathrm { d e c } } ^ { ( 0 ) } ) , \quad \mathcal { G } _ { \mathrm { d e c } } ^ { ( t + 1 ) } = \mathbf { M } \mathbf { P } _ { \mathrm { d e c } } ^ { ( t ) } ( \mathcal { G } _ { \mathrm { d e c } } ^ { ( t ) } ) , \quad \mathcal { G } _ { \mathrm { d e c } } ^ { ( T ) } = \mathbf { M } \mathbf { P } _ { \mathrm { d e c } } ^ { ( T - 1 ) } ( \mathcal { G } _ { \mathrm { d e c } } ^ { ( T - 1 ) } ) , } \\ & { \qquad \mu _ { d _ { k } } , \sigma _ { d _ { k } } ^ { 2 } = R _ { \mathrm { d e c } } ( e _ { k , \mathrm { d e c } } ^ { ( T ) } ) . } \end{array}
+$$
+The sets of parameters in the encoder and decoder, $\phi$ and $\theta$ (i.e., parameters in $F _ { \mathrm { e n c } , v } , \ F _ { \mathrm { e n c } , e }$ , $\{ \mathbf { M P } _ { \mathrm { e n c } } ^ { ( t ) } \} _ { t = 1 } ^ { T }$ , $R _ { \mathrm { e n c } }$ , $F _ { \mathrm { d e c } , v }$ , $F _ { \mathrm { d e c } , e }$ , $\{ \mathbf { M P } _ { \mathrm { d e c } } ^ { ( t ) } \} _ { t = 1 } ^ { T } , R _ { \mathrm { d e c } } )$ , respectively, are optimized by maximizing the evidence lower bound (ELBO):
+$$
+L = \mathbb { E } _ { \mathbf { z } \sim q _ { \phi } ( \mathbf { z } | \mathbf { d } , \mathcal { G } ) } [ \log p _ { \theta } ( \mathbf { d } | \mathbf { z } , \mathcal { G } ) ] - D _ { \mathrm { K L } } [ q _ { \phi } ( \mathbf { z } | \mathbf { d } , \mathcal { G } ) | | p _ { \theta } ( \mathbf { z } | \mathcal { G } ) ] ,
+$$
+where the prior $p _ { \boldsymbol { \theta } } ( \mathbf { z } | \mathcal { G } )$ consists of factorized Gaussians. The optimal values for the hyperparameters for the network dimensions, number of message passes, batch size, and learning rate of the Adam optimizer (Kingma & Ba, 2014) were tuned by maximizing the validation performance (ELBO) with a Bayesian optimizer and are reported in Appendix A.1.3.
+# 2.3 CONFORMATION GENERATION THROUGH EUCLIDEAN DISTANCE GEOMETRY
+To compute molecular properties, quantum-chemical methods need to be employed which require the input, i.e., the molecule, to be in Cartesian coordinates.3 Therefore, we use an EDG algorithm to translate the set of distances $\{ d _ { k } \} _ { k = 1 } ^ { N _ { e } }$ to a set of atomic coordinates $\{ \mathbf { r } _ { i } \} _ { i = 1 } ^ { N _ { v } }$ . 4
+EDG is the mathematical basis for a geometric theory of molecular conformation. In the field of machine learning, Weinberger & Saul (2006) used it for learning image manifolds, Tenenbaum et al. (2000) for image understanding and handwriting recognition, Jain & Saul (2004) for speech and music, and Demaine et al. (2009) for music and musical rhythms. An EDG description of a molecular system consists of a list of lower and upper bounds on the distances between pairs of atoms $\{ ( d _ { k , \operatorname* { m i n } } , d _ { k , \operatorname* { m a x } } ) \} _ { k = 1 } ^ { N _ { e } }$ . Here, $p _ { \boldsymbol { \theta } } ( \mathbf { d } | \mathbf { z } , \mathcal { G } )$ is used to model these bounds, namely, we set the bounds to $\{ ( \mu _ { d _ { k } } - \sigma _ { d _ { k } } , \mu _ { d _ { k } } + \sigma _ { d _ { k } } ) \}$ , where $\mu _ { d _ { k } }$ and $\sigma _ { d _ { k } }$ are the mean and standard deviation for each distance $d _ { k }$ given by the CVAE. Then, an EDG algorithm determines a set of Cartesian coordinates $\{ \mathbf { r } _ { i } \} _ { i = 1 } ^ { N _ { v } }$ so that these bounds are fulfilled (see Appendix A.2 for details).5 Together with the corresponding chemical elements $\{ \epsilon _ { i } \} _ { i = 1 } ^ { N _ { v } }$ , we obtain a conformation $\mathbf { x }$ .
+# 2.4 CALCULATION OF MOLECULAR PROPERTIES
+We can get an MC estimate of the expectation $\mathbb { E } _ { \mathcal { G } } [ \mathcal { O } ]$ of a property $\mathcal { O }$ (e.g., the dipole moment) for a molecule represented by $\mathcal { G }$ by drawing conformational samples $\mathbf { x } _ { i } \sim p ( \mathbf { x } | \mathcal { G } )$ and computing $\mathcal { O } ( \mathbf { x } _ { i } ) \in \mathbb { R }$ with a quantum-chemical method (e.g., density functional theory). Since we cannot draw samples from $p ( \mathbf { x } | \mathcal { G } )$ directly, we employ an IS integration scheme (Bishop, 2009) with our CVAE as the proposal distribution. We assume that we can readily evaluate the unnormalized probability of a conformation $\tilde { p } ( \mathbf { x } | \mathcal { G } ) = \exp \{ - E ( \mathbf { x } ) / k _ { B } T \}$ , where $\mathbf { x }$ must be a conformation of the molecule and the energy $E ( \mathbf { x } )$ is determined with a quantum-chemical method. Since the EDG algorithm is mapping the distribution $p _ { \boldsymbol { \theta } } ( \mathbf { d } | \mathbf { z } , \mathcal { G } )$ to a point mass in $ { \mathbb { R } } ^ { 3 N _ { v } }$ , the MC estimate for the resulting distribution $p _ { \mathrm { p r o p } } ( \mathbf { x } | \mathcal { G } )$ is given by a mixture of delta functions, each of which is centered at the $\mathbf { x } _ { i }$ resulting from mapping $p _ { \boldsymbol { \theta } } ( \mathbf { d } | \mathbf { z } _ { i } , \mathcal { G } )$ to $\mathbb { R } ^ { 3 N _ { v } }$ , where $\mathbf { z } _ { i } \sim p _ { \theta } ( \mathbf { z } | \mathcal { G } )$ , that is, $\begin{array} { r } { p _ { \mathrm { p r o p } } ( \mathbf { x } | \mathcal { G } ) \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \delta ( \mathbf { x } - \mathbf { x } _ { i } ) } \end{array}$ . The IS estimator for the expectation of $\mathcal { O }$ w. r. t. $\tilde { p } ( \mathbf { x } | \mathcal { G } )$ then reads
+$$
+\hat { \mathbb { E } } _ { \mathcal { G } } [ \mathcal { O } ] \overset { \mathrm { M C } } { \approx } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { O } ( \mathbf { x } _ { i } ) \overset { \mathrm { I S } } { = } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { O } ( \mathbf { x } _ { i } ^ { \prime } ) \frac { \tilde { p } ( \mathbf { x } _ { i } ^ { \prime } | \mathcal { G } ) } { p _ { \mathrm { p r o p } } ( \mathbf { x } _ { i } ^ { \prime } | \mathcal { G } ) } ,
+$$
+where $\mathbf { x } _ { i } \sim \tilde { p } ( \mathbf { x } _ { i } | \mathcal { G } )$ and $\mathbf { x } _ { i } ^ { \prime } \sim p _ { \mathrm { p r o p } } ( \mathbf { x } _ { i } ^ { \prime } | \mathcal { G } )$ , so that the expectation of $\mathcal { O }$ w. r. t. the normalized version of $\tilde { p } ( { \bf x } )$ is then
+$$
+\mathbb { E } _ { \mathcal { G } } [ \mathcal { O } ] = \frac { \hat { \mathbb { E } } _ { \mathcal { G } } [ \mathcal { O } ] } { \hat { \mathbb { E } } _ { \mathcal { G } } [ 1 ] } \approx \frac { 1 } { Z } \sum _ { i = 1 } ^ { N } \mathcal { O } ( \mathbf { x } _ { i } ) \tilde { p } ( \mathbf { x } _ { i } ^ { \prime } | \mathcal { G } ) ,
+$$
+where have a $\begin{array} { r } { Z \approx \sum _ { i = 1 } ^ { N } \tilde { p } ( \mathbf { x } _ { i } ^ { \prime } ) } \end{array}$ and ake s $N$ is the number of samples. When dividing two delta functions wee arbitrarily large finite value.
+# 3 RELATED WORKS
+The standard approach for generating molecular conformations is to start with one, and make small changes to it over time, e.g., by using MCMC or MD. These methods are considered the gold standard for sampling equilibrium states, but they are computationally expensive, especially if the molecule is large and the Hamiltonian is based on quantum-mechanical principles (Shim & MacKerell, 2011; Ballard et al., 2015; De Vivo et al., 2016).
+A much faster but more approximate approach for conformation generation is EDG (Havel, 2002; Blaney & Dixon, 2007; Lagorce et al., 2009; Riniker & Landrum, 2015). Lower and upper distance bounds for pairs of atoms in a molecule are fixed values based on ideal bond lengths, bond angles, and torsional angles. These values are often extracted from crystal structure databases (Allen, 2002). These methods aim to generate a low-energy conformation, not to generate unbiased samples from the underlying distribution at a certain temperature.
+There exist several machine learning approaches as well, however, they are mostly tailored towards studying protein dynamics. For example, Noe et al. (2019) trained Boltzmann generators on the ´ energy function of proteins to provide unbiased, one-shot samples from their equilibrium states. This is achieved by training an invertible neural network to learn a coordinate transformation from a system’s configurations to a latent space representation. Further, Lemke & Peter (2019) proposed a dimensionality reduction algorithm that is based on a neural network autoencoder in combination with a nonlinear distance metric to generate samples for protein structures. Both models learn protein-specific coordinate transformations that cannot be transferred to other molecules.
+AlQuraishi (2019) introduced an end-to-end differentiable recurrent geometric network for protein structure learning based on amino acid sequences. Also, Ingraham et al. (2019) proposed a neural energy simulator model for protein structure that makes use of protein sequence information. In contrast to amino acid sequences, molecular graphs are, in general, not linear but highly branched and often contain cycles. This makes them unsuitable for recurrent networks.
+Finally, Mansimov et al. (2019) presented a conditional deep generative graph neural network to generate molecular conformations given a molecular graph. Their goal is to predict the most likely conformation and not a distribution over conformations. Instead of encoding molecular environments in atomic distances, they work directly in Cartesian coordinates. As a result, the generated conformations showed significant structural differences compared to the ground-truth and required refinement through a force field, which is often employed in MD simulations.
+We argue that our model has several advantages over the approaches reviewed above:
+• It is a fast alternative to resource-intensive approaches based on MCMC or MD.
+• Our principled representation based on pair-wise distances does not restrict our approach to any particular molecular structure.
+• Since our model employs message-passing neural networks, it is transferable – it can extrapolate from only a few graphs to unseen ones.
+# 4 THE CONF17 BENCHMARK
+The CONF17 benchmark is the first benchmark for molecular conformation sampling.6 It is based on the ISO17 dataset (Schutt et al., 2017a) which consists of conformations of various molecules ¨ with the atomic composition $\mathrm { C _ { 7 } H _ { 1 0 } O _ { 2 } }$ drawn from the QM9 dataset (Ramakrishnan et al., 2014). These conformations were generated by ab initio molecular dynamics simulations at 500 Kelvin which generates trajectories of a single molecule covering a large variety of conformations. The CONF17 benchmark consists of 127 distinct molecular graphs each with 3380 conformations on average. We split this dataset into multiple training and test splits, each consisting of 107 and 20 graphs, respectively (see Appendix A.1 for more details).7
+In Fig. 4, (A), the structural formulae of a random selection of molecules from this benchmark are shown. Most molecules feature highly-strained, complex 3D structures such as rings which are typical of drug-like molecules. It is thus the structural complexity of the molecules, not their number of degrees of freedom, that makes this benchmark challenging. In Fig. 4, (B), the frequency of distances (in $\mathring \mathrm { A }$ ) in the conformations are shown for each edge type. It can be seen that the marginal distributions of the edge distances are multimodal and highly context dependent.
+![](images/46a3aa64e022fdd01a26120b31e2968b1635e75bf7b4ed5e1fb0963f320d2e6f.jpg)
+Figure 4: Overview of the CONF17 benchmark. (A) Structural formulae of a random selection of molecules. (B) Distribution of distances (in $\mathrm { \AA }$ ) grouped by edge (from left to right: $E _ { \mathrm { b o n d } }$ , $E _ { \mathrm { a n g l e } }$ , and $E _ { \mathrm { d i h e d r a l } } ,$ ) and vertex type (chemical element).
+# 5 EXPERIMENTS
+We assess the performance of our method, named Graph Distance Geometry (GRAPHDG), by comparing it with two state-of-the-art methods for molecular conformation generation: RDKIT (Riniker & Landrum, 2015), a classical EDG approach, and DL4CHEM (Mansimov et al., 2019), a machine learning approach. We trained GRAPHDG and DL4CHEM on three different training and test splits of the CONF17 benchmark using Adam (Kingma & Ba, 2014). We generated 3000 conformations with each method for molecular graphs in a test set.
+# 5.1 DISTRIBUTIONS OVER DISTANCES
+We assessed the accuracy of the distance distributions of RDKIT, DL4CHEM, and GRAPHDG by calculating the maximum mean discrepancy (MMD) (Gretton et al., 2012) to the ground-truth distribution. We compute the MMD using a Gaussian kernel, where we set the standard deviation to be the median distance between distances $\mathbf { d }$ in the aggregate sample. For this, we determined the distances in the conformations from the ground-truth and those generated by RDKIT and DL4CHEM. For each train-test split and each $\mathcal { G }$ in a test set, we compute the MMD of the joint distribution of distances between C and $\mathrm { o }$ atoms (H atoms are usually ignored), the MMDs of pair-wise distances $p ( d _ { i } , d _ { j } | \mathcal { G } )$ , and the MMDs between the marginals of individual distances $p ( d _ { i } | \mathcal { G } )$ . We aggregate the results of three train-test splits, and, finally, compute the median MMDs and average rankings. The results are summarized in Table 1. It can be seen that the samples from GRAPHDG are significantly closer to the ground-truth distribution than the other methods. RDKIT is slightly worse than GRAPHDG while DL4CHEM seems to struggle with the complexity of the molecules and the small number of graphs in the training set.
+In Fig. 5, we showcase the accuracy of our model by plotting the marginal distributions $p ( d _ { i } | \mathcal { G } )$ for distances between C and O atoms given a molecular graph from a test set. It can be seen that RDKIT consistently underestimates the marginal variances. This is because this method aims to predict the most stable conformation, i.e., the distribution’s mode. In contrast, DL4CHEM often fails to predict the correct mean. For this molecule, GRAPHDG is the most accurate, predicting the right mean and variance in most cases. Additional figures can be found in the Appendix A.4, where we also show plots for the marginal distributions $p ( d _ { i } , d _ { j } | \mathcal { G } )$ .
+Table 1: Assessment of the accuracy of the distributions over conformations generated by three models compared to the ground-truth. We compare the distributions with respect to the marginals $p ( d _ { k } | \mathcal { G } )$ , $p ( d _ { k } , d _ { l } | \mathcal { G } )$ , and the distribution over all edges between C and $\mathrm { o }$ atoms $p ( \{ d _ { k } \} | \mathcal { G } )$ . Two different metrics are used: median MMD between ground-truth conformations and generated ones, and mean ranking (1 to 3) based on the MMD. Reported are the results for molecular graphs in a test set from three train-test splits. Standard errors are given in brackets.
+<table><tr><td rowspan="2"></td><td colspan="3">Median MMD</td><td colspan="3">Mean Ranking</td></tr><tr><td>RDKIT</td><td>DL4CHEM</td><td>GRAPHDG</td><td>RDKIT</td><td>DL4CHEM</td><td>GRAPHDG</td></tr><tr><td>p(dk|9)</td><td>0.55 (0.01)</td><td>1.11 (0.01)</td><td>0.38 (0.02)</td><td>1.71 (0.03)</td><td>2.74 (0.02)</td><td>1.51 (0.03)</td></tr><tr><td>p(dk,di/9)</td><td>0.53 (0.01)</td><td>1.09 ( (0.01)</td><td>0.34 (0.01)</td><td>1.66 (0.02)</td><td>2.92 (0.01)</td><td>1.43 ( (0.02)</td></tr><tr><td>p({d}9）</td><td>0.60 (0.01)</td><td>1.07 (0.03)</td><td>0.44 (0.05)</td><td>1.58 (0.05)</td><td>2.90 (0.05)</td><td>1.45 (0.02)</td></tr></table>
+![](images/73cb33f1644a53a40561269d68794ff6ada9b90a2cebb3bc0277b7aadea252d5.jpg)
+Figure 5: Marginal distributions $p ( d _ { k } | \mathcal { G } )$ of ground-truth and predicted bond distances (in $\textrm { \AA }$ ) between C and O atoms given a molecular graph from the test set. The atoms connected by each edge $d _ { k }$ are indicated in each subplot $\left( \boldsymbol { s } _ { k } \mathrm { - } \boldsymbol { r } _ { k } \right)$ . In the 3D structure of the molecule, carbon and oxygen atoms are colored gray and red, respectively. $_ \mathrm { H }$ atoms are omitted for clarity.
+# 5.2 GENERATION OF CONFORMATIONS
+We passed the distances from our generative model to an EDG algorithm to obtain conformations. For $9 9 . 9 \%$ of the sets of distances, all triangle inequalities held. For $94 \%$ of the molecular graphs, the algorithm succeeded which is 8 pp higher than the success rate we observed for RDKIT. For each molecular graph in a test set, we generated 50 conformations with each method. This took DL4CHEM, RDKIT, and GRAPHDG on average around hundreds of milliseconds per molecule.8 In contrast, a single conformation in the ISO17 dataset takes around a minute to compute. In Fig. 6, an overlay of these conformations of six molecules generated by the different methods is shown. It can be seen that RDKIT’s conformations show too little variance, while DL4CHEM’s structures are mostly invalid, which is due in part to its failure to predict the correct interatomic angles. Our method slightly overestimates the structural variance (see, for example, Fig. 6, top row, second column), but produces conformations that are the closest to the ground-truth.
+# 5.3 CALCULATION OF MOLECULAR PROPERTIES
+We estimate expected molecular properties for molecular graphs from the test set with $N = 5 0$ conformational samples each. Due to their poor quality, we could not compute properties $\mathcal { O } ( \mathbf { x } )$ , including the energy $E ( \mathbf { x } )$ , for conformations generated with DL4CHEM, and thus, this method is excluded from this analysis. In Table 2, it can be seen that RDKIT and GRAPHDG perform
+![](images/6dbc1684f2efce71c7be4d5e7e63e8e2e0f0a2968f6b845c474caa529794805c.jpg)
+Figure 6: Overlay of 50 conformations from the ground-truth and three models based on six random molecular graphs from the test set. C, O, and H atoms are colored gray, red, and white, respectively.
+Table 2: Median difference in average properties between ground-truth and RDKIT and GRAPHDG: total electronic energy $E _ { \mathrm { e l e c } }$ (in kJ/mol), the energy of the HOMO and the LUMO LUMO and LUMO, respectively (in eV), and the dipole moment $\mu$ (in debye). Reported are the results for molecular graphs from the test set, averaged over three train-test splits. Standard errors are given in brackets.
+<table><tr><td></td><td>RDKIT</td><td>GRAPHDG</td></tr><tr><td>Eelec</td><td>42.7 (4.3)</td><td>58.0 (21.0)</td></tr><tr><td>€HOMO</td><td>0.08 (0.04)</td><td>0.10 (0.05)</td></tr><tr><td>ELUMO</td><td>0.15 (0.03)</td><td>0.09 (0.05)</td></tr><tr><td>从</td><td>0.29 (0.05)</td><td>0.33 (0.09)</td></tr></table>
+similarly well (see Appendix A.2 for computational details). However, both methods are still highly inaccurate for $E _ { \mathrm { e l e c } }$ (in practice, an accuracy of less than $5 \ \mathrm { k J / m o l }$ is required). Close inspection of the conformations shows that, even though GRAPHDG predicts the most accurate distances overall, the variances of certain strongly constrained distances (e.g., triple bonds) are overestimated so that the energies of the conformations increase drastically.
+# 6 LIMITATIONS
+The first limitation of this work is that the CVAE can sample (with low probability) invalid sets of distances for which there exists no 3D structure. Second, the CONF17 benchmark covers only a small portion of chemical space. Finally, a large set of auxiliary edges would be required to capture long-range correlations (e.g., in proteins). Future work will address these points.
+# 7 CONCLUSIONS
+We presented GRAPHDG, a transferable, generative model that allows sampling from a distribution over molecular conformations. We developed a principled learning representation of conformations that is based on distances between atoms. Then, we proposed a challenging benchmark for comparing molecular conformation generators. With this benchmark, we show experimentally that conformations generated by GRAPHDG are closer to the ground-truth than those generated by other methods. Finally, we employ our model as a proposal distribution in an IS integration scheme to estimate molecular properties. While orbital energies and the dipole moments were predicted well, a larger and more diverse dataset will be necessary for meaningful estimates of electronic energies. Further, methods have to be devised to estimate how many conformations need to be generated to ensure all important conformations have been sampled. Finally, our model could be trained on conformational distributions at different temperatures in a transfer learning-type setting.
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+Joshua B. Tenenbaum, Vin de Silva, and John C. Langford. A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science, 290(5500):2319–2323, 2000. doi: 10.1126/science.290.5500.2319.
+Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Li ´ o, and Yoshua \` Bengio. Graph Attention Networks. In International Conference on Learning Representations, 2018.
+Florian Weigend. Accurate Coulomb-fitting basis sets for H to Rn. Phys. Chem. Chem. Phys., 8(9): 1057–1065, 2006. doi: 10.1039/B515623H.
+Florian Weigend and Reinhart Ahlrichs. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys., 7(18):3297–3305, 2005. doi: 10.1039/B508541A.
+Kilian Q. Weinberger and Lawrence K. Saul. Unsupervised Learning of Image Manifolds by Semidefinite Programming. Int. J. Comput. Vision, 70(1):77–90, 2006. doi: 10.1007/ s11263-005-4939-z.
+Jiaxuan You, Bowen Liu, Zhitao Ying, Vijay Pande, and Jure Leskovec. Graph Convolutional Policy Network for Goal-Directed Molecular Graph Generation. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems 31, pp. 6410–6421. Curran Associates, Inc., 2018.
+# A APPENDIX
+# A.1 CONF17 BENCHMARK
+# A.1.1 DATA GENERATION
+The ISO17 dataset (Schutt et al., 2017a) was processed in the following way. First, conformations ¨ in which the molecular connectivity was modified (i.e., bonds were broken or new ones are formed) were discarded. For this, the tool XYZ2MOL (Jensen, 2019) was employed. Second, the molecular graphs were augmented by adding auxiliary edges for reasons described in Section 2.1. Auxiliary edges between all second neighbors were added. This can lead to a slight over-specification of the system’s geometry, however, this did not pose a problem in our experiments. In addition, auxiliary edges between third neighbors were added to fix dihedral angles. Since there are potentially many ways of specifying a dihedral angle in a molecular system, we resorted to the works of Riniker & Landrum (2015) and Guba et al. (2016) to decide where to place edges between third neighbors.
+# A.1.2 INPUT FEATURES
+Below we list the node and edges features in the CONF17 benchmark.
+Table 3: Node features.
+<table><tr><td>Feature</td><td>Data Type</td><td>Dimension</td></tr><tr><td>atomic number</td><td>integer</td><td>1</td></tr><tr><td>chiral tag</td><td>one-hot (R, S,and N/A)</td><td>3</td></tr></table>
+Table 4: Edge features.
+<table><tr><td>Feature</td><td>Data Type</td><td>Dimension</td></tr><tr><td>kind</td><td>one-hot (indicating whether e is in Ebond,Eangle,or Edihedral)</td><td>3</td></tr><tr><td>stereo chemistry</td><td>one-hot (E,Z,Any,None, and N/A)</td><td>5</td></tr><tr><td>type</td><td>integer (single, double, triple or N/A)</td><td>1</td></tr><tr><td>is aromatic</td><td>binary</td><td>1</td></tr><tr><td>is conjugated</td><td>binary</td><td>1</td></tr><tr><td>is in ring of size</td><td>one-hot (3,4,...,9) and N/A</td><td>8</td></tr></table>
+# A.1.3 MODEL ARCHITECTURE
+The full model is available online https://figshare.com/s/1b42bf865bd78c457354. In following, the hyperparameters of our model are specified:
+Activations throughout this paper: ReLU; $L _ { v }$ , $L _ { e }$ : 10; $F _ { \mathrm { e n c } , v }$ : neural network with depth 2, width 20; $F _ { \mathrm { e n c } , e }$ : neural network with depth 3, width 60; $F _ { \mathrm { d e c } , v }$ , $F _ { \mathrm { d e c } , e }$ : neural networks with depth 2, width 70; {MP(t) }T , {MP(t) }T dec t=1: MPNN width depth 1 and three multi-head attention heads, T = 3, for node and edge updates neural networks with depth 2, and width 70 were used. $R _ { \mathrm { e n c } }$ , $R _ { \mathrm { d e c } }$ : neural networks with depth 2, width 70. Batch size: 16 (conformations);
+# A.2 COMPUTATIONAL DETAILS
+# A.2.1 QUANTUM-CHEMICAL CALCULATIONS
+All quantum-chemical calculations were carried out with the PySCF program package (version 1.5) (Sun et al., 2018) employing the exchange-correlation density functional PBE (Perdew et al., 1996), and the def2-SVP (Weigend & Ahlrichs, 2005; Weigend, 2006) basis set.
+Conformations generated by DL4CHEM did not succeed as some atoms were too close to each other. Self-consistent field algorithms in quantum-chemical software such as $\operatorname { P y } \operatorname { S C F }$ do not converge for such molecular structures.
+With quantum-chemical methods, we calculate several properties that concern the states of the electrons in the conformation. These are the total electronic energy $E _ { \mathrm { e l e c } }$ , the energy of the electron in the highest occupied molecular orbital (HOMO in eV) HOMO, the energy of the lowest unoccupied molecular orbital (LUMO in eV) LUMO, and the norm of the dipole moment $\mu$ (in debye).
+# A.2.2 EUCLIDEAN DISTANCE GEOMETRY
+We refer the reader to Havel (2002) for theory on EDG, algorithms, and chemical applications. In summary, the EDG procedure consists of the following three steps:
+1. Bound smoothing: extrapolating a complete set of lower and upper limits on all the distances from the sparse set of lower and upper bounds.
+2. Embedding: choosing a random distance matrix from within these limits, and computing coordinates that are a certain best-fit to the distances.
+3. Optimization: optimizing these coordinates versus an error function which measures the total violation of the distance (and chirality) constraints.
+We use the EDG implementation found in RDKIT (Riniker & Landrum, 2015) with default settings.
+A.3 GENERATION OF CONFORMATIONS
+![](images/27951cf926cc9a59c011227fdb3757047860bb4f10db180e980c1c7f56c2a9fb.jpg)
+Figure 7: Overlay of 50 conformations from the ground-truth, RDKIT, DL4CHEM, and GRAPHDG based on two random molecular graphs from the test set. C, O, and H atoms are colored gray, red, and white, respectively.
+# A.4 DISTRIBUTIONS OVER DISTANCES
+Below, the marginal distributions of the distances for a variety of molecular graphs are shown.
+![](images/155b36793493760b7aa79975d8ddcff91c662022446556581190de48d5091e35.jpg)
+Figure 8: Marginal distributions $p ( d _ { k } | \mathcal { G } )$ of ground-truth and predicted distances (in A) between C ˚ and O atoms given a molecular graph from the test set. The atoms connected by each edge $d _ { k }$ are indicated in each subplot $\left( \boldsymbol { s } _ { k } \mathrm { - } \boldsymbol { r } _ { k } \right)$ . In the 3D structure of the molecule, carbon and oxygen atoms are colored gray and red, respectively. $_ \mathrm { H }$ atoms are omitted for clarity.
+![](images/e220b95757969ae8413448670537336f0a0200b2d7857d8c548541c7e3c21327.jpg)
+Figure 9: Marginal distributions $p ( d _ { i } , d _ { j } | \mathcal { G } )$ of ground-truth and predicted distances for a molecular graph from the test set (in $\mathring \mathrm { A }$ ). Here, $d _ { i }$ and $d _ { j }$ are restricted to edges representing bonds between C and O atoms. In the 3D structure of the molecule, carbon and oxygen atoms are colored gray and red, respectively. H atoms are omitted for clarity.
+![](images/990560565dbcd6a07e4996d7818003cb2c3bd83142a60411cb3f89d685007e5c.jpg)
+Figure 10: See caption of Fig. 8
+![](images/cf355a52b90102bd4be7140f2db8ba50e3779dbf93c62c9815ff4022643e40ce.jpg)
+Figure 11: See caption of Fig. 9
+![](images/b460530421a43a7fa1d828cbce0adb372e29c25c81e9fb137bd557fbbf59743a.jpg)
+Figure 12: See caption of Fig. 8
+![](images/3ab71d02171ec50c23dd6fda6644669702b4f5b23939b729a36135a2e37793fd.jpg)
+Figure 13: See caption of Fig. 9
+![](images/318714018966fa62ece638ab99060cdcf128629a839fab7764e5e8b5f7fa177e.jpg)
+Figure 14: See caption of Fig. 8
+![](images/046289cc08ff0e5a719e2e8d0afd1905fc1441824afab583ecca345ad569e0ab.jpg)
+Figure 15: See caption of Fig. 9

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+    {
+        "type": "text",
+        "text": "A GENERATIVE MODEL FOR MOLECULAR DISTANCE GEOMETRY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Computing equilibrium states for many-body systems, such as molecules, is a long-standing challenge. In the absence of methods for generating statistically independent samples, great computational effort is invested in simulating these systems using, for example, Markov chain Monte Carlo. We present a probabilistic model that generates such samples for molecules from their graph representations. Our model learns a low-dimensional manifold that preserves the geometry of local atomic neighborhoods through a principled learning representation that is based on Euclidean distance geometry. In a new benchmark for molecular conformation generation, we show experimentally that our generative model achieves state-ofthe-art accuracy. Finally, we show how to use our model as a proposal distribution in an importance sampling scheme to compute molecular properties. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            446,
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+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Over the last few years, many highly-effective deep learning methods generating small molecules with desired properties (e.g., novel drugs) have emerged (Gomez-Bombarelli et al., 2018; Segler ´ et al., 2018; Dai et al., 2018; Jin et al., 2018; Bradshaw et al., 2019a; Liu et al., 2018; You et al., 2018; Bradshaw et al., 2019b). These methods operate using graph representations of molecules in which nodes and edges represent atoms and bonds, respectively. A representation that is closer to the physical system is one in which a molecule is described by its geometry or conformation. A conformation $\\mathbf { x }$ of a molecule is defined by a set of atoms $\\{ ( \\boldsymbol { \\epsilon } _ { i } , \\mathbf { r } _ { i } ) \\bar  \\} _ { i = 1 } ^ { N _ { v } }$ , where $N _ { v }$ is the number of atoms in the molecule, $\\epsilon _ { i } \\in \\{ \\mathrm { H } , \\mathrm { C } , \\mathrm { O } , \\dots \\}$ is the chemical element of the atom $i$ , and $\\mathbf { r } _ { i } \\in \\mathbb { R } ^ { 3 }$ is its position in Cartesian coordinates. Importantly, the relative positions of the atoms are restricted by the bonds in the molecule and the angles between them. Due to thermal fluctuations resulting in stretching of and rotations around bonds, there exist infinitely many conformations of a molecule. A molecule’s graph representation and a set of its conformations are shown in Fig. 1. Under a wide range of conditions, the probability $p ( \\mathbf { x } )$ of a conformation $\\mathbf { x }$ , is governed by the Boltzmann distribution and is proportional to $\\exp \\{ - E ( \\mathbf { x } ) / k _ { B } T \\}$ , where $E ( \\mathbf { x } ) \\in \\mathbb { R }$ is the conformation’s energy, $k _ { B }$ is the Boltzmann constant, and $T$ is the temperature. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To compute a molecular property for a molecule, one must sample from $p ( \\mathbf { x } )$ . The main approach is to start with one conformation and make small changes to it over time, e.g., by using Markov chain Monte Carlo (MCMC) or molecular dynamics (MD). These methods can be used to accurately sample equilibrium states of molecules, but they become computationally expensive for larger ones (Shim & MacKerell, 2011; Ballard et al., 2015; De Vivo et al., 2016). Other heuristic approaches exist in which distances between atoms are set to fixed idealized values (Havel, 2002; Blaney & Dixon, 2007). Several methods based on statistical learning have also recently been developed to tackle the issue of conformation generation. However, they are mainly geared towards studying proteins and their folding dynamics (AlQuraishi, 2019). Some of these models are not targeting a distribution over conformations but the most stable folded configuration (Evans et al., 2018; Ingraham et al., 2019), while others are not transferable between different molecules (Lemke & Peter, 2019; Noe´ et al., 2019). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This work includes the following key contributions: ",
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+    },
+    {
+        "type": "text",
+        "text": "• We introduce a novel probabilistic model for learning conformational distributions of molecules with graph neural networks. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "image",
+        "img_path": "images/434d845d7c8ccab7aba98bd6f5f40a3ddc2c0a355bf36f4ceacd18ef6cbb7a0b.jpg",
+        "image_caption": [
+            "Figure 1: Standard graph representation of a molecule (left) with a set of possible conformations $\\{ { \\bf { x } } _ { i } \\}$ (right). Hydrogen (H), carbon (C), and oxygen (O) atoms are colored white, gray, and red, respectively. Conformations feature the same atom types and bonds but the atoms are arranged differently in space. These differences arise from rotations around and stretching of bonds in the molecule. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "• We create a new, challenging benchmark for conformation generation, which is made publicly available. To the best of our knowledge, this is the first benchmark of this kind.   \n• By combining a conditional variational autoencoder (CVAE) with an Euclidean distance geometry (EDG) algorithm we present a state-of-the-art approach for generating one-shot samples of molecular conformations for unseen molecules that is independent of their size and shape.   \n• We develop a rigorous experimental approach for evaluating and comparing the accuracy of conformation generation methods based on the mean maximum deviation distance metric.   \n• We show how this generative model can be used as a proposal distribution in an importance sampling (IS) scheme to estimate molecular properties. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to build a statistical model that generates molecular conformations in a one-shot fashion from a molecule’s graph representation. First, we describe how a molecule’s conformation can be represented by a set of pairwise distances between atoms and why this presentation is advantageous over one in Cartesian coordinates (Section 2.1). Second, we present a generative model in Section 2.2 that will generate sets of atomic distances for a given molecular graph. Third, we explain in Section 2.3 how a set of predicted distances can be transformed into a molecular conformation and why this transformation is necessary. Finally, we detail in Section 2.4 how our generative model can be used as a proposal distribution in an IS scheme to estimate molecular properties. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 EXTENDED MOLECULAR GRAPHS AND DISTANCE GEOMETRY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this study, a molecule is represented by an undirected graph which is defined as a tuple $\\mathcal { G } =$ $( V , E )$ . $V \\stackrel { \\cdot } { = } \\{ v _ { i } \\} _ { i = 1 } ^ { N _ { v } }$ is the set of nodes representing atoms, where each $v _ { i } \\in \\mathbb { R } ^ { F _ { v } }$ holds atomic attributes (e.g., the element type $\\epsilon _ { i }$ ). $E = \\{ ( e _ { k } , r _ { k } , s _ { k } ) \\} _ { k = 1 } ^ { N _ { e } }$ is the set of edges, where each $e _ { k } \\in \\mathbb { R } ^ { F _ { e } }$ holds an edge’s attributes (e.g., the bond type), and $r _ { k }$ and $s _ { k }$ are the nodes an edge is connecting. Here, $E$ represents the molecular bonds (and the auxiliary edges which are explained below) in the molecule. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We assume that, givof atomic distances r graph , where $\\mathcal { G }$ ent one of its conformations  is the Euclidean distance b $\\mathbf { x }$ by a setween the $\\mathbf { d } = \\{ d _ { k } \\} _ { k = 1 } ^ { N _ { e } }$ $d _ { k } = | \\mathbf { r } _ { r _ { k } } - \\mathbf { r } _ { s _ { k } } |$   \n$r _ { k }$ $s _ { k }$   \n$( E _ { \\mathrm { b o n d } } )$ alone would not suffice to describe a conformation, we expand the traditional graph representation of a molecule by adding auxiliary edges. Auxiliary edges between atoms that are second neighbors in the original graph fix angles between atoms, and those between third neighbors fix dihedral angles (denoted $E _ { \\mathrm { a n g l e } }$ and $E _ { \\mathrm { d i h e d r a l } }$ , respectively). In this work, $E _ { \\mathrm { a n g l e } }$ consists of edges between all second neighbors in the original graph. Edges between third neighbors are added according to a heuristic (see Appendix A.1). From now on we are always referring to this extended molecular graph when talking about molecular graphs. In Fig. 2, the process of extending the molecular graph and the extraction of $\\mathbf { d }$ from $\\mathbf { x }$ and $\\mathcal { G }$ are illustrated. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/08c1b3e3b35a33713718a18e75ad2f38d7311ffb7d495eaa5cf2fd437d3fe2a8.jpg",
+        "image_caption": [
+            "Figure 2: A) The structural formula of a molecule is converted to an extended molecular graph $\\mathcal { G }$ consisting of nodes representing atoms (circles, e.g., $v _ { 1 }$ ) and edges representing molecular bonds (solid lines, e.g., $e _ { 1 } \\in E _ { \\mathrm { b o n d } } )$ ) and auxiliary edges (dotted lines, e.g., $e _ { 2 } \\in E _ { \\mathrm { a n g l e } }$ and $e _ { 3 } \\in E _ { \\mathrm { d i h e d r a l } } )$ . B) The distances $\\mathbf { d }$ are extracted from a conformation $\\mathbf { x }$ based on the edges $E$ . C) Graphical model of the variational autoencoder: generative model $p _ { \\theta } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } ) p _ { \\theta } ( \\mathbf { z } | \\mathcal { G } )$ (solid lines) and variational approximation $q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } )$ (dashed lines). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            97,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A key advantage of a representation in terms of distances is its invariance to rotation and translation; by contrast, Cartesian coordinates depend on the (arbitrary) choice of origin, for example. In addition, it reflects pair-wise physical interactions and their generally local nature. Auxiliary edges can be placed between higher-order neighbors depending on how far the physical interactions dominating the potential energy of the system reach. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "samples We have a set of $\\{ \\mathbf { x } _ { l , j } \\} _ { j = 1 } ^ { S _ { l } }$ $N _ { \\mathcal { G } }$ l=1from the ground-truth distribution resulting in molecular graphs $\\{ \\mathcal { G } _ { l } \\} _ { l = 1 } ^ { N _ { g } }$ . Further, for each $S _ { l }$ $\\mathcal { G } _ { l }$ , we have sets of distances $S _ { l }$ conformational $\\{ \\mathbf { d } _ { l , j } \\} _ { j = 1 } ^ { S _ { l } }$ With this data, we will train a generative model which we detail in the following section. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 GENERATIVE MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            497,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We employ a CVAE (Kingma & Welling, 2014; Pagnoni et al., 2018) to model the distribution over distances d given a molecular graph $\\mathcal { G }$ . A CVAE first encodes $\\mathcal { G }$ together with d into a latent space $\\mathbf { z } ~ \\in ~ \\mathbb { R } ^ { k N _ { v } }$ , where $k \\in \\mathbb { N } ^ { + }$ , with an encoder $q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } )$ . Subsequently, the decoder $p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )$ decodes $\\mathbf { z }$ back into a set of distances. A graphical model is shown in Fig. 2 C). ",
+        "bbox": [
+            174,
+            525,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A conformation has, in general, $3 N _ { v } - 6$ spatial degrees of freedom (dofs): one dof per spacial dimension per atom minus three translational and three rotational dofs. Therefore, the latent space should be proportional to the number of atoms in the molecule. In addition, the latent space should be smaller than $3 N _ { v }$ as it is the role of the encoder to project the conformation into a lower-dimensional space. As a result, we set $k = 1$ to avoid overfitting. ",
+        "bbox": [
+            174,
+            587,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here, $q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } )$ and $p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )$ are Gaussian distributions, the mean and variance of which are modeled by two artificial neural networks. At the center of this model are message-passing neural networks (MPNNs) (Gilmer et al., 2017) with multi-head attention (Velickovi ˇ c et al., 2018). In short, ´ an MPNN is a convolutional neural network that allows end-to-end learning of prediction pipelines whose inputs are graphs of arbitrary size and shape. In a convolution, neighboring nodes exchange so-called messages between neighbors to update their attributes. Edges update their attributes with the features of the nodes they are connecting. The MPNN is a well-studied technique that achieves state-of-the-art performance in representation learning for molecules (Kipf & Welling, 2017; Duvenaud et al., 2015; Kearnes et al., 2016; Schutt et al., 2017b; Gilmer et al., 2017; Kusner et al., 2017; ¨ Bradshaw et al., 2019a). ",
+        "bbox": [
+            173,
+            664,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the following, we describe the details of the model.2 In Fig. 3, an illustration of the model is shown. In the encoder $q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } )$ , each $d _ { k }$ is concatenated with the respective edge feature $e _ { k }$ to give $e _ { k } ^ { \\prime } \\in \\mathbb { R } ^ { F _ { e } + 1 }$ . Then, each $v _ { i }$ and each $\\boldsymbol { e } _ { k } ^ { \\prime }$ are passed to $F _ { \\mathrm { e n c , } v }$ and $F _ { \\mathrm { e n c } , e }$ (two multilayer perceptrons, ",
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f0738038473b071de697991b53166ab0b2ba48a790fd8a700a36244723f5ab6.jpg",
+        "image_caption": [
+            "Figure 3: The molecular graph $\\mathcal { G }$ together with the distances $\\mathbf { d }$ are passed through the model consisting of an encoder $q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } )$ and a decoder $p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )$ . See the main text for details. "
+        ],
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+    {
+        "type": "text",
+        "text": "MLPs), respectively, to give $\\mathcal { G } _ { \\mathrm { e n c } } ^ { ( 0 ) }$ , where $\\mathcal { G } _ { \\mathrm { e n c } } ^ { ( t ) } = ( \\{ v _ { i , \\mathrm { e n c } } ^ { ( t ) } \\} _ { i = 1 } ^ { N _ { v } } , \\{ ( e _ { k , \\mathrm { e n c } } ^ { ( t ) } , r _ { k } , s _ { k } ) \\} _ { k = 1 } ^ { N _ { e } } ) , v _ { i } ^ { ( }$ $v _ { i , \\mathrm { e n c } } ^ { ( t ) } \\in \\mathbb { R } ^ { L _ { v } }$ and $e _ { k , \\mathrm { e n c } } ^ { ( t ) } \\in \\mathbb { R } ^ { L _ { \\mathrm { c } } }$ . Then, $T$ MPNNs of depth 1, $\\{ \\mathbf { M P } _ { \\mathrm { e n c } } ^ { ( t ) } \\} _ { t = 1 } ^ { T }$ , are consecutively applied to obtain $\\mathcal { G } _ { \\mathrm { e n c } } ^ { ( T ) }$ . Finally, the read-out function $R _ { \\mathrm { e n c } }$ (an MLP) takes each $v _ { \\mathrm { i , e n c } } ^ { ( T ) }$ to predict the mean $\\mu _ { z _ { i } } \\in \\mathbb { R }$ and the variance $\\sigma _ { z _ { i } } ^ { 2 } \\in \\mathbb { R }$ of the Gaussian distribution for $z _ { i }$ . The so-called reparametrization trick is employed to draw a sample for $z _ { i }$ . In summary, ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea2e8d68fcb9089882d112d8b9533da68ffa6c1a89f445f3334ff8e10ef1b1cd.jpg",
+        "text": "$$\n\\begin{array} { r l } { v _ { i , \\mathrm { e n c } } ^ { ( 0 ) } = F _ { \\mathrm { e n c } , v } ( v _ { i } ) , } & { e _ { k , \\mathrm { e n c } } ^ { ( 0 ) } = F _ { \\mathrm { e n c } , e } ( e _ { i } ^ { \\prime } ) , } \\\\ { \\mathcal { G } _ { \\mathrm { e n c } } ^ { ( 1 ) } = \\mathsf { M P } _ { \\mathrm { e n c } } ^ { ( 0 ) } ( \\mathcal { G } _ { \\mathrm { e n c } } ^ { ( 0 ) } ) , } & { \\mathcal { G } _ { \\mathrm { e n c } } ^ { ( t + 1 ) } = \\mathsf { M P } _ { \\mathrm { e n c } } ^ { ( t ) } ( \\mathcal { G } _ { \\mathrm { e n c } } ^ { ( t ) } ) , \\quad \\mathcal { G } _ { \\mathrm { e n c } } ^ { ( T ) } = \\mathsf { M P } _ { \\mathrm { e n c } } ^ { ( T - 1 ) } ( \\mathcal { G } _ { \\mathrm { e n c } } ^ { ( T - 1 ) } ) , } \\\\ { \\mu _ { z _ { i } } , \\sigma _ { z _ { i } } ^ { 2 } = R _ { \\mathrm { e n c } } ( v _ { i , \\mathrm { e n c } } ^ { ( T ) } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "decode. Each $p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )$ ach are $z _ { i }$ is concassed to d wand respective node feature (two MLPs), respective $v _ { i }$ to give  to give $v _ { i } ^ { \\prime } \\in$   \nwhere $\\mathbb { R } ^ { F _ { v } + 1 }$ $\\mathcal { G } _ { \\mathrm { d e c } } ^ { ( t ) } = ( \\{ v _ { i , \\mathrm { d e c } } ^ { ( t ) } \\} _ { i = 1 } ^ { N _ { v } } , \\{ ( e _ { k , \\mathrm { d e c } } ^ { ( t ) } , r _ { k } , s _ { k } ) \\} _ { k = 1 } ^ { N _ { e } } )$ $\\boldsymbol { v } _ { i } ^ { \\prime }$ $e _ { k }$ $F _ { \\mathrm { d e c } , v }$ , $v _ { i , \\mathrm { d e c } } ^ { ( t ) } \\in \\mathbb { R } ^ { L _ { v } }$ $F _ { \\mathrm { d e c } , e }$ , and $e _ { k , \\mathrm { d e c } } ^ { ( t ) } \\in \\mathbb { R } ^ { L _ { \\mathrm { e } } }$ . Then, dec MPNNs   \nof depth 1, {MP(t)dec}Tt= , are consecutively applied to obtain $\\mathcal { G } _ { \\mathrm { d e c } } ^ { ( T ) }$ . Finally, the read-out function   \n(an MLP) takes ch $e _ { \\mathbf { k } , \\mathrm { d e c } } ^ { ( T ) }$ to predict the mean $\\mu _ { d _ { k } } \\in \\mathbb { R }$ dec  and the variance $\\sigma _ { d _ { k } } ^ { 2 } \\in \\mathbb { R }$ decof the Gaussian $d _ { k }$ . In summary, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e7ec360161c0d1f4679e4d5d60fd4312149118e0ca7b0c158ca322b33120af1f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { v _ { i , \\mathrm { d e c } } ^ { ( 0 ) } = F _ { \\mathrm { d e c } , v } ( v _ { i } ^ { \\prime } ) , \\quad e _ { k , \\mathrm { d e c } } ^ { ( 0 ) } = F _ { \\mathrm { d e c } , e } ( e _ { i } ) , } \\\\ & { \\mathcal { G } _ { \\mathrm { d e c } } ^ { ( 1 ) } = \\mathbf { M } \\mathbf { P } _ { \\mathrm { d e c } } ^ { ( 0 ) } ( \\mathcal { G } _ { \\mathrm { d e c } } ^ { ( 0 ) } ) , \\quad \\mathcal { G } _ { \\mathrm { d e c } } ^ { ( t + 1 ) } = \\mathbf { M } \\mathbf { P } _ { \\mathrm { d e c } } ^ { ( t ) } ( \\mathcal { G } _ { \\mathrm { d e c } } ^ { ( t ) } ) , \\quad \\mathcal { G } _ { \\mathrm { d e c } } ^ { ( T ) } = \\mathbf { M } \\mathbf { P } _ { \\mathrm { d e c } } ^ { ( T - 1 ) } ( \\mathcal { G } _ { \\mathrm { d e c } } ^ { ( T - 1 ) } ) , } \\\\ & { \\qquad \\mu _ { d _ { k } } , \\sigma _ { d _ { k } } ^ { 2 } = R _ { \\mathrm { d e c } } ( e _ { k , \\mathrm { d e c } } ^ { ( T ) } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The sets of parameters in the encoder and decoder, $\\phi$ and $\\theta$ (i.e., parameters in $F _ { \\mathrm { e n c } , v } , \\ F _ { \\mathrm { e n c } , e }$ , $\\{ \\mathbf { M P } _ { \\mathrm { e n c } } ^ { ( t ) } \\} _ { t = 1 } ^ { T }$ , $R _ { \\mathrm { e n c } }$ , $F _ { \\mathrm { d e c } , v }$ , $F _ { \\mathrm { d e c } , e }$ , $\\{ \\mathbf { M P } _ { \\mathrm { d e c } } ^ { ( t ) } \\} _ { t = 1 } ^ { T } , R _ { \\mathrm { d e c } } )$ , respectively, are optimized by maximizing the evidence lower bound (ELBO): ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb3995a22c9be21758e58b6bc58241eb8f72ff8949d461d17b313ade99a43b37.jpg",
+        "text": "$$\nL = \\mathbb { E } _ { \\mathbf { z } \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } ) } [ \\log p _ { \\theta } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } ) ] - D _ { \\mathrm { K L } } [ q _ { \\phi } ( \\mathbf { z } | \\mathbf { d } , \\mathcal { G } ) | | p _ { \\theta } ( \\mathbf { z } | \\mathcal { G } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            835,
+            705,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the prior $p _ { \\boldsymbol { \\theta } } ( \\mathbf { z } | \\mathcal { G } )$ consists of factorized Gaussians. The optimal values for the hyperparameters for the network dimensions, number of message passes, batch size, and learning rate of the Adam optimizer (Kingma & Ba, 2014) were tuned by maximizing the validation performance (ELBO) with a Bayesian optimizer and are reported in Appendix A.1.3. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 CONFORMATION GENERATION THROUGH EUCLIDEAN DISTANCE GEOMETRY",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To compute molecular properties, quantum-chemical methods need to be employed which require the input, i.e., the molecule, to be in Cartesian coordinates.3 Therefore, we use an EDG algorithm to translate the set of distances $\\{ d _ { k } \\} _ { k = 1 } ^ { N _ { e } }$ to a set of atomic coordinates $\\{ \\mathbf { r } _ { i } \\} _ { i = 1 } ^ { N _ { v } }$ . 4 ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "EDG is the mathematical basis for a geometric theory of molecular conformation. In the field of machine learning, Weinberger & Saul (2006) used it for learning image manifolds, Tenenbaum et al. (2000) for image understanding and handwriting recognition, Jain & Saul (2004) for speech and music, and Demaine et al. (2009) for music and musical rhythms. An EDG description of a molecular system consists of a list of lower and upper bounds on the distances between pairs of atoms $\\{ ( d _ { k , \\operatorname* { m i n } } , d _ { k , \\operatorname* { m a x } } ) \\} _ { k = 1 } ^ { N _ { e } }$ . Here, $p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )$ is used to model these bounds, namely, we set the bounds to $\\{ ( \\mu _ { d _ { k } } - \\sigma _ { d _ { k } } , \\mu _ { d _ { k } } + \\sigma _ { d _ { k } } ) \\}$ , where $\\mu _ { d _ { k } }$ and $\\sigma _ { d _ { k } }$ are the mean and standard deviation for each distance $d _ { k }$ given by the CVAE. Then, an EDG algorithm determines a set of Cartesian coordinates $\\{ \\mathbf { r } _ { i } \\} _ { i = 1 } ^ { N _ { v } }$ so that these bounds are fulfilled (see Appendix A.2 for details).5 Together with the corresponding chemical elements $\\{ \\epsilon _ { i } \\} _ { i = 1 } ^ { N _ { v } }$ , we obtain a conformation $\\mathbf { x }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.4 CALCULATION OF MOLECULAR PROPERTIES",
+        "text_level": 1,
+        "bbox": [
+            178,
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+    },
+    {
+        "type": "text",
+        "text": "We can get an MC estimate of the expectation $\\mathbb { E } _ { \\mathcal { G } } [ \\mathcal { O } ]$ of a property $\\mathcal { O }$ (e.g., the dipole moment) for a molecule represented by $\\mathcal { G }$ by drawing conformational samples $\\mathbf { x } _ { i } \\sim p ( \\mathbf { x } | \\mathcal { G } )$ and computing $\\mathcal { O } ( \\mathbf { x } _ { i } ) \\in \\mathbb { R }$ with a quantum-chemical method (e.g., density functional theory). Since we cannot draw samples from $p ( \\mathbf { x } | \\mathcal { G } )$ directly, we employ an IS integration scheme (Bishop, 2009) with our CVAE as the proposal distribution. We assume that we can readily evaluate the unnormalized probability of a conformation $\\tilde { p } ( \\mathbf { x } | \\mathcal { G } ) = \\exp \\{ - E ( \\mathbf { x } ) / k _ { B } T \\}$ , where $\\mathbf { x }$ must be a conformation of the molecule and the energy $E ( \\mathbf { x } )$ is determined with a quantum-chemical method. Since the EDG algorithm is mapping the distribution $p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } , \\mathcal { G } )$ to a point mass in $ { \\mathbb { R } } ^ { 3 N _ { v } }$ , the MC estimate for the resulting distribution $p _ { \\mathrm { p r o p } } ( \\mathbf { x } | \\mathcal { G } )$ is given by a mixture of delta functions, each of which is centered at the $\\mathbf { x } _ { i }$ resulting from mapping $p _ { \\boldsymbol { \\theta } } ( \\mathbf { d } | \\mathbf { z } _ { i } , \\mathcal { G } )$ to $\\mathbb { R } ^ { 3 N _ { v } }$ , where $\\mathbf { z } _ { i } \\sim p _ { \\theta } ( \\mathbf { z } | \\mathcal { G } )$ , that is, $\\begin{array} { r } { p _ { \\mathrm { p r o p } } ( \\mathbf { x } | \\mathcal { G } ) \\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\delta ( \\mathbf { x } - \\mathbf { x } _ { i } ) } \\end{array}$ . The IS estimator for the expectation of $\\mathcal { O }$ w. r. t. $\\tilde { p } ( \\mathbf { x } | \\mathcal { G } )$ then reads ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fafd8c965057b48a86acdf60b98e8b8b8279987871ddc24d58456b76197d09cb.jpg",
+        "text": "$$\n\\hat { \\mathbb { E } } _ { \\mathcal { G } } [ \\mathcal { O } ] \\overset { \\mathrm { M C } } { \\approx } \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathcal { O } ( \\mathbf { x } _ { i } ) \\overset { \\mathrm { I S } } { = } \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathcal { O } ( \\mathbf { x } _ { i } ^ { \\prime } ) \\frac { \\tilde { p } ( \\mathbf { x } _ { i } ^ { \\prime } | \\mathcal { G } ) } { p _ { \\mathrm { p r o p } } ( \\mathbf { x } _ { i } ^ { \\prime } | \\mathcal { G } ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            529,
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+            573
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { x } _ { i } \\sim \\tilde { p } ( \\mathbf { x } _ { i } | \\mathcal { G } )$ and $\\mathbf { x } _ { i } ^ { \\prime } \\sim p _ { \\mathrm { p r o p } } ( \\mathbf { x } _ { i } ^ { \\prime } | \\mathcal { G } )$ , so that the expectation of $\\mathcal { O }$ w. r. t. the normalized version of $\\tilde { p } ( { \\bf x } )$ is then ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d1bfdb172990ccca20a4060ab8f7b2c464e206d95b17e86baa539c360b4a59c1.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathcal { G } } [ \\mathcal { O } ] = \\frac { \\hat { \\mathbb { E } } _ { \\mathcal { G } } [ \\mathcal { O } ] } { \\hat { \\mathbb { E } } _ { \\mathcal { G } } [ 1 ] } \\approx \\frac { 1 } { Z } \\sum _ { i = 1 } ^ { N } \\mathcal { O } ( \\mathbf { x } _ { i } ) \\tilde { p } ( \\mathbf { x } _ { i } ^ { \\prime } | \\mathcal { G } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            603,
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+            647
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where have a $\\begin{array} { r } { Z \\approx \\sum _ { i = 1 } ^ { N } \\tilde { p } ( \\mathbf { x } _ { i } ^ { \\prime } ) } \\end{array}$ and ake s $N$ is the number of samples. When dividing two delta functions wee arbitrarily large finite value. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            699,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The standard approach for generating molecular conformations is to start with one, and make small changes to it over time, e.g., by using MCMC or MD. These methods are considered the gold standard for sampling equilibrium states, but they are computationally expensive, especially if the molecule is large and the Hamiltonian is based on quantum-mechanical principles (Shim & MacKerell, 2011; Ballard et al., 2015; De Vivo et al., 2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A much faster but more approximate approach for conformation generation is EDG (Havel, 2002; Blaney & Dixon, 2007; Lagorce et al., 2009; Riniker & Landrum, 2015). Lower and upper distance bounds for pairs of atoms in a molecule are fixed values based on ideal bond lengths, bond angles, and torsional angles. These values are often extracted from crystal structure databases (Allen, 2002). These methods aim to generate a low-energy conformation, not to generate unbiased samples from the underlying distribution at a certain temperature. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "There exist several machine learning approaches as well, however, they are mostly tailored towards studying protein dynamics. For example, Noe et al. (2019) trained Boltzmann generators on the ´ energy function of proteins to provide unbiased, one-shot samples from their equilibrium states. This is achieved by training an invertible neural network to learn a coordinate transformation from a system’s configurations to a latent space representation. Further, Lemke & Peter (2019) proposed a dimensionality reduction algorithm that is based on a neural network autoencoder in combination with a nonlinear distance metric to generate samples for protein structures. Both models learn protein-specific coordinate transformations that cannot be transferred to other molecules. ",
+        "bbox": [
+            174,
+            166,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "AlQuraishi (2019) introduced an end-to-end differentiable recurrent geometric network for protein structure learning based on amino acid sequences. Also, Ingraham et al. (2019) proposed a neural energy simulator model for protein structure that makes use of protein sequence information. In contrast to amino acid sequences, molecular graphs are, in general, not linear but highly branched and often contain cycles. This makes them unsuitable for recurrent networks. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, Mansimov et al. (2019) presented a conditional deep generative graph neural network to generate molecular conformations given a molecular graph. Their goal is to predict the most likely conformation and not a distribution over conformations. Instead of encoding molecular environments in atomic distances, they work directly in Cartesian coordinates. As a result, the generated conformations showed significant structural differences compared to the ground-truth and required refinement through a force field, which is often employed in MD simulations. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We argue that our model has several advantages over the approaches reviewed above: ",
+        "bbox": [
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+            452,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• It is a fast alternative to resource-intensive approaches based on MCMC or MD.   \n• Our principled representation based on pair-wise distances does not restrict our approach to any particular molecular structure.   \n• Since our model employs message-passing neural networks, it is transferable – it can extrapolate from only a few graphs to unseen ones. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 THE CONF17 BENCHMARK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            607,
+            431,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The CONF17 benchmark is the first benchmark for molecular conformation sampling.6 It is based on the ISO17 dataset (Schutt et al., 2017a) which consists of conformations of various molecules ¨ with the atomic composition $\\mathrm { C _ { 7 } H _ { 1 0 } O _ { 2 } }$ drawn from the QM9 dataset (Ramakrishnan et al., 2014). These conformations were generated by ab initio molecular dynamics simulations at 500 Kelvin which generates trajectories of a single molecule covering a large variety of conformations. The CONF17 benchmark consists of 127 distinct molecular graphs each with 3380 conformations on average. We split this dataset into multiple training and test splits, each consisting of 107 and 20 graphs, respectively (see Appendix A.1 for more details).7 ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 4, (A), the structural formulae of a random selection of molecules from this benchmark are shown. Most molecules feature highly-strained, complex 3D structures such as rings which are typical of drug-like molecules. It is thus the structural complexity of the molecules, not their number of degrees of freedom, that makes this benchmark challenging. In Fig. 4, (B), the frequency of distances (in $\\mathring \\mathrm { A }$ ) in the conformations are shown for each edge type. It can be seen that the marginal distributions of the edge distances are multimodal and highly context dependent. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/46a3aa64e022fdd01a26120b31e2968b1635e75bf7b4ed5e1fb0963f320d2e6f.jpg",
+        "image_caption": [
+            "Figure 4: Overview of the CONF17 benchmark. (A) Structural formulae of a random selection of molecules. (B) Distribution of distances (in $\\mathrm { \\AA }$ ) grouped by edge (from left to right: $E _ { \\mathrm { b o n d } }$ , $E _ { \\mathrm { a n g l e } }$ , and $E _ { \\mathrm { d i h e d r a l } } ,$ ) and vertex type (chemical element). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            465,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We assess the performance of our method, named Graph Distance Geometry (GRAPHDG), by comparing it with two state-of-the-art methods for molecular conformation generation: RDKIT (Riniker & Landrum, 2015), a classical EDG approach, and DL4CHEM (Mansimov et al., 2019), a machine learning approach. We trained GRAPHDG and DL4CHEM on three different training and test splits of the CONF17 benchmark using Adam (Kingma & Ba, 2014). We generated 3000 conformations with each method for molecular graphs in a test set. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 DISTRIBUTIONS OVER DISTANCES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "We assessed the accuracy of the distance distributions of RDKIT, DL4CHEM, and GRAPHDG by calculating the maximum mean discrepancy (MMD) (Gretton et al., 2012) to the ground-truth distribution. We compute the MMD using a Gaussian kernel, where we set the standard deviation to be the median distance between distances $\\mathbf { d }$ in the aggregate sample. For this, we determined the distances in the conformations from the ground-truth and those generated by RDKIT and DL4CHEM. For each train-test split and each $\\mathcal { G }$ in a test set, we compute the MMD of the joint distribution of distances between C and $\\mathrm { o }$ atoms (H atoms are usually ignored), the MMDs of pair-wise distances $p ( d _ { i } , d _ { j } | \\mathcal { G } )$ , and the MMDs between the marginals of individual distances $p ( d _ { i } | \\mathcal { G } )$ . We aggregate the results of three train-test splits, and, finally, compute the median MMDs and average rankings. The results are summarized in Table 1. It can be seen that the samples from GRAPHDG are significantly closer to the ground-truth distribution than the other methods. RDKIT is slightly worse than GRAPHDG while DL4CHEM seems to struggle with the complexity of the molecules and the small number of graphs in the training set. ",
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+    {
+        "type": "text",
+        "text": "In Fig. 5, we showcase the accuracy of our model by plotting the marginal distributions $p ( d _ { i } | \\mathcal { G } )$ for distances between C and O atoms given a molecular graph from a test set. It can be seen that RDKIT consistently underestimates the marginal variances. This is because this method aims to predict the most stable conformation, i.e., the distribution’s mode. In contrast, DL4CHEM often fails to predict the correct mean. For this molecule, GRAPHDG is the most accurate, predicting the right mean and variance in most cases. Additional figures can be found in the Appendix A.4, where we also show plots for the marginal distributions $p ( d _ { i } , d _ { j } | \\mathcal { G } )$ . ",
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+    {
+        "type": "text",
+        "text": "Table 1: Assessment of the accuracy of the distributions over conformations generated by three models compared to the ground-truth. We compare the distributions with respect to the marginals $p ( d _ { k } | \\mathcal { G } )$ , $p ( d _ { k } , d _ { l } | \\mathcal { G } )$ , and the distribution over all edges between C and $\\mathrm { o }$ atoms $p ( \\{ d _ { k } \\} | \\mathcal { G } )$ . Two different metrics are used: median MMD between ground-truth conformations and generated ones, and mean ranking (1 to 3) based on the MMD. Reported are the results for molecular graphs in a test set from three train-test splits. Standard errors are given in brackets. ",
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+    {
+        "type": "table",
+        "img_path": "images/e83dbdf3c3547b9f3252fec33f50c007fb77c3c8439737f6fe4f14904c384495.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Median MMD</td><td colspan=\"3\">Mean Ranking</td></tr><tr><td>RDKIT</td><td>DL4CHEM</td><td>GRAPHDG</td><td>RDKIT</td><td>DL4CHEM</td><td>GRAPHDG</td></tr><tr><td>p(dk|9)</td><td>0.55 (0.01)</td><td>1.11 (0.01)</td><td>0.38 (0.02)</td><td>1.71 (0.03)</td><td>2.74 (0.02)</td><td>1.51 (0.03)</td></tr><tr><td>p(dk,di/9)</td><td>0.53 (0.01)</td><td>1.09 ( (0.01)</td><td>0.34 (0.01)</td><td>1.66 (0.02)</td><td>2.92 (0.01)</td><td>1.43 ( (0.02)</td></tr><tr><td>p({d}9）</td><td>0.60 (0.01)</td><td>1.07 (0.03)</td><td>0.44 (0.05)</td><td>1.58 (0.05)</td><td>2.90 (0.05)</td><td>1.45 (0.02)</td></tr></table>",
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+        "type": "image",
+        "img_path": "images/73cb33f1644a53a40561269d68794ff6ada9b90a2cebb3bc0277b7aadea252d5.jpg",
+        "image_caption": [
+            "Figure 5: Marginal distributions $p ( d _ { k } | \\mathcal { G } )$ of ground-truth and predicted bond distances (in $\\textrm { \\AA }$ ) between C and O atoms given a molecular graph from the test set. The atoms connected by each edge $d _ { k }$ are indicated in each subplot $\\left( \\boldsymbol { s } _ { k } \\mathrm { - } \\boldsymbol { r } _ { k } \\right)$ . In the 3D structure of the molecule, carbon and oxygen atoms are colored gray and red, respectively. $_ \\mathrm { H }$ atoms are omitted for clarity. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "5.2 GENERATION OF CONFORMATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We passed the distances from our generative model to an EDG algorithm to obtain conformations. For $9 9 . 9 \\%$ of the sets of distances, all triangle inequalities held. For $94 \\%$ of the molecular graphs, the algorithm succeeded which is 8 pp higher than the success rate we observed for RDKIT. For each molecular graph in a test set, we generated 50 conformations with each method. This took DL4CHEM, RDKIT, and GRAPHDG on average around hundreds of milliseconds per molecule.8 In contrast, a single conformation in the ISO17 dataset takes around a minute to compute. In Fig. 6, an overlay of these conformations of six molecules generated by the different methods is shown. It can be seen that RDKIT’s conformations show too little variance, while DL4CHEM’s structures are mostly invalid, which is due in part to its failure to predict the correct interatomic angles. Our method slightly overestimates the structural variance (see, for example, Fig. 6, top row, second column), but produces conformations that are the closest to the ground-truth. ",
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+    {
+        "type": "text",
+        "text": "5.3 CALCULATION OF MOLECULAR PROPERTIES",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We estimate expected molecular properties for molecular graphs from the test set with $N = 5 0$ conformational samples each. Due to their poor quality, we could not compute properties $\\mathcal { O } ( \\mathbf { x } )$ , including the energy $E ( \\mathbf { x } )$ , for conformations generated with DL4CHEM, and thus, this method is excluded from this analysis. In Table 2, it can be seen that RDKIT and GRAPHDG perform ",
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+    {
+        "type": "image",
+        "img_path": "images/6dbc1684f2efce71c7be4d5e7e63e8e2e0f0a2968f6b845c474caa529794805c.jpg",
+        "image_caption": [
+            "Figure 6: Overlay of 50 conformations from the ground-truth and three models based on six random molecular graphs from the test set. C, O, and H atoms are colored gray, red, and white, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Median difference in average properties between ground-truth and RDKIT and GRAPHDG: total electronic energy $E _ { \\mathrm { e l e c } }$ (in kJ/mol), the energy of the HOMO and the LUMO \u000fLUMO and \u000fLUMO, respectively (in eV), and the dipole moment $\\mu$ (in debye). Reported are the results for molecular graphs from the test set, averaged over three train-test splits. Standard errors are given in brackets. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/28fdddd3902b37a3a8fedd86b4e37bef3145c3d79b8a57f2f80e6e8539ef3e2e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>RDKIT</td><td>GRAPHDG</td></tr><tr><td>Eelec</td><td>42.7 (4.3)</td><td>58.0 (21.0)</td></tr><tr><td>€HOMO</td><td>0.08 (0.04)</td><td>0.10 (0.05)</td></tr><tr><td>ELUMO</td><td>0.15 (0.03)</td><td>0.09 (0.05)</td></tr><tr><td>从</td><td>0.29 (0.05)</td><td>0.33 (0.09)</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "similarly well (see Appendix A.2 for computational details). However, both methods are still highly inaccurate for $E _ { \\mathrm { e l e c } }$ (in practice, an accuracy of less than $5 \\ \\mathrm { k J / m o l }$ is required). Close inspection of the conformations shows that, even though GRAPHDG predicts the most accurate distances overall, the variances of certain strongly constrained distances (e.g., triple bonds) are overestimated so that the energies of the conformations increase drastically. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 LIMITATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The first limitation of this work is that the CVAE can sample (with low probability) invalid sets of distances for which there exists no 3D structure. Second, the CONF17 benchmark covers only a small portion of chemical space. Finally, a large set of auxiliary edges would be required to capture long-range correlations (e.g., in proteins). Future work will address these points. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We presented GRAPHDG, a transferable, generative model that allows sampling from a distribution over molecular conformations. We developed a principled learning representation of conformations that is based on distances between atoms. Then, we proposed a challenging benchmark for comparing molecular conformation generators. With this benchmark, we show experimentally that conformations generated by GRAPHDG are closer to the ground-truth than those generated by other methods. Finally, we employ our model as a proposal distribution in an IS integration scheme to estimate molecular properties. While orbital energies and the dipole moments were predicted well, a larger and more diverse dataset will be necessary for meaningful estimates of electronic energies. Further, methods have to be devised to estimate how many conformations need to be generated to ensure all important conformations have been sampled. Finally, our model could be trained on conformational distributions at different temperatures in a transfer learning-type setting. ",
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+        ],
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+    {
+        "type": "text",
+        "text": "Jiaxuan You, Bowen Liu, Zhitao Ying, Vijay Pande, and Jure Leskovec. Graph Convolutional Policy Network for Goal-Directed Molecular Graph Generation. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems 31, pp. 6410–6421. Curran Associates, Inc., 2018. ",
+        "bbox": [
+            173,
+            859,
+            825,
+            916
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 CONF17 BENCHMARK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            372,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1.1 DATA GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            161,
+            367,
+            175
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The ISO17 dataset (Schutt et al., 2017a) was processed in the following way. First, conformations ¨ in which the molecular connectivity was modified (i.e., bonds were broken or new ones are formed) were discarded. For this, the tool XYZ2MOL (Jensen, 2019) was employed. Second, the molecular graphs were augmented by adding auxiliary edges for reasons described in Section 2.1. Auxiliary edges between all second neighbors were added. This can lead to a slight over-specification of the system’s geometry, however, this did not pose a problem in our experiments. In addition, auxiliary edges between third neighbors were added to fix dihedral angles. Since there are potentially many ways of specifying a dihedral angle in a molecular system, we resorted to the works of Riniker & Landrum (2015) and Guba et al. (2016) to decide where to place edges between third neighbors. ",
+        "bbox": [
+            174,
+            185,
+            825,
+            311
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1.2 INPUT FEATURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            329,
+            352,
+            343
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Below we list the node and edges features in the CONF17 benchmark. ",
+        "bbox": [
+            174,
+            353,
+            632,
+            368
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/6cc45a2dc156f39e2d8557d3cd25a25765682fea3ba914e5518419e42a8c79e1.jpg",
+        "table_caption": [
+            "Table 3: Node features. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Feature</td><td>Data Type</td><td>Dimension</td></tr><tr><td>atomic number</td><td>integer</td><td>1</td></tr><tr><td>chiral tag</td><td>one-hot (R, S,and N/A)</td><td>3</td></tr></table>",
+        "bbox": [
+            300,
+            414,
+            692,
+            467
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/fb0acd7c9379ea76c3ccaec9d8ecacbf5f6bdd2b6892ea6826fb0e5c5149c7eb.jpg",
+        "table_caption": [
+            "Table 4: Edge features. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Feature</td><td>Data Type</td><td>Dimension</td></tr><tr><td>kind</td><td>one-hot (indicating whether e is in Ebond,Eangle,or Edihedral)</td><td>3</td></tr><tr><td>stereo chemistry</td><td>one-hot (E,Z,Any,None, and N/A)</td><td>5</td></tr><tr><td>type</td><td>integer (single, double, triple or N/A)</td><td>1</td></tr><tr><td>is aromatic</td><td>binary</td><td>1</td></tr><tr><td>is conjugated</td><td>binary</td><td>1</td></tr><tr><td>is in ring of size</td><td>one-hot (3,4,...,9) and N/A</td><td>8</td></tr></table>",
+        "bbox": [
+            189,
+            536,
+            810,
+            646
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1.3 MODEL ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            676,
+            401,
+            691
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The full model is available online https://figshare.com/s/1b42bf865bd78c457354. In following, the hyperparameters of our model are specified: ",
+        "bbox": [
+            173,
+            702,
+            821,
+            729
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Activations throughout this paper: ReLU; $L _ { v }$ , $L _ { e }$ : 10; $F _ { \\mathrm { e n c } , v }$ : neural network with depth 2, width 20; $F _ { \\mathrm { e n c } , e }$ : neural network with depth 3, width 60; $F _ { \\mathrm { d e c } , v }$ , $F _ { \\mathrm { d e c } , e }$ : neural networks with depth 2, width 70; {MP(t) }T , {MP(t) }T dec t=1: MPNN width depth 1 and three multi-head attention heads, T = 3, for node and edge updates neural networks with depth 2, and width 70 were used. $R _ { \\mathrm { e n c } }$ , $R _ { \\mathrm { d e c } }$ : neural networks with depth 2, width 70. Batch size: 16 (conformations); ",
+        "bbox": [
+            174,
+            736,
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+            810
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 COMPUTATIONAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            829,
+            405,
+            844
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2.1 QUANTUM-CHEMICAL CALCULATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            856,
+            503,
+            871
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "All quantum-chemical calculations were carried out with the PySCF program package (version 1.5) (Sun et al., 2018) employing the exchange-correlation density functional PBE (Perdew et al., 1996), and the def2-SVP (Weigend & Ahlrichs, 2005; Weigend, 2006) basis set. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Conformations generated by DL4CHEM did not succeed as some atoms were too close to each other. Self-consistent field algorithms in quantum-chemical software such as $\\operatorname { P y } \\operatorname { S C F }$ do not converge for such molecular structures. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "With quantum-chemical methods, we calculate several properties that concern the states of the electrons in the conformation. These are the total electronic energy $E _ { \\mathrm { e l e c } }$ , the energy of the electron in the highest occupied molecular orbital (HOMO in eV) \u000fHOMO, the energy of the lowest unoccupied molecular orbital (LUMO in eV) \u000fLUMO, and the norm of the dipole moment $\\mu$ (in debye). ",
+        "bbox": [
+            174,
+            152,
+            825,
+            209
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2.2 EUCLIDEAN DISTANCE GEOMETRY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            250,
+            475,
+            265
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We refer the reader to Havel (2002) for theory on EDG, algorithms, and chemical applications. In summary, the EDG procedure consists of the following three steps: ",
+        "bbox": [
+            174,
+            284,
+            823,
+            313
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1. Bound smoothing: extrapolating a complete set of lower and upper limits on all the distances from the sparse set of lower and upper bounds.   \n2. Embedding: choosing a random distance matrix from within these limits, and computing coordinates that are a certain best-fit to the distances.   \n3. Optimization: optimizing these coordinates versus an error function which measures the total violation of the distance (and chirality) constraints. ",
+        "bbox": [
+            207,
+            335,
+            826,
+            476
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We use the EDG implementation found in RDKIT (Riniker & Landrum, 2015) with default settings. ",
+        "bbox": [
+            176,
+            498,
+            821,
+            513
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3 GENERATION OF CONFORMATIONS ",
+        "bbox": [
+            174,
+            556,
+            460,
+            570
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/27951cf926cc9a59c011227fdb3757047860bb4f10db180e980c1c7f56c2a9fb.jpg",
+        "image_caption": [
+            "Figure 7: Overlay of 50 conformations from the ground-truth, RDKIT, DL4CHEM, and GRAPHDG based on two random molecular graphs from the test set. C, O, and H atoms are colored gray, red, and white, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            316,
+            617,
+            673,
+            765
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4 DISTRIBUTIONS OVER DISTANCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            872,
+            455,
+            887
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Below, the marginal distributions of the distances for a variety of molecular graphs are shown. ",
+        "bbox": [
+            168,
+            909,
+            789,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/155b36793493760b7aa79975d8ddcff91c662022446556581190de48d5091e35.jpg",
+        "image_caption": [
+            "Figure 8: Marginal distributions $p ( d _ { k } | \\mathcal { G } )$ of ground-truth and predicted distances (in A) between C ˚ and O atoms given a molecular graph from the test set. The atoms connected by each edge $d _ { k }$ are indicated in each subplot $\\left( \\boldsymbol { s } _ { k } \\mathrm { - } \\boldsymbol { r } _ { k } \\right)$ . In the 3D structure of the molecule, carbon and oxygen atoms are colored gray and red, respectively. $_ \\mathrm { H }$ atoms are omitted for clarity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            102,
+            816,
+            450
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/e220b95757969ae8413448670537336f0a0200b2d7857d8c548541c7e3c21327.jpg",
+        "image_caption": [
+            "Figure 9: Marginal distributions $p ( d _ { i } , d _ { j } | \\mathcal { G } )$ of ground-truth and predicted distances for a molecular graph from the test set (in $\\mathring \\mathrm { A }$ ). Here, $d _ { i }$ and $d _ { j }$ are restricted to edges representing bonds between C and O atoms. In the 3D structure of the molecule, carbon and oxygen atoms are colored gray and red, respectively. H atoms are omitted for clarity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            229,
+            818,
+            710
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/990560565dbcd6a07e4996d7818003cb2c3bd83142a60411cb3f89d685007e5c.jpg",
+        "image_caption": [
+            "Figure 10: See caption of Fig. 8 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            323,
+            802,
+            659
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/cf355a52b90102bd4be7140f2db8ba50e3779dbf93c62c9815ff4022643e40ce.jpg",
+        "image_caption": [
+            "Figure 11: See caption of Fig. 9 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
+            252,
+            805,
+            732
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/b460530421a43a7fa1d828cbce0adb372e29c25c81e9fb137bd557fbbf59743a.jpg",
+        "image_caption": [
+            "Figure 12: See caption of Fig. 8 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            321,
+            818,
+            669
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/3ab71d02171ec50c23dd6fda6644669702b4f5b23939b729a36135a2e37793fd.jpg",
+        "image_caption": [
+            "Figure 13: See caption of Fig. 9 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            253,
+            797,
+            727
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/318714018966fa62ece638ab99060cdcf128629a839fab7764e5e8b5f7fa177e.jpg",
+        "image_caption": [
+            "Figure 14: See caption of Fig. 8 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            321,
+            818,
+            669
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/046289cc08ff0e5a719e2e8d0afd1905fc1441824afab583ecca345ad569e0ab.jpg",
+        "image_caption": [
+            "Figure 15: See caption of Fig. 9 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
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+            800,
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+        ],
+        "page_idx": 21
+    }
+]